diff --git a/main/opengeometry/src/brep/builder/boundary.rs b/main/opengeometry/src/brep/builder/boundary.rs
index 35668587..718a4a54 100644
--- a/main/opengeometry/src/brep/builder/boundary.rs
+++ b/main/opengeometry/src/brep/builder/boundary.rs
@@ -1,7 +1,8 @@
 use super::brep_builder::{Builder, Use};
 use crate::brep::error::GeometryError;
 use crate::brep::frame::Frame3;
-use crate::brep::geometry::CurveGeometry;
+use crate::brep::frame::UV;
+use crate::brep::geometry::{Curve, CurveGeometry, Surface, SurfaceGeometry};
 use crate::brep::pcurve::PcurveGeometry;
 use crate::brep::topology::{EdgeGeometry, Orientation};
 use crate::math::{dot, scale, sub, Point3};
@@ -15,7 +16,7 @@ pub(crate) fn dimensions(values: &[f64]) -> Result<(), GeometryError> {
     Ok(())
 }
 
-pub(crate) fn plane_boundary(
+pub fn plane_boundary(
     builder: &Builder,
     frame: Frame3,
     edge: u32,
@@ -50,16 +51,71 @@ pub(crate) fn plane_boundary(
     Ok(boundary(edge, from, to, sense, pcurve))
 }
 
-pub(crate) fn uv_line(origin: [f64; 2], direction: [f64; 2]) -> PcurveGeometry {
-    PcurveGeometry::Line2 { origin, direction }
-}
-pub(crate) fn boundary(
+const PROJECTION_SAMPLES: usize = 64;
+
+pub fn projected_boundary(
+    builder: &Builder,
+    surface: &SurfaceGeometry,
     edge: u32,
     from: u32,
     to: u32,
     sense: Orientation,
-    pcurve: PcurveGeometry,
-) -> Use {
+    hint: Option<UV>,
+) -> Result<Use, GeometryError> {
+    let geometry = &builder
+        .brep
+        .topology
+        .edges
+        .get(edge as usize)
+        .ok_or_else(|| GeometryError::InvalidTopology("builder references missing edge".into()))?
+        .geometry;
+    let EdgeGeometry::Curve { curve, range } = *geometry else {
+        return Err(GeometryError::InvalidGeometry(
+            "collapsed projected boundary".into(),
+        ));
+    };
+    let evaluator = builder.brep.geometry.curve(curve)?;
+    let at = |i: usize| range.lo + range.width() * i as f64 / PROJECTION_SAMPLES as f64;
+    let mut uv: Vec<UV> = Vec::with_capacity(PROJECTION_SAMPLES + 1);
+    for i in 0..=PROJECTION_SAMPLES {
+        let previous = uv.last().copied().or(hint);
+        uv.push(surface.project(evaluator.point_at(at(i))?, previous)?);
+    }
+    let (start, end) = (uv[0], uv[PROJECTION_SAMPLES]);
+    let width = range.width();
+    let rate: UV = std::array::from_fn(|axis| {
+        if width > 0.0 {
+            (end[axis] - start[axis]) / width
+        } else {
+            0.0
+        }
+    });
+    let periods = surface.charts()[0].periods;
+    for (i, sample) in uv.iter().enumerate() {
+        for axis in 0..2 {
+            let guess = start[axis] + rate[axis] * (at(i) - range.lo);
+            if periods[axis].is_some_and(|period| (guess - sample[axis]).abs() >= period / 4.0) {
+                return Err(GeometryError::UnsupportedGeometry(
+                    "projected boundary winds unevenly around a periodic surface".into(),
+                ));
+            }
+        }
+    }
+    let pcurve = PcurveGeometry::ProjectedCurve {
+        curve,
+        surface: builder.brep.geometry.surfaces.len() as u32,
+        chart: 0,
+        uv_hint: start,
+        uv_rate: rate,
+        parameter_origin: range.lo,
+    };
+    Ok(boundary(edge, from, to, sense, pcurve))
+}
+
+pub fn uv_line(origin: [f64; 2], direction: [f64; 2]) -> PcurveGeometry {
+    PcurveGeometry::Line2 { origin, direction }
+}
+pub fn boundary(edge: u32, from: u32, to: u32, sense: Orientation, pcurve: PcurveGeometry) -> Use {
     Use {
         edge,
         from,
diff --git a/main/opengeometry/src/brep/builder/brep_builder.rs b/main/opengeometry/src/brep/builder/brep_builder.rs
index 568f4bcc..b1b6f140 100644
--- a/main/opengeometry/src/brep/builder/brep_builder.rs
+++ b/main/opengeometry/src/brep/builder/brep_builder.rs
@@ -8,7 +8,7 @@ use crate::brep::topology::{
 };
 use crate::math::{Interval, Point3};
 
-pub(crate) struct Use {
+pub struct Use {
     pub(super) edge: u32,
     pub(super) from: u32,
     pub(super) to: u32,
@@ -16,16 +16,19 @@ pub(crate) struct Use {
     pub(super) pcurve: PcurveGeometry,
 }
 
-pub(crate) struct Builder {
+pub struct Builder {
     pub(crate) brep: BrepEnvelope,
 }
 impl Builder {
-    pub(crate) fn new(id: String, accuracy: Accuracy) -> Result<Self, GeometryError> {
+    pub fn new(id: String, accuracy: Accuracy) -> Result<Self, GeometryError> {
         Ok(Self {
             brep: BrepEnvelope::new(id, accuracy)?,
         })
     }
-    pub(crate) fn vertex(&mut self, p: Point3) -> u32 {
+    pub fn brep(&self) -> &BrepEnvelope {
+        &self.brep
+    }
+    pub fn vertex(&mut self, p: Point3) -> u32 {
         let id = self.brep.topology.vertices.len() as u32;
         self.brep.topology.vertices.push(Vertex {
             id,
@@ -35,7 +38,7 @@ impl Builder {
         });
         id
     }
-    pub(crate) fn edge(&mut self, curve: CurveGeometry, range: Interval, seam: bool) -> u32 {
+    pub fn edge(&mut self, curve: CurveGeometry, range: Interval, seam: bool) -> u32 {
         let curve_id = self.brep.geometry.curves.len() as u32;
         self.brep.geometry.curves.push(curve);
         self.edge_geometry(
@@ -129,24 +132,29 @@ impl Builder {
         });
         Ok(loop_id)
     }
-    pub(crate) fn face(
+    pub fn face(
         &mut self,
         key: &str,
         surface: SurfaceGeometry,
         bounds: [[f64; 2]; 2],
         uses: Vec<Use>,
-    ) -> Result<(), GeometryError> {
+    ) -> Result<u32, GeometryError> {
         self.face_with_holes(key, surface, bounds, uses, Vec::new())
     }
-    pub(crate) fn face_with_holes(
+    pub fn face_with_holes(
         &mut self,
         key: &str,
         surface: SurfaceGeometry,
         bounds: [[f64; 2]; 2],
-        outer_uses: Vec<Use>,
-        hole_uses: Vec<Vec<Use>>,
-    ) -> Result<(), GeometryError> {
+        mut outer_uses: Vec<Use>,
+        mut hole_uses: Vec<Vec<Use>>,
+    ) -> Result<u32, GeometryError> {
         let surface_id = self.brep.geometry.surfaces.len() as u32;
+        for u in outer_uses.iter_mut().chain(hole_uses.iter_mut().flatten()) {
+            if let PcurveGeometry::ProjectedCurve { surface, .. } = &mut u.pcurve {
+                *surface = surface_id;
+            }
+        }
         self.brep.geometry.surfaces.push(surface);
         let face = self.brep.topology.faces.len() as u32;
         let outer = self.face_loop(face, false, outer_uses)?;
@@ -180,9 +188,21 @@ impl Builder {
                 reversed: false,
             },
         });
+        Ok(face)
+    }
+    pub fn set_face_sense(&mut self, face: u32, sense: Orientation) -> Result<(), GeometryError> {
+        self.brep
+            .topology
+            .faces
+            .get_mut(face as usize)
+            .ok_or(GeometryError::MissingReference {
+                kind: "face".into(),
+                index: face,
+            })?
+            .sense = sense;
         Ok(())
     }
-    pub(crate) fn finish_solid(mut self) -> Result<BrepEnvelope, GeometryError> {
+    pub fn finish_solid(mut self) -> Result<BrepEnvelope, GeometryError> {
         let faces = (0..self.brep.topology.faces.len() as u32).collect();
         self.brep.topology.shells.push(Shell {
             id: 0,
@@ -193,6 +213,7 @@ impl Builder {
             outer_shell: 0,
             cavity_shells: Vec::new(),
         });
+        self.brep.orient_loops()?;
         self.brep.validate()?;
         Ok(self.brep)
     }
diff --git a/main/opengeometry/src/brep/builder/fixtures.rs b/main/opengeometry/src/brep/builder/fixtures.rs
new file mode 100644
index 00000000..5cf3d585
--- /dev/null
+++ b/main/opengeometry/src/brep/builder/fixtures.rs
@@ -0,0 +1,299 @@
+use super::boundary::{boundary, plane_boundary, uv_line};
+use super::brep_builder::{Builder, Use};
+use crate::brep::accuracy::Accuracy;
+use crate::brep::frame::Frame3;
+use crate::brep::geometry::{CurveGeometry, SurfaceGeometry, Wrap};
+use crate::brep::topology::{BrepEnvelope, Orientation};
+use crate::math::{Interval, Point3};
+
+pub(crate) struct Sector {
+    pub(crate) inner: f64,
+    pub(crate) thickness: f64,
+    pub(crate) wrap_radius: f64,
+    pub(crate) length: f64,
+    pub(crate) rise: f64,
+    pub(crate) width: f64,
+}
+
+pub(crate) const SECTOR: Sector = Sector {
+    inner: 2.0,
+    thickness: 2.0,
+    wrap_radius: 2.8,
+    length: 4.0,
+    rise: 1.0,
+    width: 10.0,
+};
+
+const SECTOR_ACCURACY: Accuracy = Accuracy {
+    geometric: 1e-6,
+    intersection: 1e-7,
+    tessellation: 1e-3,
+    exchange: 1e-5,
+};
+
+impl Sector {
+    pub(crate) fn wrap(&self) -> Wrap {
+        Wrap {
+            frame: Frame3::from_axis([1.0, 2.0, 3.0], [0.2, 1.0, 0.1], [1.0, 0.0, 0.3])
+                .expect("sector frame"),
+            radius: self.wrap_radius,
+        }
+    }
+
+    pub(crate) fn volume(&self) -> f64 {
+        let area = self.length * self.width - self.length * self.rise / 2.0;
+        let outer = self.inner + self.thickness;
+        area * (outer * outer - self.inner * self.inner) / (2.0 * self.wrap_radius)
+    }
+
+    fn slant(&self) -> f64 {
+        self.length.hypot(self.rise)
+    }
+
+    fn levels(&self) -> [f64; 2] {
+        [self.inner, self.inner + self.thickness]
+    }
+
+    fn corners(&self) -> [[f64; 2]; 4] {
+        [
+            [0.0, 0.0],
+            [self.length, self.rise],
+            [self.length, self.width],
+            [0.0, self.width],
+        ]
+    }
+}
+
+struct Edges {
+    vertices: Vec<[u32; 2]>,
+    ab: [u32; 2],
+    bc: [u32; 2],
+    cd: [u32; 2],
+    da: [u32; 2],
+    up: Vec<u32>,
+}
+
+fn line(b: &mut Builder, origin: Point3, direction: Point3, length: f64) -> u32 {
+    b.edge(
+        CurveGeometry::Line { origin, direction },
+        Interval::new(0.0, length).expect("line range"),
+        false,
+    )
+}
+
+fn level_edges(b: &mut Builder, s: &Sector, z: f64) -> [u32; 4] {
+    let w = s.wrap();
+    let f = w.frame;
+    let d = [s.length / s.slant(), s.rise / s.slant(), 0.0];
+    let ab = b.edge(
+        CurveGeometry::Wrapped {
+            wrap: w,
+            curve: Box::new(CurveGeometry::Line {
+                origin: [0.0, 0.0, z],
+                direction: d,
+            }),
+        },
+        Interval::new(0.0, s.slant()).expect("slant range"),
+        false,
+    );
+    let bc = line(b, w.point([s.length, s.rise, z]), f.z, s.width - s.rise);
+    let da = line(b, w.point([0.0, 0.0, z]), f.z, s.width);
+    let cd = b.edge(
+        CurveGeometry::Circle {
+            frame: Frame3 {
+                origin: f.point([0.0, 0.0, s.width]),
+                ..f
+            },
+            radius: z,
+        },
+        Interval::new(0.0, s.length / s.wrap_radius).expect("arc range"),
+        false,
+    );
+    [ab, bc, cd, da]
+}
+
+fn edges(b: &mut Builder, s: &Sector) -> Edges {
+    let w = s.wrap();
+    let vertices = s
+        .corners()
+        .iter()
+        .map(|c| s.levels().map(|z| b.vertex(w.point([c[0], c[1], z]))))
+        .collect();
+    let [inner, outer] = s.levels().map(|z| level_edges(b, s, z));
+    let radial = |x: f64| {
+        w.frame
+            .vector([(x / s.wrap_radius).cos(), (x / s.wrap_radius).sin(), 0.0])
+    };
+    let up = s
+        .corners()
+        .iter()
+        .map(|c| line(b, w.point([c[0], c[1], s.inner]), radial(c[0]), s.thickness))
+        .collect();
+    Edges {
+        vertices,
+        ab: [inner[0], outer[0]],
+        bc: [inner[1], outer[1]],
+        cd: [inner[2], outer[2]],
+        da: [inner[3], outer[3]],
+        up,
+    }
+}
+
+fn cylinder_faces(b: &mut Builder, s: &Sector, e: &Edges) {
+    let (fwd, rev) = (Orientation::Forward, Orientation::Reverse);
+    let end = s.length / s.wrap_radius;
+    let ab = uv_line(
+        [0.0, 0.0],
+        [s.length / s.slant() / s.wrap_radius, s.rise / s.slant()],
+    );
+    let bc = uv_line([end, s.rise], [0.0, 1.0]);
+    let cd = uv_line([0.0, s.width], [1.0, 0.0]);
+    let da = uv_line([0.0, 0.0], [0.0, 1.0]);
+    let v = |corner: usize, level: usize| e.vertices[corner][level];
+    let inner = vec![
+        boundary(e.da[0], v(0, 0), v(3, 0), fwd, da.clone()),
+        boundary(e.cd[0], v(3, 0), v(2, 0), fwd, cd.clone()),
+        boundary(e.bc[0], v(2, 0), v(1, 0), rev, bc.clone()),
+        boundary(e.ab[0], v(1, 0), v(0, 0), rev, ab.clone()),
+    ];
+    let outer = vec![
+        boundary(e.ab[1], v(0, 1), v(1, 1), fwd, ab),
+        boundary(e.bc[1], v(1, 1), v(2, 1), fwd, bc),
+        boundary(e.cd[1], v(2, 1), v(3, 1), rev, cd),
+        boundary(e.da[1], v(3, 1), v(0, 1), rev, da),
+    ];
+    let pad = 1e-3;
+    for (level, uses) in [inner, outer].into_iter().enumerate() {
+        let face = b.brep.topology.faces.len();
+        b.face(
+            if level == 0 { "inner" } else { "outer" },
+            SurfaceGeometry::Cylinder {
+                frame: s.wrap().frame,
+                radius: s.levels()[level],
+            },
+            [[-pad, end + pad], [-pad, s.width + pad]],
+            uses,
+        )
+        .expect("cylinder face");
+        if level == 0 {
+            b.brep.topology.faces[face].sense = rev;
+        }
+    }
+}
+
+fn plane(b: &mut Builder, key: &str, frame: Frame3, uses: &[(u32, u32, u32, Orientation)]) {
+    let uses: Vec<Use> = uses
+        .iter()
+        .map(|&(e, from, to, sense)| {
+            plane_boundary(b, frame, e, from, to, sense).expect("plane boundary")
+        })
+        .collect();
+    let big = 4.0 * (SECTOR.inner + SECTOR.thickness + SECTOR.width);
+    b.face(
+        key,
+        SurfaceGeometry::Plane { frame },
+        [[-big, big], [-big, big]],
+        uses,
+    )
+    .expect("plane face");
+}
+
+fn plane_faces(b: &mut Builder, s: &Sector, e: &Edges) {
+    let (fwd, rev) = (Orientation::Forward, Orientation::Reverse);
+    let f = s.wrap().frame;
+    let v = |corner: usize, level: usize| e.vertices[corner][level];
+    let end = s.length / s.wrap_radius;
+    let start = Frame3::from_axis(f.origin, f.vector([0.0, -1.0, 0.0]), f.x).expect("start");
+    let radial = f.vector([end.cos(), end.sin(), 0.0]);
+    let finish =
+        Frame3::from_axis(f.origin, f.vector([-end.sin(), end.cos(), 0.0]), radial).expect("end");
+    let start_uses = [
+        (e.up[0], v(0, 0), v(0, 1), fwd),
+        (e.da[1], v(0, 1), v(3, 1), fwd),
+        (e.up[3], v(3, 1), v(3, 0), rev),
+        (e.da[0], v(3, 0), v(0, 0), rev),
+    ];
+    plane(b, "start", start, &start_uses);
+    let end_uses = [
+        (e.bc[0], v(1, 0), v(2, 0), fwd),
+        (e.up[2], v(2, 0), v(2, 1), fwd),
+        (e.bc[1], v(2, 1), v(1, 1), rev),
+        (e.up[1], v(1, 1), v(1, 0), rev),
+    ];
+    plane(b, "end", finish, &end_uses);
+    let far = Frame3 {
+        origin: f.point([0.0, 0.0, s.width]),
+        ..f
+    };
+    let uses = [
+        (e.cd[0], v(2, 0), v(3, 0), rev),
+        (e.up[3], v(3, 0), v(3, 1), fwd),
+        (e.cd[1], v(3, 1), v(2, 1), fwd),
+        (e.up[2], v(2, 1), v(2, 0), rev),
+    ];
+    plane(b, "far", far, &uses);
+}
+
+fn slanted_face(b: &mut Builder, s: &Sector, e: &Edges) {
+    let (fwd, rev) = (Orientation::Forward, Orientation::Reverse);
+    let slant = s.slant();
+    let [r, top] = s.levels();
+    let flat = Frame3 {
+        origin: [0.0; 3],
+        x: [s.length / slant, s.rise / slant, 0.0],
+        y: [0.0, 0.0, 1.0],
+        z: [s.rise / slant, -s.length / slant, 0.0],
+    };
+    let v = |corner: usize, level: usize| e.vertices[corner][level];
+    let uses = vec![
+        boundary(
+            e.ab[0],
+            v(0, 0),
+            v(1, 0),
+            fwd,
+            uv_line([0.0, r], [1.0, 0.0]),
+        ),
+        boundary(
+            e.up[1],
+            v(1, 0),
+            v(1, 1),
+            fwd,
+            uv_line([slant, r], [0.0, 1.0]),
+        ),
+        boundary(
+            e.ab[1],
+            v(1, 1),
+            v(0, 1),
+            rev,
+            uv_line([0.0, top], [1.0, 0.0]),
+        ),
+        boundary(
+            e.up[0],
+            v(0, 1),
+            v(0, 0),
+            rev,
+            uv_line([0.0, r], [0.0, 1.0]),
+        ),
+    ];
+    let pad = 1e-3;
+    b.face(
+        "slant",
+        SurfaceGeometry::Wrapped {
+            wrap: s.wrap(),
+            surface: Box::new(SurfaceGeometry::Plane { frame: flat }),
+        },
+        [[-pad, slant + pad], [r - pad, top + pad]],
+        uses,
+    )
+    .expect("slanted face");
+}
+
+pub(crate) fn wrapped_sector() -> BrepEnvelope {
+    let s = &SECTOR;
+    let mut b = Builder::new("sector".into(), SECTOR_ACCURACY).expect("builder");
+    let e = edges(&mut b, s);
+    cylinder_faces(&mut b, s, &e);
+    plane_faces(&mut b, s, &e);
+    slanted_face(&mut b, s, &e);
+    b.finish_solid().expect("wrapped sector")
+}
diff --git a/main/opengeometry/src/brep/builder/mod.rs b/main/opengeometry/src/brep/builder/mod.rs
index dfb958c2..95b23609 100644
--- a/main/opengeometry/src/brep/builder/mod.rs
+++ b/main/opengeometry/src/brep/builder/mod.rs
@@ -1,9 +1,14 @@
 mod boundary;
 mod brep_builder;
 #[cfg(test)]
+mod fixtures;
+#[cfg(test)]
 mod tests;
 mod wire;
 
-pub(crate) use boundary::{boundary, dimensions, padded_uv_bounds, plane_boundary, uv_line};
-pub(crate) use brep_builder::{Builder, Use};
+pub use boundary::{boundary, plane_boundary, projected_boundary, uv_line};
+pub(crate) use boundary::{dimensions, padded_uv_bounds};
+pub use brep_builder::{Builder, Use};
+#[cfg(test)]
+pub(crate) use fixtures::{wrapped_sector, SECTOR};
 pub(crate) use wire::WireUse;
diff --git a/main/opengeometry/src/brep/error.rs b/main/opengeometry/src/brep/error.rs
index 2f504256..e4bafe8b 100644
--- a/main/opengeometry/src/brep/error.rs
+++ b/main/opengeometry/src/brep/error.rs
@@ -50,6 +50,12 @@ pub(super) fn positive(value: f64) -> Result<(), GeometryError> {
     }
     Ok(())
 }
+pub(crate) fn unsupported_wrapped(operation: &str) -> GeometryError {
+    GeometryError::UnsupportedGeometry(format!("{operation} of wrapped geometry"))
+}
+pub(crate) fn unsupported_conoid(operation: &str) -> GeometryError {
+    GeometryError::UnsupportedGeometry(format!("{operation} of conoid surfaces"))
+}
 pub(crate) fn coverage_gap(a: &str, b: &str) -> GeometryError {
     GeometryError::CoverageGap {
         families: [a.into(), b.into()],
diff --git a/main/opengeometry/src/brep/geometry/conoid.rs b/main/opengeometry/src/brep/geometry/conoid.rs
new file mode 100644
index 00000000..ac54ce47
--- /dev/null
+++ b/main/opengeometry/src/brep/geometry/conoid.rs
@@ -0,0 +1,125 @@
+use super::curve::CurveGeometry;
+use super::turned::turned_axis;
+use crate::brep::error::GeometryError;
+use crate::brep::frame::{Frame3, UV};
+use crate::math::{add, cross, dot, scale, sub, Interval, Point3};
+
+const NEWTON_STEPS: usize = 32;
+
+type LocalJet = (Point3, Point3, Point3, Point3, Point3, Point3);
+
+pub(super) fn height(c: &[f64; 3], u: f64) -> [f64; 3] {
+    let (s, co) = u.sin_cos();
+    let h = c[0] + c[1] * co + c[2] * s;
+    [h, c[2] * co - c[1] * s, c[0] - h]
+}
+
+pub(super) fn local_jet(c: &[f64; 3], uv: UV) -> LocalJet {
+    let (s, co) = uv[0].sin_cos();
+    let (radial, tangent, z) = ([co, s, 0.0], [-s, co, 0.0], [0.0, 0.0, 1.0]);
+    let [h, slope, curvature] = height(c, uv[0]);
+    (
+        add(scale(radial, uv[1]), scale(z, h)),
+        add(scale(tangent, uv[1]), scale(z, slope)),
+        radial,
+        add(scale(radial, -uv[1]), scale(z, curvature)),
+        tangent,
+        [0.0; 3],
+    )
+}
+
+pub(super) fn validate(c: &[f64; 3]) -> Result<(), GeometryError> {
+    if c.iter().all(|v| v.is_finite()) {
+        Ok(())
+    } else {
+        Err(GeometryError::InvalidGeometry(
+            "conoid coefficients must be finite".into(),
+        ))
+    }
+}
+
+pub fn conoid_ellipse(
+    frame: Frame3,
+    c: [f64; 3],
+    radius: f64,
+) -> Result<(CurveGeometry, f64), GeometryError> {
+    frame.validate()?;
+    validate(&c)?;
+    if !radius.is_finite() || radius <= 0.0 {
+        return Err(GeometryError::InvalidGeometry(
+            "conoid ellipse radius must be positive".into(),
+        ));
+    }
+    let slope = [c[1] / radius, c[2] / radius];
+    let phase = slope[1].atan2(slope[0]);
+    let (s, co) = phase.sin_cos();
+    let rise = slope[0].hypot(slope[1]);
+    let stretch = rise.hypot(1.0);
+    let x = frame.vector([co / stretch, s / stretch, rise / stretch]);
+    let y = frame.vector([-s, co, 0.0]);
+    let curve = CurveGeometry::Ellipse {
+        frame: Frame3::from_axis(frame.point([0.0, 0.0, c[0]]), cross(x, y), x)?,
+        major_radius: radius * stretch,
+        minor_radius: radius,
+    };
+    curve.validate()?;
+    Ok((curve, phase))
+}
+
+pub(super) fn local_normal(c: &[f64; 3], uv: UV) -> Result<Point3, GeometryError> {
+    let (s, co) = uv[0].sin_cos();
+    let slope = height(c, uv[0])[1];
+    let length = slope.hypot(uv[1]);
+    if length == 0.0 || !length.is_finite() {
+        return Err(GeometryError::SingularParameterization);
+    }
+    Ok([-slope * s / length, slope * co / length, -uv[1] / length])
+}
+
+pub(super) fn local_bounds(
+    c: &[f64; 3],
+    domain: [Interval; 2],
+) -> Result<[Interval; 3], GeometryError> {
+    let [u, v] = domain;
+    let (co, s) = (u.cos()?, u.sin()?);
+    let z = turned_axis([c[0], 0.0], [c[1], c[2]], u)?;
+    Ok([co.mul_interval(v)?, s.mul_interval(v)?, z])
+}
+
+pub(super) fn closest(c: &[f64; 3], p: Point3, start: UV) -> Result<UV, GeometryError> {
+    let mut uv = start;
+    for _ in 0..NEWTON_STEPS {
+        let (point, du, dv, duu, duv, dvv) = local_jet(c, uv);
+        let r = sub(point, p);
+        let g = [dot(du, r), dot(dv, r)];
+        let metric = [dot(du, du), dot(du, dv), dot(dv, dv)];
+        let full = [
+            metric[0] + dot(duu, r),
+            metric[1] + dot(duv, r),
+            metric[2] + dot(dvv, r),
+        ];
+        let det = |h: [f64; 3]| h[0] * h[2] - h[1] * h[1];
+        let h = if full[0] > 0.0 && det(full) > 0.0 {
+            full
+        } else {
+            metric
+        };
+        let d = det(h);
+        if d <= 0.0 || !d.is_finite() {
+            return Err(GeometryError::SingularParameterization);
+        }
+        let step = [
+            (h[2] * g[0] - h[1] * g[1]) / d,
+            (h[0] * g[1] - h[1] * g[0]) / d,
+        ];
+        if !step.iter().all(|s| s.is_finite()) {
+            return Err(GeometryError::SingularParameterization);
+        }
+        uv = [uv[0] - step[0].clamp(-0.5, 0.5), uv[1] - step[1]];
+        let small = |s: f64, x: f64| s.abs() <= 4.0 * f64::EPSILON * (1.0 + x.abs());
+        if small(step[0], uv[0]) && small(step[1], uv[1]) {
+            break;
+        }
+    }
+    Ok(uv)
+}
diff --git a/main/opengeometry/src/brep/geometry/curve.rs b/main/opengeometry/src/brep/geometry/curve.rs
index 934a4719..98bcf9c2 100644
--- a/main/opengeometry/src/brep/geometry/curve.rs
+++ b/main/opengeometry/src/brep/geometry/curve.rs
@@ -1,4 +1,5 @@
 use super::turned::{narrowed, turned_axis};
+use super::wrap::Wrap;
 use crate::brep::bounds::PatchBounds;
 use crate::brep::error::GeometryError;
 use crate::brep::frame::{checked, Frame3};
@@ -24,6 +25,10 @@ pub enum CurveGeometry {
     Intersection {
         definition: u32,
     },
+    Wrapped {
+        wrap: Wrap,
+        curve: Box<CurveGeometry>,
+    },
 }
 
 pub(crate) trait Curve {
@@ -65,10 +70,39 @@ impl CurveGeometry {
                 }
             }
             Self::Intersection { .. } => {}
+            Self::Wrapped { wrap, curve } => {
+                wrap.validate()?;
+                if matches!(**curve, Self::Wrapped { .. } | Self::Intersection { .. }) {
+                    return Err(GeometryError::InvalidGeometry(
+                        "a wrapped curve must wrap a line, circle or ellipse".into(),
+                    ));
+                }
+                curve.validate()?;
+            }
         };
         Ok(())
     }
 
+    pub(crate) fn derivative_bounds(&self, range: Interval) -> Result<(f64, f64), GeometryError> {
+        Ok(match self {
+            Self::Line { .. } => (1.0, 0.0),
+            Self::Circle { radius, .. } => (*radius, *radius),
+            Self::Ellipse { major_radius, .. } => (*major_radius, *major_radius),
+            Self::Intersection { definition } => {
+                return Err(GeometryError::MissingReference {
+                    kind: "intersection".into(),
+                    index: *definition,
+                })
+            }
+            Self::Wrapped { wrap, curve } => {
+                let (speed, acceleration) = curve.derivative_bounds(range)?;
+                let z = curve.elementary_bounds(range)?.axes[2];
+                let (first, second) = wrap.derivative_bounds(z.lo.abs().max(z.hi.abs()));
+                (first * speed, second * speed * speed + first * acceleration)
+            }
+        })
+    }
+
     pub(crate) fn elementary_point(&self, t: f64) -> Result<Point3, GeometryError> {
         self.validate()?;
         finite(t)?;
@@ -82,6 +116,7 @@ impl CurveGeometry {
                 major_radius: a,
                 minor_radius: b,
             } => frame.point([a * t.cos(), b * t.sin(), 0.0]),
+            Self::Wrapped { wrap, curve } => wrap.point(curve.elementary_point(t)?),
             Self::Intersection { definition } => {
                 return Err(GeometryError::MissingReference {
                     kind: "intersection".into(),
@@ -104,6 +139,9 @@ impl CurveGeometry {
                 major_radius: a,
                 minor_radius: b,
             } => frame.vector([-a * t.sin(), b * t.cos(), 0.0]),
+            Self::Wrapped { wrap, curve } => {
+                wrap.push(curve.elementary_point(t)?, curve.elementary_tangent(t)?)
+            }
             Self::Intersection { definition } => {
                 return Err(GeometryError::MissingReference {
                     kind: "intersection".into(),
@@ -132,6 +170,7 @@ impl CurveGeometry {
                 major_radius: a,
                 minor_radius: b,
             } => conic_bounds(frame, [*a, *b], range),
+            Self::Wrapped { wrap, curve } => wrap.enclose(curve.elementary_bounds(range)?),
             Self::Intersection { definition } => Err(GeometryError::MissingReference {
                 kind: "intersection".into(),
                 index: *definition,
@@ -153,18 +192,12 @@ impl CurveGeometry {
                 "deflection must be finite and positive".into(),
             ));
         }
-        let bound = match self {
-            Self::Line { .. } => 0.0,
-            Self::Circle { radius, .. } => *radius,
-            Self::Ellipse { major_radius, .. } => *major_radius,
-            Self::Intersection { definition } => {
-                return Err(GeometryError::MissingReference {
-                    kind: "intersection".into(),
-                    index: *definition,
-                })
-            }
+        let bound = self.derivative_bounds(range)?.1;
+        let closed = match self {
+            Self::Wrapped { curve, .. } => !matches!(**curve, Self::Line { .. }),
+            _ => true,
         };
-        let minimum = if bound > 0.0 && range.width() >= std::f64::consts::TAU - 1e-12 {
+        let minimum = if closed && bound > 0.0 && range.width() >= std::f64::consts::TAU - 1e-12 {
             3.0
         } else {
             1.0
diff --git a/main/opengeometry/src/brep/geometry/mod.rs b/main/opengeometry/src/brep/geometry/mod.rs
index a3e64e14..c140c643 100644
--- a/main/opengeometry/src/brep/geometry/mod.rs
+++ b/main/opengeometry/src/brep/geometry/mod.rs
@@ -1,4 +1,5 @@
 mod chord;
+mod conoid;
 mod curve;
 mod intersection_definition;
 mod store;
@@ -7,8 +8,10 @@ mod surface;
 mod tests;
 mod trace;
 mod turned;
+mod wrap;
 
 pub(crate) use chord::{bounds_chord_deviation, point_segment_distance};
+pub use conoid::conoid_ellipse;
 pub(crate) use curve::Curve;
 pub use curve::CurveGeometry;
 pub(crate) use intersection_definition::{IntersectionDefinition, TraceAnchor};
@@ -16,3 +19,4 @@ pub use store::GeometryStore;
 pub use surface::{Surface, SurfaceChart, SurfaceGeometry, SurfaceJet2};
 pub use trace::IntersectionPoint;
 pub(crate) use trace::SsiBudget;
+pub use wrap::Wrap;
diff --git a/main/opengeometry/src/brep/geometry/store.rs b/main/opengeometry/src/brep/geometry/store.rs
index 22a53bcc..13ef3b98 100644
--- a/main/opengeometry/src/brep/geometry/store.rs
+++ b/main/opengeometry/src/brep/geometry/store.rs
@@ -31,6 +31,20 @@ impl GeometryStore {
             intersections: Vec::new(),
         }
     }
+    pub(crate) fn has_wrapped(&self) -> bool {
+        self.surfaces
+            .iter()
+            .any(|s| matches!(s, SurfaceGeometry::Wrapped { .. }))
+            || self
+                .curves
+                .iter()
+                .any(|c| matches!(c, CurveGeometry::Wrapped { .. }))
+    }
+    pub(crate) fn has_conoid(&self) -> bool {
+        self.surfaces
+            .iter()
+            .any(|s| matches!(s, SurfaceGeometry::Conoid { .. }))
+    }
     pub fn surface(&self, id: u32) -> Result<&SurfaceGeometry, GeometryError> {
         self.surfaces
             .get(id as usize)
diff --git a/main/opengeometry/src/brep/geometry/surface.rs b/main/opengeometry/src/brep/geometry/surface.rs
index 9566cd8a..709c25df 100644
--- a/main/opengeometry/src/brep/geometry/surface.rs
+++ b/main/opengeometry/src/brep/geometry/surface.rs
@@ -1,7 +1,9 @@
-use super::turned::{narrowed, reached, ring_axis, swept_axis};
+use super::conoid;
+use super::turned::{lift, narrowed, reached, ring_axis, swept_axis};
+use super::wrap::Wrap;
 use crate::brep::bounds::PatchBounds;
 use crate::brep::error::GeometryError;
-use crate::brep::frame::{checked, Frame3, UVBox, UV};
+use crate::brep::frame::{checked, unit, Frame3, UVBox, UV};
 use crate::math::{add, finite, norm, scale, Interval, Point3};
 use serde::{Deserialize, Serialize};
 
@@ -29,6 +31,10 @@ const CONE_CHART: [SurfaceChart; 1] = [SurfaceChart {
     domain: [[0.0, std::f64::consts::TAU], [0.0, f64::MAX]],
     periods: [Some(std::f64::consts::TAU), None],
 }];
+const CONOID_CHART: [SurfaceChart; 1] = [SurfaceChart {
+    domain: [[0.0, std::f64::consts::TAU], [0.0, f64::MAX]],
+    periods: [Some(std::f64::consts::TAU), None],
+}];
 const TORUS_CHART: [SurfaceChart; 1] = [SurfaceChart {
     domain: [[0.0, std::f64::consts::TAU]; 2],
     periods: [Some(std::f64::consts::TAU); 2],
@@ -57,6 +63,14 @@ pub enum SurfaceGeometry {
         major_radius: f64,
         minor_radius: f64,
     },
+    Wrapped {
+        wrap: Wrap,
+        surface: Box<SurfaceGeometry>,
+    },
+    Conoid {
+        frame: Frame3,
+        c: [f64; 3],
+    },
 }
 
 #[derive(Clone, Copy, Debug)]
@@ -85,7 +99,9 @@ impl SurfaceGeometry {
             | Self::Sphere { frame, .. }
             | Self::Cylinder { frame, .. }
             | Self::Cone { frame, .. }
-            | Self::Torus { frame, .. } => frame,
+            | Self::Torus { frame, .. }
+            | Self::Conoid { frame, .. } => frame,
+            Self::Wrapped { wrap, .. } => &wrap.frame,
         }
     }
     pub(crate) fn validate(&self) -> Result<(), GeometryError> {
@@ -110,6 +126,8 @@ impl SurfaceGeometry {
                     && *minor_radius > 0.0
                     && major_radius > minor_radius
             }
+            Self::Wrapped { wrap, surface } => return wrapped_validate(wrap, surface),
+            Self::Conoid { c, .. } => return conoid::validate(c),
         };
         if !valid {
             return Err(GeometryError::InvalidGeometry(
@@ -120,6 +138,10 @@ impl SurfaceGeometry {
     }
 
     fn check_uv(&self, uv: UV) -> Result<(), GeometryError> {
+        if let Self::Wrapped { wrap, surface } = self {
+            wrap.validate()?;
+            return surface.check_uv(uv);
+        }
         self.validate()?;
         for x in uv {
             finite(x)?;
@@ -135,18 +157,15 @@ impl SurfaceGeometry {
     }
 
     pub(crate) fn parameter_in_domain(&self, uv: UV) -> bool {
+        if let Self::Wrapped { surface, .. } = self {
+            return surface.parameter_in_domain(uv);
+        }
         uv.into_iter().all(f64::is_finite)
             && (!matches!(self, Self::Sphere { .. }) || uv[1].abs() <= std::f64::consts::FRAC_PI_2)
             && (!matches!(self, Self::Cone { .. }) || uv[1] >= 0.0)
     }
 }
 
-fn lift(angle: f64, hint: Option<f64>) -> f64 {
-    hint.map_or(angle, |h| {
-        angle + ((h - angle) / std::f64::consts::TAU).round() * std::f64::consts::TAU
-    })
-}
-
 impl Surface for SurfaceGeometry {
     fn point_at(&self, uv: UV) -> Result<Point3, GeometryError> {
         Ok(self.derivatives(uv)?.point)
@@ -215,6 +234,8 @@ impl Surface for SurfaceGeometry {
                     add(scale(radial, -a * cv), scale(z, -a * sv)),
                 )
             }
+            Self::Wrapped { wrap, surface } => return wrapped_derivatives(wrap, surface, uv),
+            Self::Conoid { c, .. } => conoid::local_jet(c, uv),
         };
         let f = self.frame();
         Ok(SurfaceJet2 {
@@ -245,6 +266,8 @@ impl Surface for SurfaceGeometry {
                     -semi_angle.sin(),
                 ]
             }
+            Self::Wrapped { wrap, surface } => return wrapped_normal(wrap, surface, uv),
+            Self::Conoid { c, .. } => conoid::local_normal(c, uv)?,
         };
         checked(self.frame().vector(n))
     }
@@ -257,6 +280,7 @@ impl Surface for SurfaceGeometry {
                 finite(v)?;
             }
         }
+        let world = p;
         let p = checked(self.frame().local(p))?;
         let radial = p[0].hypot(p[1]);
         let u = lift(p[1].atan2(p[0]), hint.map(|h| h[0]));
@@ -290,6 +314,13 @@ impl Surface for SurfaceGeometry {
                     lift(p[2].atan2(radial - major_radius), hint.map(|h| h[1])),
                 ]
             }
+            Self::Wrapped { wrap, surface } => return wrapped_project(wrap, surface, world, hint),
+            Self::Conoid { c, .. } => {
+                if radial == 0.0 {
+                    return Err(GeometryError::SingularParameterization);
+                }
+                conoid::closest(c, p, [u, radial])?
+            }
         };
         for x in result {
             finite(x)?;
@@ -304,6 +335,8 @@ impl Surface for SurfaceGeometry {
             Self::Cylinder { .. } => &CYLINDER_CHART,
             Self::Cone { .. } => &CONE_CHART,
             Self::Torus { .. } => &TORUS_CHART,
+            Self::Wrapped { surface, .. } => surface.charts(),
+            Self::Conoid { .. } => &CONOID_CHART,
         }
     }
 
@@ -351,6 +384,8 @@ impl Surface for SurfaceGeometry {
                     v.sin()?.mul_interval(a)?,
                 ]
             }
+            Self::Wrapped { wrap, surface } => return wrap.enclose(surface.enclose(domain)?),
+            Self::Conoid { c, .. } => conoid::local_bounds(c, domain)?,
         };
         let bounds = self.frame().bounds(local)?;
         self.tightened(bounds, domain)
@@ -390,3 +425,52 @@ impl SurfaceGeometry {
         Ok(bounds)
     }
 }
+
+fn wrapped_validate(wrap: &Wrap, surface: &SurfaceGeometry) -> Result<(), GeometryError> {
+    wrap.validate()?;
+    if matches!(surface, SurfaceGeometry::Wrapped { .. }) {
+        return Err(GeometryError::InvalidGeometry(
+            "a wrapped surface cannot wrap another wrapped surface".into(),
+        ));
+    }
+    surface.validate()
+}
+
+fn wrapped_derivatives(
+    wrap: &Wrap,
+    surface: &SurfaceGeometry,
+    uv: UV,
+) -> Result<SurfaceJet2, GeometryError> {
+    let f = surface.derivatives(uv)?;
+    let p = f.point;
+    let second = |a, b, d| add(wrap.second(p, a, b), wrap.push(p, d));
+    Ok(SurfaceJet2 {
+        point: checked(wrap.point(p))?,
+        du: checked(wrap.push(p, f.du))?,
+        dv: checked(wrap.push(p, f.dv))?,
+        duu: checked(second(f.du, f.du, f.duu))?,
+        duv: checked(second(f.du, f.dv, f.duv))?,
+        dvv: checked(second(f.dv, f.dv, f.dvv))?,
+    })
+}
+
+fn wrapped_normal(wrap: &Wrap, surface: &SurfaceGeometry, uv: UV) -> Result<Point3, GeometryError> {
+    let p = surface.point_at(uv)?;
+    if p[2] <= 0.0 {
+        return Err(GeometryError::SingularParameterization);
+    }
+    unit(wrap.push_normal(p, surface.normal_at(uv)?))
+}
+
+fn wrapped_project(
+    wrap: &Wrap,
+    surface: &SurfaceGeometry,
+    p: Point3,
+    hint: Option<UV>,
+) -> Result<UV, GeometryError> {
+    let hint_x = match hint {
+        Some(h) => Some(surface.point_at(h)?[0]),
+        None => None,
+    };
+    surface.project(wrap.flatten(p, hint_x)?, hint)
+}
diff --git a/main/opengeometry/src/brep/geometry/tests.rs b/main/opengeometry/src/brep/geometry/tests.rs
index 817998be..503ce4ae 100644
--- a/main/opengeometry/src/brep/geometry/tests.rs
+++ b/main/opengeometry/src/brep/geometry/tests.rs
@@ -1,8 +1,10 @@
+use super::conoid::conoid_ellipse;
 use super::curve::CurveGeometry;
 use super::surface::{Surface, SurfaceGeometry};
+use super::wrap::Wrap;
 use crate::brep::error::GeometryError;
 use crate::brep::frame::Frame3;
-use crate::math::{dot, norm, scale, sub, Interval};
+use crate::math::{cross, dot, norm, scale, sub, Interval};
 
 fn surfaces() -> Vec<SurfaceGeometry> {
     let frame = Frame3::from_axis([3.0, -2.0, 7.0], [1.0, 2.0, 3.0], [0.0, 1.0, 0.0]).unwrap();
@@ -19,9 +21,115 @@ fn surfaces() -> Vec<SurfaceGeometry> {
             major_radius: 3.0,
             minor_radius: 1.0,
         },
+        conoid(),
+        SurfaceGeometry::Wrapped {
+            wrap: wrap(),
+            surface: Box::new(SurfaceGeometry::Cylinder {
+                frame: Frame3 {
+                    origin: [1.0, 0.5, 3.0],
+                    ..Frame3::IDENTITY
+                },
+                radius: 0.8,
+            }),
+        },
+        SurfaceGeometry::Wrapped {
+            wrap: wrap(),
+            surface: Box::new(SurfaceGeometry::Plane {
+                frame: Frame3::from_axis([0.3, -0.2, 2.5], [1.0, 2.0, 0.3], [0.0, 0.0, 1.0])
+                    .unwrap(),
+            }),
+        },
     ]
 }
 
+fn conoid() -> SurfaceGeometry {
+    SurfaceGeometry::Conoid {
+        frame: Frame3::from_axis([3.0, -2.0, 7.0], [1.0, 2.0, 3.0], [0.0, 1.0, 0.0]).unwrap(),
+        c: [1.5, 0.8, -0.6],
+    }
+}
+
+fn wrap() -> Wrap {
+    Wrap {
+        frame: Frame3::from_axis([3.0, -2.0, 7.0], [1.0, 2.0, 3.0], [0.0, 1.0, 0.0]).unwrap(),
+        radius: 2.6,
+    }
+}
+
+#[test]
+fn a_wrap_keeps_arc_length_on_its_radius_and_round_trips() {
+    let w = wrap();
+    let p = [1.3, -0.4, 2.6];
+    let chord = norm(sub(w.point(p), w.point([0.0, -0.4, 2.6])));
+    assert!((chord - 2.0 * 2.6 * (1.3_f64 / 5.2).sin()).abs() < 1e-12);
+    let back = w.flatten(w.point(p), None).unwrap();
+    assert!(norm(sub(back, p)) < 1e-12);
+    let far = [2.6 * 4.0, 0.0, 3.0];
+    assert!(norm(sub(w.flatten(w.point(far), Some(far[0])).unwrap(), far)) < 1e-12);
+    let curve = CurveGeometry::Wrapped {
+        wrap: w,
+        curve: Box::new(CurveGeometry::Line {
+            origin: [0.0, 0.0, 2.0],
+            direction: [0.6, 0.8, 0.0],
+        }),
+    };
+    let speed = |t: f64| norm(curve.elementary_tangent(t).unwrap());
+    assert!((speed(0.1) - speed(3.0)).abs() < 1e-12);
+    let h = 1e-6;
+    let fd = scale(
+        sub(
+            curve.elementary_point(1.0 + h).unwrap(),
+            curve.elementary_point(1.0 - h).unwrap(),
+        ),
+        0.5 / h,
+    );
+    assert!(norm(sub(fd, curve.elementary_tangent(1.0).unwrap())) < 1e-8);
+    let bounds = curve
+        .elementary_bounds(Interval::new(0.0, 4.0).unwrap())
+        .unwrap();
+    for i in 0..=40 {
+        assert!(bounds.contains(curve.elementary_point(i as f64 / 10.0).unwrap()));
+    }
+    let nested = CurveGeometry::Wrapped {
+        wrap: w,
+        curve: Box::new(curve),
+    };
+    assert!(nested.validate().is_err());
+    let nested_surface = SurfaceGeometry::Wrapped {
+        wrap: w,
+        surface: Box::new(surfaces().pop().unwrap()),
+    };
+    assert!(nested_surface.validate().is_err());
+}
+
+#[test]
+fn wrapped_circle_samples_meet_their_chord_error() {
+    let curve = CurveGeometry::Wrapped {
+        wrap: wrap(),
+        curve: Box::new(CurveGeometry::Circle {
+            frame: Frame3 {
+                origin: [1.0, 0.5, 3.0],
+                ..Frame3::IDENTITY
+            },
+            radius: 0.8,
+        }),
+    };
+    let range = Interval::new(0.0, std::f64::consts::TAU).unwrap();
+    let samples = curve.sample_elementary(range, 1e-3, 100_000).unwrap();
+    let n = samples.len() - 1;
+    for i in 0..n {
+        for k in 1..8 {
+            let t = range.width() * (i as f64 + k as f64 / 8.0) / n as f64;
+            let p = curve.elementary_point(t).unwrap();
+            let (a, b) = (samples[i], samples[i + 1]);
+            let ab = sub(b, a);
+            let along = dot(sub(p, a), ab) / dot(ab, ab);
+            let foot = [0, 1, 2].map(|j| a[j] + along * ab[j]);
+            assert!(norm(sub(p, foot)) <= 1e-3);
+        }
+    }
+}
+
 #[test]
 fn derivatives_projection_and_normals_for_all_families() {
     let uv = [0.7, 0.4];
@@ -170,3 +278,77 @@ fn partial_cylinder_and_cone_patches_on_a_skew_frame_get_their_exact_box() {
         }
     }
 }
+
+#[test]
+fn conoid_rulings_are_radial_and_constant_radius_curves_are_planar() {
+    let surface = conoid();
+    let frame = *surface.frame();
+    let SurfaceGeometry::Conoid { c, .. } = surface else {
+        unreachable!()
+    };
+    for u in [0.0, 0.9, 2.5, 4.0] {
+        let a = surface.point_at([u, 1.0]).unwrap();
+        let b = surface.point_at([u, 3.0]).unwrap();
+        assert!(dot(sub(b, a), frame.z).abs() < 1e-12);
+        let mid = surface.point_at([u, 2.0]).unwrap();
+        assert!(norm(sub(mid, scale(add3(a, b), 0.5))) < 1e-12);
+        for r in [0.5, 2.0] {
+            let local = frame.local(surface.point_at([u, r]).unwrap());
+            assert!((local[0].hypot(local[1]) - r).abs() < 1e-12);
+            let plane = c[0] + c[1] / r * local[0] + c[2] / r * local[1];
+            assert!((local[2] - plane).abs() < 1e-12);
+        }
+    }
+    let jet = surface.derivatives([0.7, 2.0]).unwrap();
+    let n = surface.normal_at([0.7, 2.0]).unwrap();
+    let expected = cross(jet.du, jet.dv);
+    assert!(norm(sub(n, scale(expected, 1.0 / norm(expected)))) < 1e-12);
+}
+
+#[test]
+fn conoid_projection_finds_the_closest_point_and_respects_the_hint() {
+    let surface = conoid();
+    for (u, v, offset) in [(0.3, 1.2, 0.2), (2.0, 2.5, -0.4), (5.5, 0.8, 0.05)] {
+        let p = surface.point_at([u, v]).unwrap();
+        let off = add3(p, scale(surface.normal_at([u, v]).unwrap(), offset));
+        let uv = surface.project(off, Some([u, v])).unwrap();
+        assert!(
+            (uv[0] - u).abs() < 1e-9 && (uv[1] - v).abs() < 1e-9,
+            "{uv:?}"
+        );
+        let lifted = surface
+            .project(p, Some([u + std::f64::consts::TAU, v]))
+            .unwrap();
+        assert!((lifted[0] - u - std::f64::consts::TAU).abs() < 1e-12);
+    }
+    let bad = SurfaceGeometry::Conoid {
+        frame: Frame3::IDENTITY,
+        c: [0.0, f64::NAN, 0.0],
+    };
+    assert!(bad.validate().is_err());
+}
+
+fn add3(a: [f64; 3], b: [f64; 3]) -> [f64; 3] {
+    std::array::from_fn(|i| a[i] + b[i])
+}
+
+#[test]
+fn conoid_ellipses_trace_the_constant_radius_curves() {
+    let surface = conoid();
+    let SurfaceGeometry::Conoid { frame, c } = surface else {
+        unreachable!()
+    };
+    for radius in [0.5, 2.0, 7.0] {
+        let (ellipse, phase) = conoid_ellipse(frame, c, radius).unwrap();
+        for t in [-1.0, 0.0, 0.4, 2.5] {
+            let on_surface = surface.point_at([t + phase, radius]).unwrap();
+            assert!(norm(sub(ellipse.elementary_point(t).unwrap(), on_surface)) < 1e-12);
+        }
+    }
+    let (circle, _) = conoid_ellipse(frame, [c[0], 0.0, 0.0], 1.0).unwrap();
+    assert!(matches!(
+        circle,
+        CurveGeometry::Ellipse { major_radius, minor_radius, .. } if major_radius == minor_radius
+    ));
+    assert!(conoid_ellipse(frame, c, 0.0).is_err());
+}
diff --git a/main/opengeometry/src/brep/geometry/turned.rs b/main/opengeometry/src/brep/geometry/turned.rs
index c19ddfe1..5f0953cf 100644
--- a/main/opengeometry/src/brep/geometry/turned.rs
+++ b/main/opengeometry/src/brep/geometry/turned.rs
@@ -2,6 +2,12 @@ use crate::brep::error::GeometryError;
 use crate::brep::frame::{Frame3, UVBox};
 use crate::math::Interval;
 
+pub(super) fn lift(angle: f64, hint: Option<f64>) -> f64 {
+    hint.map_or(angle, |h| {
+        angle + ((h - angle) / std::f64::consts::TAU).round() * std::f64::consts::TAU
+    })
+}
+
 pub(super) fn reached(frame: &Frame3, axis: usize) -> usize {
     [frame.x[axis], frame.y[axis], frame.z[axis]]
         .iter()
diff --git a/main/opengeometry/src/brep/geometry/wrap.rs b/main/opengeometry/src/brep/geometry/wrap.rs
new file mode 100644
index 00000000..f8038810
--- /dev/null
+++ b/main/opengeometry/src/brep/geometry/wrap.rs
@@ -0,0 +1,86 @@
+use super::turned::lift;
+use crate::brep::bounds::PatchBounds;
+use crate::brep::error::GeometryError;
+use crate::brep::frame::{checked, Frame3};
+use crate::math::{add, scale, Interval, Point3};
+use serde::{Deserialize, Serialize};
+
+#[derive(Clone, Copy, Debug, PartialEq, Serialize, Deserialize)]
+#[serde(deny_unknown_fields)]
+pub struct Wrap {
+    pub frame: Frame3,
+    pub radius: f64,
+}
+
+impl Wrap {
+    pub(crate) fn validate(&self) -> Result<(), GeometryError> {
+        self.frame.validate()?;
+        (self.radius.is_finite() && self.radius > 0.0)
+            .then_some(())
+            .ok_or_else(|| GeometryError::InvalidGeometry("wrap radius must be positive".into()))
+    }
+
+    fn axes(&self, p: Point3) -> [Point3; 3] {
+        let (s, c) = (p[0] / self.radius).sin_cos();
+        [[-s, c, 0.0], [0.0, 0.0, 1.0], [c, s, 0.0]]
+    }
+
+    pub fn point(&self, p: Point3) -> Point3 {
+        let (s, c) = (p[0] / self.radius).sin_cos();
+        self.frame.point([p[2] * c, p[2] * s, p[1]])
+    }
+
+    pub(crate) fn push(&self, p: Point3, v: Point3) -> Point3 {
+        let [t, a, r] = self.axes(p);
+        let local = add(
+            add(scale(t, v[0] * p[2] / self.radius), scale(a, v[1])),
+            scale(r, v[2]),
+        );
+        self.frame.vector(local)
+    }
+
+    pub(crate) fn second(&self, p: Point3, a: Point3, b: Point3) -> Point3 {
+        let [t, _, r] = self.axes(p);
+        let xx = scale(r, -p[2] / (self.radius * self.radius));
+        let xz = scale(t, 1.0 / self.radius);
+        self.frame.vector(add(
+            scale(xx, a[0] * b[0]),
+            scale(xz, a[0] * b[2] + a[2] * b[0]),
+        ))
+    }
+
+    pub(crate) fn push_normal(&self, p: Point3, n: Point3) -> Point3 {
+        let [t, a, r] = self.axes(p);
+        let k = p[2] / self.radius;
+        self.frame.vector(add(
+            add(scale(t, n[0]), scale(a, k * n[1])),
+            scale(r, k * n[2]),
+        ))
+    }
+
+    pub fn flatten(&self, q: Point3, hint_x: Option<f64>) -> Result<Point3, GeometryError> {
+        let l = self.frame.local(q);
+        let z = l[0].hypot(l[1]);
+        if z == 0.0 {
+            return Err(GeometryError::SingularParameterization);
+        }
+        let angle = lift(l[1].atan2(l[0]), hint_x.map(|x| x / self.radius));
+        checked([angle * self.radius, l[2], z])
+    }
+
+    pub(crate) fn enclose(&self, flat: PatchBounds) -> Result<PatchBounds, GeometryError> {
+        let [x, y, z] = flat.axes;
+        let angle = x.mul_interval(Interval::point(1.0 / self.radius)?)?;
+        let (c, s) = (angle.cos()?, angle.sin()?);
+        self.frame
+            .bounds([c.mul_interval(z)?, s.mul_interval(z)?, y])
+    }
+
+    pub(crate) fn derivative_bounds(&self, z_max: f64) -> (f64, f64) {
+        let z = z_max.abs();
+        (
+            (z / self.radius).max(1.0),
+            z / (self.radius * self.radius) + 1.0 / self.radius,
+        )
+    }
+}
diff --git a/main/opengeometry/src/brep/mod.rs b/main/opengeometry/src/brep/mod.rs
index 3978682a..1b5be0f3 100644
--- a/main/opengeometry/src/brep/mod.rs
+++ b/main/opengeometry/src/brep/mod.rs
@@ -18,11 +18,12 @@ mod validate;
 pub use accuracy::Accuracy;
 pub use body_type::BodyType;
 pub use bounds::PatchBounds;
-pub(super) use builder::{
-    boundary, dimensions, padded_uv_bounds, plane_boundary, uv_line, Builder, Use, WireUse,
-};
-pub(super) use error::coverage_gap;
+pub use builder::{boundary, plane_boundary, projected_boundary, uv_line, Builder, Use};
+pub(super) use builder::{dimensions, padded_uv_bounds, WireUse};
+#[cfg(test)]
+pub(super) use builder::{wrapped_sector, SECTOR};
 pub use error::GeometryError;
+pub(super) use error::{coverage_gap, unsupported_conoid, unsupported_wrapped};
 pub use frame::Frame3;
 pub(super) use frame::{checked, unit, UVBox, GROUND, UV};
 pub(super) use geometry::{
@@ -30,18 +31,19 @@ pub(super) use geometry::{
     TraceAnchor,
 };
 pub use geometry::{
-    CurveGeometry, GeometryStore, IntersectionPoint, Surface, SurfaceChart, SurfaceGeometry,
-    SurfaceJet2,
+    conoid_ellipse, CurveGeometry, GeometryStore, IntersectionPoint, Surface, SurfaceChart,
+    SurfaceGeometry, SurfaceJet2, Wrap,
 };
-pub(super) use pcurve::{period_lifted_uv, IntersectionSide, PcurveGeometry};
+pub(super) use pcurve::period_lifted_uv;
+pub use pcurve::{IntersectionSide, PcurveGeometry};
 pub use placement::{mirror::mirrored, placed};
 pub use similarity::Similarity3;
 #[cfg(test)]
 pub(super) use topology::{circular_face, coarse_accuracy, fine_accuracy};
 pub use topology::{
-    BrepEnvelope, Edge, Face, FaceProvenance, FaceRole, FaceSource, HalfEdge, Topology, Vertex,
+    BrepEnvelope, Edge, Face, FaceProvenance, FaceRole, FaceSource, HalfEdge, Orientation,
+    Topology, Vertex,
 };
 pub(super) use topology::{
-    EdgeGeometry, GeometryQuality, HalfEdgeGeometryUse, Loop, Orientation, Shell, SolidRegion,
-    TrimRegion, Wire,
+    EdgeGeometry, GeometryQuality, HalfEdgeGeometryUse, Loop, Shell, SolidRegion, TrimRegion, Wire,
 };
diff --git a/main/opengeometry/src/brep/pcurve.rs b/main/opengeometry/src/brep/pcurve.rs
index e0046c5c..39f2919a 100644
--- a/main/opengeometry/src/brep/pcurve.rs
+++ b/main/opengeometry/src/brep/pcurve.rs
@@ -3,7 +3,7 @@ use serde::{Deserialize, Serialize};
 
 #[derive(Clone, Debug, PartialEq, Serialize, Deserialize)]
 #[serde(tag = "kind", deny_unknown_fields)]
-pub(crate) enum PcurveGeometry {
+pub enum PcurveGeometry {
     Line2 {
         origin: UV,
         direction: UV,
@@ -27,7 +27,7 @@ pub(crate) enum PcurveGeometry {
     },
 }
 #[derive(Clone, Copy, Debug, Eq, PartialEq, Serialize, Deserialize)]
-pub(crate) enum IntersectionSide {
+pub enum IntersectionSide {
     A,
     B,
 }
diff --git a/main/opengeometry/src/brep/placement/mirror.rs b/main/opengeometry/src/brep/placement/mirror.rs
index b277334c..2bf76457 100644
--- a/main/opengeometry/src/brep/placement/mirror.rs
+++ b/main/opengeometry/src/brep/placement/mirror.rs
@@ -1,5 +1,5 @@
 use super::check_coordinate_precision;
-use crate::brep::error::GeometryError;
+use crate::brep::error::{unsupported_wrapped, GeometryError};
 use crate::brep::frame::{checked, unit, Frame3, UV};
 use crate::brep::geometry::{CurveGeometry, IntersectionDefinition, SurfaceGeometry};
 use crate::brep::pcurve::PcurveGeometry;
@@ -107,8 +107,8 @@ pub fn mirrored(
         .revision
         .checked_add(1)
         .ok_or_else(|| GeometryError::LimitExceeded("BRep revision overflow".into()))?;
-    reflect_surfaces(&mut output.geometry.surfaces, reflection);
-    reflect_curves(&mut output.geometry.curves, reflection);
+    reflect_surfaces(&mut output.geometry.surfaces, reflection)?;
+    reflect_curves(&mut output.geometry.curves, reflection)?;
     for vertex in &mut output.topology.vertices {
         vertex.position = reflection.point(vertex.position);
     }
@@ -129,7 +129,10 @@ pub fn mirrored(
     Ok(output)
 }
 
-fn reflect_surfaces(surfaces: &mut [SurfaceGeometry], reflection: Reflection) {
+fn reflect_surfaces(
+    surfaces: &mut [SurfaceGeometry],
+    reflection: Reflection,
+) -> Result<(), GeometryError> {
     for surface in surfaces {
         match surface {
             SurfaceGeometry::Plane { frame }
@@ -137,11 +140,20 @@ fn reflect_surfaces(surfaces: &mut [SurfaceGeometry], reflection: Reflection) {
             | SurfaceGeometry::Cylinder { frame, .. }
             | SurfaceGeometry::Cone { frame, .. }
             | SurfaceGeometry::Torus { frame, .. } => *frame = reflection.surface_frame(*frame),
+            SurfaceGeometry::Wrapped { .. } => return Err(unsupported_wrapped("mirroring")),
+            SurfaceGeometry::Conoid { frame, c } => {
+                *frame = reflection.surface_frame(*frame);
+                c[2] = -c[2];
+            }
         }
     }
+    Ok(())
 }
 
-fn reflect_curves(curves: &mut [CurveGeometry], reflection: Reflection) {
+fn reflect_curves(
+    curves: &mut [CurveGeometry],
+    reflection: Reflection,
+) -> Result<(), GeometryError> {
     for curve in curves {
         match curve {
             CurveGeometry::Line { origin, direction } => {
@@ -152,8 +164,10 @@ fn reflect_curves(curves: &mut [CurveGeometry], reflection: Reflection) {
                 *frame = reflection.curve_frame(*frame)
             }
             CurveGeometry::Intersection { .. } => {}
+            CurveGeometry::Wrapped { .. } => return Err(unsupported_wrapped("mirroring")),
         }
     }
+    Ok(())
 }
 
 fn remap_pcurves(input: &BrepEnvelope, output: &mut BrepEnvelope) {
diff --git a/main/opengeometry/src/brep/placement/mod.rs b/main/opengeometry/src/brep/placement/mod.rs
index 3356992b..814c0621 100644
--- a/main/opengeometry/src/brep/placement/mod.rs
+++ b/main/opengeometry/src/brep/placement/mod.rs
@@ -2,7 +2,7 @@ pub(super) mod mirror;
 #[cfg(test)]
 mod tests;
 
-use super::error::GeometryError;
+use super::error::{unsupported_wrapped, GeometryError};
 use super::frame::Frame3;
 use super::geometry::{CurveGeometry, IntersectionDefinition, SurfaceGeometry};
 use super::pcurve::PcurveGeometry;
@@ -21,6 +21,9 @@ pub fn placed(
             "placement scale must be positive and finite".into(),
         ));
     }
+    if scale != 1.0 && input.geometry.has_wrapped() {
+        return Err(unsupported_wrapped("scaling"));
+    }
     let mut output = input.clone();
     output.revision = input
         .revision
@@ -82,7 +85,9 @@ pub fn placed(
 fn metric(surface: &SurfaceGeometry, scale: f64) -> [f64; 2] {
     match surface {
         SurfaceGeometry::Plane { .. } => [scale; 2],
-        SurfaceGeometry::Cylinder { .. } | SurfaceGeometry::Cone { .. } => [1.0, scale],
+        SurfaceGeometry::Cylinder { .. }
+        | SurfaceGeometry::Cone { .. }
+        | SurfaceGeometry::Conoid { .. } => [1.0, scale],
         _ => [1.0; 2],
     }
 }
@@ -107,6 +112,13 @@ fn place_surfaces(surfaces: &mut [SurfaceGeometry], placement: Frame3, scale: f6
                 *major_radius *= scale;
                 *minor_radius *= scale;
             }
+            SurfaceGeometry::Wrapped { wrap, .. } => {
+                transform_frame(&mut wrap.frame, placement, 1.0)
+            }
+            SurfaceGeometry::Conoid { frame, c } => {
+                transform_frame(frame, placement, scale);
+                *c = c.map(|value| value * scale);
+            }
         }
     }
 }
@@ -139,6 +151,7 @@ fn place_curves(curves: &mut [CurveGeometry], placement: Frame3, scale: f64) {
                 *minor_radius *= scale;
             }
             CurveGeometry::Intersection { .. } => {}
+            CurveGeometry::Wrapped { wrap, .. } => transform_frame(&mut wrap.frame, placement, 1.0),
         }
     }
 }
diff --git a/main/opengeometry/src/brep/placement/tests.rs b/main/opengeometry/src/brep/placement/tests.rs
index 2afa0728..04bd1780 100644
--- a/main/opengeometry/src/brep/placement/tests.rs
+++ b/main/opengeometry/src/brep/placement/tests.rs
@@ -1,6 +1,7 @@
 use super::*;
 use crate::brep::geometry::{Curve, Surface};
 use crate::brep::topology::coarse_accuracy;
+use crate::brep::wrapped_sector;
 use crate::math::sub;
 use crate::primitives;
 
@@ -125,3 +126,31 @@ fn invalid_scale_revision_overflow_and_precision_are_structured_errors() {
         Err(GeometryError::LimitExceeded(_))
     ));
 }
+
+#[test]
+fn wrapped_geometry_moves_rigidly_and_refuses_scaling_and_mirroring() {
+    let brep = wrapped_sector();
+    let placement = Frame3::from_axis([10.0, -3.0, 5.0], [1.0, 2.0, 3.0], [0.0, 1.0, 0.0]).unwrap();
+    let moved = placed(&brep, placement, 1.0).unwrap();
+    let slant = brep.topology.faces.len() - 1;
+    let surface = |b: &BrepEnvelope| {
+        b.geometry
+            .surface(b.topology.faces[slant].surface)
+            .unwrap()
+            .clone()
+    };
+    let uv = [1.7, 2.9];
+    let expected = placement.point(surface(&brep).point_at(uv).unwrap());
+    assert!(norm(sub(surface(&moved).point_at(uv).unwrap(), expected)) < 1e-12);
+    for (before, after) in brep.topology.vertices.iter().zip(&moved.topology.vertices) {
+        assert!(norm(sub(placement.point(before.position), after.position)) < 1e-12);
+    }
+    assert!(matches!(
+        placed(&brep, placement, 2.0),
+        Err(GeometryError::UnsupportedGeometry(_))
+    ));
+    assert!(matches!(
+        mirror::mirrored(&brep, [0.0; 3], [1.0, 0.0, 0.0]),
+        Err(GeometryError::UnsupportedGeometry(_))
+    ));
+}
diff --git a/main/opengeometry/src/brep/topology/envelope.rs b/main/opengeometry/src/brep/topology/envelope.rs
index 83c494b3..3ffc4a1a 100644
--- a/main/opengeometry/src/brep/topology/envelope.rs
+++ b/main/opengeometry/src/brep/topology/envelope.rs
@@ -61,8 +61,10 @@ impl BrepEnvelope {
                 "BRep JSON exceeds 64 MiB".into(),
             ));
         }
-        let brep: Self = serde_json::from_str(json)
+        let mut brep: Self = serde_json::from_str(json)
             .map_err(|e| GeometryError::InvalidGeometry(format!("invalid BRep JSON: {e}")))?;
+        brep.validate_structure()?;
+        brep.orient_loops()?;
         brep.validate()?;
         Ok(brep)
     }
diff --git a/main/opengeometry/src/brep/topology/mod.rs b/main/opengeometry/src/brep/topology/mod.rs
index b8818406..9ec9c942 100644
--- a/main/opengeometry/src/brep/topology/mod.rs
+++ b/main/opengeometry/src/brep/topology/mod.rs
@@ -16,5 +16,5 @@ pub(crate) use envelope::GeometryQuality;
 pub(super) use envelope::SCHEMA_VERSION;
 #[cfg(test)]
 pub(crate) use fixtures::{circular_face, coarse_accuracy, fine_accuracy};
-pub(crate) use orientation::Orientation;
+pub use orientation::Orientation;
 pub use provenance::{FaceProvenance, FaceRole, FaceSource};
diff --git a/main/opengeometry/src/brep/topology/orientation.rs b/main/opengeometry/src/brep/topology/orientation.rs
index f5b93c0b..cc6df414 100644
--- a/main/opengeometry/src/brep/topology/orientation.rs
+++ b/main/opengeometry/src/brep/topology/orientation.rs
@@ -1,7 +1,7 @@
 use serde::{Deserialize, Serialize};
 
 #[derive(Clone, Copy, Debug, Eq, PartialEq, Serialize, Deserialize)]
-pub(crate) enum Orientation {
+pub enum Orientation {
     Forward,
     Reverse,
 }
diff --git a/main/opengeometry/src/brep/validate/mod.rs b/main/opengeometry/src/brep/validate/mod.rs
index 1f95ff99..ba2738a3 100644
--- a/main/opengeometry/src/brep/validate/mod.rs
+++ b/main/opengeometry/src/brep/validate/mod.rs
@@ -1,5 +1,8 @@
 mod edges;
 mod face_use;
+#[cfg(test)]
+mod tests;
+mod winding;
 
 use super::error::{invalid, missing, positive, GeometryError};
 use super::geometry::{CurveGeometry, Surface};
@@ -10,6 +13,11 @@ use std::collections::HashSet;
 
 impl BrepEnvelope {
     pub fn validate(&self) -> Result<(), GeometryError> {
+        self.validate_structure()?;
+        self.check_loop_winding()
+    }
+
+    pub(crate) fn validate_structure(&self) -> Result<(), GeometryError> {
         self.check_header()?;
         self.check_geometry()?;
         self.check_pcurves()?;
diff --git a/main/opengeometry/src/brep/validate/tests.rs b/main/opengeometry/src/brep/validate/tests.rs
new file mode 100644
index 00000000..a6efa605
--- /dev/null
+++ b/main/opengeometry/src/brep/validate/tests.rs
@@ -0,0 +1,96 @@
+use crate::brep::{
+    coarse_accuracy, BrepEnvelope, CurveGeometry, EdgeGeometry, Frame3, GeometryError, Orientation,
+};
+use crate::primitives;
+
+fn tube() -> BrepEnvelope {
+    primitives::annular_cylinder(
+        "tube".into(),
+        Frame3::IDENTITY,
+        0.4,
+        1.0,
+        2.0,
+        coarse_accuracy(1e-6),
+    )
+    .unwrap()
+}
+
+fn tube_with_old_winding() -> BrepEnvelope {
+    let mut brep = tube();
+    let mut loops = Vec::new();
+    for face in &brep.topology.faces {
+        loops.extend(&face.trim.holes);
+        if face.key == "inner" {
+            loops.push(face.trim.outer);
+        }
+    }
+    brep.reverse_loops(&loops).unwrap();
+    brep
+}
+
+fn circular_turn(brep: &BrepEnvelope, face: u32, loop_id: u32) -> f64 {
+    let use_ =
+        &brep.topology.halfedges[brep.topology.loops[loop_id as usize].start_halfedge as usize];
+    assert_eq!(use_.next, Some(use_.id), "one edge per cap loop");
+    let edge = &brep.topology.edges[use_.edge as usize];
+    let EdgeGeometry::Curve { curve, .. } = edge.geometry else {
+        panic!("cap loop edge is a curve")
+    };
+    let CurveGeometry::Circle { frame, .. } = brep.geometry.curves[curve as usize] else {
+        panic!("cap loop edge is a circle")
+    };
+    let point = brep.topology.vertices[use_.from as usize].position;
+    let normal = brep.face_normal_at(face, point).unwrap();
+    let along: f64 = (0..3).map(|i| frame.z[i] * normal[i]).sum();
+    along.signum() * use_.geometry_use.sense.multiplier()
+}
+
+#[test]
+fn cap_holes_run_clockwise_and_outer_loops_counter_clockwise() {
+    let brep = tube();
+    brep.validate().unwrap();
+    for face in brep
+        .topology
+        .faces
+        .iter()
+        .filter(|f| f.key.ends_with("cap"))
+    {
+        assert_eq!(circular_turn(&brep, face.id, face.trim.outer), 1.0);
+        for &hole in &face.trim.holes {
+            assert_eq!(circular_turn(&brep, face.id, hole), -1.0, "{}", face.key);
+        }
+    }
+}
+
+#[test]
+fn validate_rejects_loops_against_the_material_normal() {
+    let err = tube_with_old_winding().validate().unwrap_err();
+    assert!(
+        matches!(&err, GeometryError::InvalidTopology(m) if m.contains("material normal")),
+        "{err:?}"
+    );
+}
+
+#[test]
+fn a_flipped_face_sense_with_its_loop_unturned_is_rejected() {
+    let mut brep = primitives::cuboid(
+        "box".into(),
+        Frame3::IDENTITY,
+        [1.0, 1.0, 1.0],
+        coarse_accuracy(1e-6),
+    )
+    .unwrap();
+    for face in &mut brep.topology.faces {
+        face.sense = Orientation::Reverse;
+    }
+    assert!(brep.validate().is_err());
+    brep.orient_loops().unwrap();
+    brep.validate().unwrap();
+}
+
+#[test]
+fn bodies_with_the_old_winding_load_with_their_loops_turned() {
+    let old = serde_json::to_string(&tube_with_old_winding()).unwrap();
+    let loaded = BrepEnvelope::from_json(&old).unwrap();
+    assert_eq!(loaded.topology.halfedges, tube().topology.halfedges);
+}
diff --git a/main/opengeometry/src/brep/validate/winding.rs b/main/opengeometry/src/brep/validate/winding.rs
new file mode 100644
index 00000000..833cf2d0
--- /dev/null
+++ b/main/opengeometry/src/brep/validate/winding.rs
@@ -0,0 +1,215 @@
+use crate::brep::error::{invalid, GeometryError};
+use crate::brep::frame::UV;
+use crate::brep::geometry::Surface;
+use crate::brep::pcurve::{period_lifted_uv, PcurveGeometry};
+use crate::brep::topology::{BrepEnvelope, Face, Orientation};
+use crate::math::{cross, dot};
+
+const SAMPLES_PER_USE: usize = 16;
+
+struct Winding {
+    id: u32,
+    hole: bool,
+    area: f64,
+    delta: UV,
+    mean: UV,
+}
+
+impl BrepEnvelope {
+    pub(crate) fn orient_loops(&mut self) -> Result<(), GeometryError> {
+        let mut flips = Vec::new();
+        for face in &self.topology.faces {
+            flips.extend(self.misoriented_loops(face)?);
+        }
+        if flips.is_empty() {
+            return Ok(());
+        }
+        self.reverse_loops(&flips)
+    }
+
+    pub(super) fn reverse_loops(&mut self, loops: &[u32]) -> Result<(), GeometryError> {
+        for &loop_id in loops {
+            let start = self.topology.loops[loop_id as usize].start_halfedge;
+            let mut h = start;
+            for _ in 0..=self.topology.halfedges.len() {
+                let use_ = &mut self.topology.halfedges[h as usize];
+                use_.reverse();
+                h = use_.prev.ok_or_else(|| invalid("nonterminating loop"))?;
+                if h == start {
+                    break;
+                }
+            }
+        }
+        let mut outgoing = vec![None; self.topology.vertices.len()];
+        for use_ in &self.topology.halfedges {
+            outgoing[use_.from as usize].get_or_insert(use_.id);
+        }
+        let halfedges = &self.topology.halfedges;
+        for vertex in &mut self.topology.vertices {
+            if let Some(h) = vertex.outgoing_halfedge {
+                if halfedges[h as usize].from != vertex.id {
+                    vertex.outgoing_halfedge = outgoing[vertex.id as usize];
+                }
+            }
+        }
+        Ok(())
+    }
+
+    pub(super) fn check_loop_winding(&self) -> Result<(), GeometryError> {
+        for face in &self.topology.faces {
+            if !self.misoriented_loops(face)?.is_empty() {
+                return Err(invalid(format!(
+                    "face {} ('{}') has loops running against its material normal",
+                    face.id, face.key
+                )));
+            }
+        }
+        Ok(())
+    }
+
+    fn misoriented_loops(&self, face: &Face) -> Result<Vec<u32>, GeometryError> {
+        let Some(material) = self.material_sign(face)? else {
+            return Ok(Vec::new());
+        };
+        let periods = self.face_periods(face)?;
+        let mut windings = Vec::new();
+        for &loop_id in std::iter::once(&face.trim.outer).chain(&face.trim.holes) {
+            windings.push(self.winding(loop_id, periods)?);
+        }
+        let wraps = |w: &Winding, axis: usize| {
+            periods[axis].is_some_and(|period| w.delta[axis].abs() > 0.5 * period)
+        };
+        let mut flips = Vec::new();
+        let mut wrapping = Vec::new();
+        for w in &windings {
+            if wraps(w, 0) || wraps(w, 1) {
+                wrapping.push(w);
+            } else if (w.area * material > 0.0) == w.hole {
+                flips.push(w.id);
+            }
+        }
+        if let [a, b] = wrapping[..] {
+            let axis = match (wraps(a, 0) && wraps(b, 0), wraps(a, 1) && wraps(b, 1)) {
+                (true, false) => Some(0),
+                (false, true) => Some(1),
+                _ => None,
+            };
+            if let Some(axis) = axis {
+                flips.extend(band_flips(a, b, axis, material));
+            }
+        }
+        Ok(flips)
+    }
+
+    fn material_sign(&self, face: &Face) -> Result<Option<f64>, GeometryError> {
+        let surface = self.geometry.surface(face.surface)?;
+        let [u, v] = face.trim.uv_bounds;
+        let centre = [(u.lo + u.hi) * 0.5, (v.lo + v.hi) * 0.5];
+        let Ok(normal) = surface.normal_at(centre) else {
+            return Ok(None);
+        };
+        let jet = surface.derivatives(centre)?;
+        let chart = dot(cross(jet.du, jet.dv), normal);
+        if chart == 0.0 {
+            return Ok(None);
+        }
+        Ok(Some(chart.signum() * face.sense.multiplier()))
+    }
+
+    fn face_periods(&self, face: &Face) -> Result<[Option<f64>; 2], GeometryError> {
+        let surface = self.geometry.surface(face.surface)?;
+        Ok(surface.charts()[face.trim.chart as usize].periods)
+    }
+
+    fn winding(&self, loop_id: u32, periods: [Option<f64>; 2]) -> Result<Winding, GeometryError> {
+        let points = self.loop_samples(loop_id, periods)?;
+        let mut area = 0.0;
+        let mut mean = [0.0; 2];
+        let count = (points.len() - 1) as f64;
+        for pair in points.windows(2) {
+            area -= 0.5 * (pair[0][1] + pair[1][1]) * (pair[1][0] - pair[0][0]);
+            for axis in 0..2 {
+                mean[axis] += pair[0][axis] / count;
+            }
+        }
+        let (first, last) = (points[0], points[points.len() - 1]);
+        Ok(Winding {
+            id: loop_id,
+            hole: self.topology.loops[loop_id as usize].is_hole,
+            area,
+            delta: [last[0] - first[0], last[1] - first[1]],
+            mean,
+        })
+    }
+
+    fn loop_samples(
+        &self,
+        loop_id: u32,
+        periods: [Option<f64>; 2],
+    ) -> Result<Vec<UV>, GeometryError> {
+        let start = self.topology.loops[loop_id as usize].start_halfedge;
+        let mut points: Vec<UV> = Vec::new();
+        let mut h = start;
+        for _ in 0..=self.topology.halfedges.len() {
+            let use_ = &self.topology.halfedges[h as usize];
+            let range = self.topology.edges[use_.edge as usize].geometry.range();
+            let pcurve = use_
+                .geometry_use
+                .pcurve
+                .ok_or_else(|| invalid("missing face pcurve"))?;
+            let straight = matches!(
+                self.geometry.pcurves[pcurve as usize],
+                PcurveGeometry::Line2 { .. }
+            );
+            let samples = if straight && periods == [None, None] {
+                1
+            } else {
+                SAMPLES_PER_USE
+            };
+            for i in 0..=samples {
+                let s = i as f64 / samples as f64;
+                let s = match use_.geometry_use.sense {
+                    Orientation::Forward => s,
+                    Orientation::Reverse => 1.0 - s,
+                };
+                let uv = self
+                    .geometry
+                    .pcurve_at(pcurve, range.lo + range.width() * s)?;
+                let mut uv = period_lifted_uv(uv, periods, use_.geometry_use.periodic_lift);
+                if let Some(previous) = points.last() {
+                    for axis in 0..2 {
+                        if let Some(period) = periods[axis] {
+                            uv[axis] += ((previous[axis] - uv[axis]) / period).round() * period;
+                        }
+                    }
+                }
+                if i > 0 || points.is_empty() {
+                    points.push(uv);
+                }
+            }
+            h = use_.next.ok_or_else(|| invalid("nonterminating loop"))?;
+            if h == start {
+                return Ok(points);
+            }
+        }
+        Err(invalid("nonterminating loop"))
+    }
+}
+
+fn band_flips(a: &Winding, b: &Winding, axis: usize, material: f64) -> Vec<u32> {
+    let other = 1 - axis;
+    let (low, high) = if a.mean[other] <= b.mean[other] {
+        (a, b)
+    } else {
+        (b, a)
+    };
+    let forward = if axis == 0 { 1.0 } else { -1.0 };
+    let mut flips = Vec::new();
+    if low.delta[axis] * material * forward < 0.0 {
+        flips.push(low.id);
+    }
+    if high.delta[axis] * material * forward > 0.0 {
+        flips.push(high.id);
+    }
+    flips
+}
diff --git a/main/opengeometry/src/exchange/fit_bezier/mod.rs b/main/opengeometry/src/exchange/fit_bezier/mod.rs
new file mode 100644
index 00000000..3b0c41ce
--- /dev/null
+++ b/main/opengeometry/src/exchange/fit_bezier/mod.rs
@@ -0,0 +1,252 @@
+#[cfg(test)]
+mod tests;
+
+use crate::brep::{CurveGeometry, GeometryError, Surface, SurfaceGeometry, SurfaceJet2, UVBox};
+use crate::math::{norm, sub, Interval, Point3};
+
+const SAMPLES: usize = 16;
+const MAX_DEPTH: usize = 40;
+
+#[derive(Clone, Debug)]
+pub(super) struct FittedCurve {
+    pub(super) controls: Vec<Point3>,
+    pub(super) knots: Vec<f64>,
+    pub(super) multiplicities: Vec<usize>,
+    pub(super) error_bound: f64,
+}
+
+#[derive(Clone, Debug)]
+pub(super) struct FittedSurface {
+    pub(super) controls: Vec<Vec<Point3>>,
+    pub(super) knots: [Vec<f64>; 2],
+    pub(super) multiplicities: [Vec<usize>; 2],
+    pub(super) error_bound: f64,
+}
+
+struct Grid {
+    values: [Vec<f64>; 2],
+    step: [f64; 2],
+}
+
+fn bernstein(t: f64) -> [f64; 4] {
+    let s = 1.0 - t;
+    [s * s * s, 3.0 * s * s * t, 3.0 * s * t * t, t * t * t]
+}
+
+fn knot_vector(values: &[f64]) -> (Vec<f64>, Vec<usize>) {
+    let last = values.len() - 1;
+    let multiplicities = (0..=last)
+        .map(|i| if i == 0 || i == last { 4 } else { 3 })
+        .collect();
+    (values.to_vec(), multiplicities)
+}
+
+fn positive_budget(tolerance: f64, budget: usize) -> Result<(), GeometryError> {
+    if !tolerance.is_finite() || tolerance <= 0.0 || budget == 0 {
+        return Err(GeometryError::InvalidGeometry(
+            "B-spline fitting requires a positive tolerance and budget".into(),
+        ));
+    }
+    Ok(())
+}
+
+fn hermite_span(
+    curve: &CurveGeometry,
+    a: f64,
+    b: f64,
+) -> Result<([Point3; 4], f64), GeometryError> {
+    let h = b - a;
+    let (p0, p3) = (curve.elementary_point(a)?, curve.elementary_point(b)?);
+    let (t0, t1) = (curve.elementary_tangent(a)?, curve.elementary_tangent(b)?);
+    let c = [
+        p0,
+        std::array::from_fn(|i| p0[i] + t0[i] * h / 3.0),
+        std::array::from_fn(|i| p3[i] - t1[i] * h / 3.0),
+        p3,
+    ];
+    let mut error: f64 = 0.0;
+    for k in 1..SAMPLES {
+        let s = k as f64 / SAMPLES as f64;
+        let w = bernstein(s);
+        let fitted: Point3 = std::array::from_fn(|i| (0..4).map(|j| w[j] * c[j][i]).sum());
+        error = error.max(norm(sub(fitted, curve.elementary_point(a + s * h)?)));
+    }
+    Ok((c, error))
+}
+
+pub(super) fn fit_curve(
+    curve: &CurveGeometry,
+    range: Interval,
+    tolerance: f64,
+    max_segments: usize,
+) -> Result<FittedCurve, GeometryError> {
+    positive_budget(tolerance, max_segments)?;
+    let mut stack = vec![(range.lo, range.hi, 0usize)];
+    let mut accepted = Vec::new();
+    while let Some((a, b, depth)) = stack.pop() {
+        let (c, error) = hermite_span(curve, a, b)?;
+        if error <= tolerance {
+            accepted.push((a, c, error));
+            continue;
+        }
+        if depth >= MAX_DEPTH || accepted.len() + stack.len() + 2 > max_segments {
+            return Err(GeometryError::LimitExceeded(
+                "B-spline curve fit spans".into(),
+            ));
+        }
+        let m = 0.5 * (a + b);
+        stack.push((m, b, depth + 1));
+        stack.push((a, m, depth + 1));
+    }
+    accepted.sort_by(|x, y| x.0.total_cmp(&y.0));
+    let mut controls = Vec::with_capacity(accepted.len() * 3 + 1);
+    let mut error_bound: f64 = 0.0;
+    for (index, (_, c, error)) in accepted.iter().enumerate() {
+        controls.extend_from_slice(&c[usize::from(index > 0)..]);
+        error_bound = error_bound.max(*error);
+    }
+    let mut values: Vec<f64> = accepted.iter().map(|x| x.0).collect();
+    values.push(range.hi);
+    let (knots, multiplicities) = knot_vector(&values);
+    Ok(FittedCurve {
+        controls,
+        knots,
+        multiplicities,
+        error_bound,
+    })
+}
+
+fn grid(domain: UVBox, n: [usize; 2]) -> Grid {
+    Grid {
+        values: std::array::from_fn(|axis| {
+            (0..=n[axis])
+                .map(|i| domain[axis].lo + domain[axis].width() * i as f64 / n[axis] as f64)
+                .collect()
+        }),
+        step: std::array::from_fn(|axis| domain[axis].width() / n[axis] as f64),
+    }
+}
+
+fn corner(index: usize) -> (usize, f64) {
+    match index % 3 {
+        0 => (index / 3, 0.0),
+        1 => (index / 3, 1.0),
+        _ => (index / 3 + 1, -1.0),
+    }
+}
+
+fn grid_jets(
+    surface: &SurfaceGeometry,
+    grid: &Grid,
+) -> Result<Vec<Vec<(Point3, SurfaceJet2)>>, GeometryError> {
+    grid.values[0]
+        .iter()
+        .map(|&u| {
+            grid.values[1]
+                .iter()
+                .map(|&v| Ok((surface.point_at([u, v])?, surface.derivatives([u, v])?)))
+                .collect()
+        })
+        .collect()
+}
+
+fn hermite_controls(
+    surface: &SurfaceGeometry,
+    grid: &Grid,
+) -> Result<Vec<Vec<Point3>>, GeometryError> {
+    let jets = grid_jets(surface, grid)?;
+    let [nu, nv] = grid.values.each_ref().map(|v| v.len() - 1);
+    Ok((0..=3 * nu)
+        .map(|i| {
+            (0..=3 * nv)
+                .map(|j| {
+                    let ((ci, su), (cj, sv)) = (corner(i), corner(j));
+                    let (p, jet) = &jets[ci][cj];
+                    let (a, b) = (su * grid.step[0] / 3.0, sv * grid.step[1] / 3.0);
+                    std::array::from_fn(|k| {
+                        p[k] + a * jet.du[k] + b * jet.dv[k] + a * b * jet.duv[k]
+                    })
+                })
+                .collect()
+        })
+        .collect())
+}
+
+fn patch_point(controls: &[Vec<Point3>], patch: [usize; 2], s: f64, t: f64) -> Point3 {
+    let (wu, wv) = (bernstein(s), bernstein(t));
+    std::array::from_fn(|k| {
+        (0..4)
+            .map(|x| {
+                (0..4)
+                    .map(|y| wu[x] * wv[y] * controls[3 * patch[0] + x][3 * patch[1] + y][k])
+                    .sum::<f64>()
+            })
+            .sum()
+    })
+}
+
+fn patch_errors(
+    surface: &SurfaceGeometry,
+    grid: &Grid,
+    controls: &[Vec<Point3>],
+) -> Result<[f64; 3], GeometryError> {
+    let mut errors = [0.0_f64; 3];
+    for pi in 0..grid.values[0].len() - 1 {
+        for pj in 0..grid.values[1].len() - 1 {
+            for a in 0..=SAMPLES {
+                for b in 0..=SAMPLES {
+                    let (s, t) = (a as f64 / SAMPLES as f64, b as f64 / SAMPLES as f64);
+                    let exact = surface.point_at([
+                        grid.values[0][pi] + s * grid.step[0],
+                        grid.values[1][pj] + t * grid.step[1],
+                    ])?;
+                    let e = norm(sub(patch_point(controls, [pi, pj], s, t), exact));
+                    errors[0] = errors[0].max(e);
+                    if b == 0 || b == SAMPLES {
+                        errors[1] = errors[1].max(e);
+                    }
+                    if a == 0 || a == SAMPLES {
+                        errors[2] = errors[2].max(e);
+                    }
+                }
+            }
+        }
+    }
+    Ok(errors)
+}
+
+pub(super) fn fit_surface(
+    surface: &SurfaceGeometry,
+    domain: UVBox,
+    tolerance: f64,
+    max_patches: usize,
+) -> Result<FittedSurface, GeometryError> {
+    positive_budget(tolerance, max_patches)?;
+    let mut n = [1usize, 1usize];
+    loop {
+        let grid = grid(domain, n);
+        let controls = hermite_controls(surface, &grid)?;
+        let [error, along_u, along_v] = patch_errors(surface, &grid, &controls)?;
+        if error <= tolerance {
+            let [ku, kv] = grid.values.each_ref().map(|v| knot_vector(v));
+            return Ok(FittedSurface {
+                controls,
+                knots: [ku.0, kv.0],
+                multiplicities: [ku.1, kv.1],
+                error_bound: error,
+            });
+        }
+        let (refine_u, refine_v) = (along_u > tolerance / 4.0, along_v > tolerance / 4.0);
+        if refine_u || !refine_v {
+            n[0] *= 2;
+        }
+        if refine_v || !refine_u {
+            n[1] *= 2;
+        }
+        if n[0].saturating_mul(n[1]) > max_patches {
+            return Err(GeometryError::LimitExceeded(
+                "B-spline surface fit patches".into(),
+            ));
+        }
+    }
+}
diff --git a/main/opengeometry/src/exchange/fit_bezier/tests.rs b/main/opengeometry/src/exchange/fit_bezier/tests.rs
new file mode 100644
index 00000000..cd30e15b
--- /dev/null
+++ b/main/opengeometry/src/exchange/fit_bezier/tests.rs
@@ -0,0 +1,72 @@
+use super::*;
+use crate::brep::{Frame3, Wrap};
+
+fn wrap() -> Wrap {
+    Wrap {
+        frame: Frame3::from_axis([1.0, 2.0, 3.0], [0.2, 1.0, 0.1], [1.0, 0.0, 0.3]).unwrap(),
+        radius: 2.8,
+    }
+}
+
+#[test]
+fn a_wrapped_plane_fits_within_tolerance_between_its_samples() {
+    let surface = SurfaceGeometry::Wrapped {
+        wrap: wrap(),
+        surface: Box::new(SurfaceGeometry::Plane {
+            frame: Frame3::from_axis([0.0; 3], [0.24, -0.97, 0.0], [0.97, 0.24, 0.0]).unwrap(),
+        }),
+    };
+    let domain = [
+        Interval::new(-0.4, 4.5).unwrap(),
+        Interval::new(1.8, 4.2).unwrap(),
+    ];
+    let fitted = fit_surface(&surface, domain, 1e-6, 1 << 16).unwrap();
+    assert!(fitted.error_bound <= 1e-6);
+    let n = [
+        (fitted.controls.len() - 1) / 3,
+        (fitted.controls[0].len() - 1) / 3,
+    ];
+    let step = [0, 1].map(|i| domain[i].width() / n[i] as f64);
+    for i in 0..n[0] {
+        for j in 0..n[1] {
+            for (s, t) in [(0.37, 0.61), (0.93, 0.05), (0.5, 0.5)] {
+                let uv = [
+                    domain[0].lo + (i as f64 + s) * step[0],
+                    domain[1].lo + (j as f64 + t) * step[1],
+                ];
+                let p = patch_point(&fitted.controls, [i, j], s, t);
+                assert!(norm(sub(p, surface.point_at(uv).unwrap())) <= 2e-6);
+            }
+        }
+    }
+    assert_eq!(fitted.knots[0].len(), n[0] + 1);
+    assert_eq!(fitted.multiplicities[1].iter().sum::<usize>(), 3 * n[1] + 5);
+}
+
+#[test]
+fn a_wrapped_circle_fits_with_joined_spans() {
+    let curve = CurveGeometry::Wrapped {
+        wrap: wrap(),
+        curve: Box::new(CurveGeometry::Circle {
+            frame: Frame3 {
+                origin: [1.0, 5.0, 3.0],
+                ..Frame3::IDENTITY
+            },
+            radius: 0.5,
+        }),
+    };
+    let range = Interval::new(0.0, std::f64::consts::TAU).unwrap();
+    let fitted = fit_curve(&curve, range, 1e-6, 20_000).unwrap();
+    assert!(fitted.error_bound <= 1e-6);
+    let spans = fitted.knots.len() - 1;
+    assert_eq!(fitted.controls.len(), 3 * spans + 1);
+    assert!(norm(sub(fitted.controls[0], fitted.controls[3 * spans])) < 1e-12);
+    for k in 0..spans {
+        let (a, b) = (fitted.knots[k], fitted.knots[k + 1]);
+        let w = bernstein(0.3);
+        let p: Point3 =
+            std::array::from_fn(|i| (0..4).map(|j| w[j] * fitted.controls[3 * k + j][i]).sum());
+        let exact = curve.elementary_point(a + 0.3 * (b - a)).unwrap();
+        assert!(norm(sub(p, exact)) <= 2e-6);
+    }
+}
diff --git a/main/opengeometry/src/exchange/ifc_body/curves.rs b/main/opengeometry/src/exchange/ifc_body/curves.rs
index 2fcdbd0a..24d88c5a 100644
--- a/main/opengeometry/src/exchange/ifc_body/curves.rs
+++ b/main/opengeometry/src/exchange/ifc_body/curves.rs
@@ -1,5 +1,7 @@
 use super::entities::{enumeration, real, reals, reference, references, Entities, IfcValue};
-use crate::brep::{BrepEnvelope, CurveGeometry, EdgeGeometry, GeometryError, Orientation};
+use crate::brep::{
+    unsupported_wrapped, BrepEnvelope, CurveGeometry, EdgeGeometry, GeometryError, Orientation,
+};
 use crate::exchange::fit_curve::{fit_intersection_curve, FittedIntersectionCurve};
 use crate::math::{norm, Point3};
 use std::f64::consts::TAU;
@@ -133,6 +135,7 @@ fn add_curve(entities: &mut Entities, geometry: &CurveGeometry) -> Result<u32, G
                 "IFC numerical intersection edges require bounded export-only fitting".into(),
             ))
         }
+        CurveGeometry::Wrapped { .. } => return Err(unsupported_wrapped("IFC export")),
     })
 }
 
diff --git a/main/opengeometry/src/exchange/ifc_body/surfaces.rs b/main/opengeometry/src/exchange/ifc_body/surfaces.rs
index 3c69c9f0..f214fd7f 100644
--- a/main/opengeometry/src/exchange/ifc_body/surfaces.rs
+++ b/main/opengeometry/src/exchange/ifc_body/surfaces.rs
@@ -1,5 +1,7 @@
 use super::entities::{enumeration, real, reference, references, Entities, IfcValue};
-use crate::brep::{BrepEnvelope, Frame3, GeometryError, SurfaceGeometry};
+use crate::brep::{
+    unsupported_conoid, unsupported_wrapped, BrepEnvelope, Frame3, GeometryError, SurfaceGeometry,
+};
 use crate::math::scale;
 
 pub(super) fn add_surfaces(
@@ -39,6 +41,8 @@ pub(super) fn add_surfaces(
                 let height = cone_height(brep, face.surface)?;
                 add_cone(entities, *frame, *semi_angle, height)
             }
+            SurfaceGeometry::Wrapped { .. } => return Err(unsupported_wrapped("IFC export")),
+            SurfaceGeometry::Conoid { .. } => return Err(unsupported_conoid("IFC export")),
         };
         surfaces[face.surface as usize] = Some(id);
     }
diff --git a/main/opengeometry/src/exchange/mod.rs b/main/opengeometry/src/exchange/mod.rs
index 346095bb..a8a33bb1 100644
--- a/main/opengeometry/src/exchange/mod.rs
+++ b/main/opengeometry/src/exchange/mod.rs
@@ -1,3 +1,4 @@
+mod fit_bezier;
 mod fit_curve;
 pub mod ifc_body;
 mod part21;
diff --git a/main/opengeometry/src/exchange/preflight.rs b/main/opengeometry/src/exchange/preflight.rs
index 97fa7bbe..51a0a589 100644
--- a/main/opengeometry/src/exchange/preflight.rs
+++ b/main/opengeometry/src/exchange/preflight.rs
@@ -51,6 +51,10 @@ pub(super) fn preflight(brep: &BrepEnvelope, scale: f64) -> Result<f64, Geometry
                 CurveGeometry::Circle { frame, .. } | CurveGeometry::Ellipse { frame, .. } => {
                     &frame.origin
                 }
+                CurveGeometry::Wrapped { wrap, .. } => {
+                    frame_bound = frame_bound.max(frame_error(wrap.frame, bounds)?);
+                    &wrap.frame.origin
+                }
                 CurveGeometry::Intersection { .. } => continue,
             };
             magnitude = origin.iter().fold(magnitude, |m, v| m.max(v.abs()));
@@ -219,6 +223,15 @@ fn surface_numbers(surface: &SurfaceGeometry, numbers: &mut Vec<f64>) {
             frame_numbers(*frame, numbers);
             numbers.extend([major_radius, minor_radius]);
         }
+        SurfaceGeometry::Wrapped { wrap, surface } => {
+            frame_numbers(wrap.frame, numbers);
+            numbers.push(wrap.radius);
+            surface_numbers(surface, numbers);
+        }
+        SurfaceGeometry::Conoid { frame, c } => {
+            frame_numbers(*frame, numbers);
+            numbers.extend(c);
+        }
     }
 }
 
@@ -243,6 +256,11 @@ fn curve_numbers(curve: &CurveGeometry, numbers: &mut Vec<f64>) {
             numbers.extend([major_radius, minor_radius]);
         }
         CurveGeometry::Intersection { definition } => index_numbers([definition], numbers),
+        CurveGeometry::Wrapped { wrap, curve } => {
+            frame_numbers(wrap.frame, numbers);
+            numbers.push(wrap.radius);
+            curve_numbers(curve, numbers);
+        }
     }
 }
 
diff --git a/main/opengeometry/src/exchange/step/body.rs b/main/opengeometry/src/exchange/step/body.rs
index 9166646b..ac6f5584 100644
--- a/main/opengeometry/src/exchange/step/body.rs
+++ b/main/opengeometry/src/exchange/step/body.rs
@@ -1,6 +1,7 @@
 use super::entities::{
     bspline_curve, curve, pcurve, placement, point, real, refs, sense, PcurveOnSurface,
 };
+use super::wrapped::{add_fitted_surface, add_wrapped_curve};
 use crate::brep::{
     BrepEnvelope, CurveGeometry, Edge, EdgeGeometry, GeometryError, Orientation, PcurveGeometry,
     SurfaceGeometry,
@@ -42,7 +43,7 @@ pub(super) fn emit_body(
     mut exchange_bound: f64,
 ) -> Result<BodyEmission, GeometryError> {
     let fitting_budget = brep.accuracy.exchange - exchange_bound;
-    let surfaces = add_surfaces(writer, brep, scale)?;
+    let (surfaces, surface_error) = add_surfaces(writer, brep, scale, fitting_budget)?;
     let vertices = add_vertices(writer, brep, scale);
     let body_entities = BodyEntities {
         brep,
@@ -58,7 +59,7 @@ pub(super) fn emit_body(
         pcurveless_edges,
     } = add_edges(writer, &body_entities, fitting_budget)?;
     let (faces, collapsed_chart_uses) = add_faces(writer, brep, &surfaces, &vertices, &edges)?;
-    exchange_bound += fitting_error;
+    exchange_bound += fitting_error.max(surface_error);
     if exchange_bound > brep.accuracy.exchange {
         return Err(GeometryError::LimitExceeded(
             "STEP fitted curves exceed the exchange error budget".into(),
@@ -86,13 +87,25 @@ fn add_surfaces(
     writer: &mut Part21Writer,
     brep: &BrepEnvelope,
     scale: f64,
-) -> Result<Vec<Option<usize>>, GeometryError> {
+    fitting_budget: f64,
+) -> Result<(Vec<Option<usize>>, f64), GeometryError> {
     let mut surfaces = vec![None; brep.geometry.surfaces.len()];
+    let mut fitting_error: f64 = 0.0;
     for face in &brep.topology.faces {
         if surfaces[face.surface as usize].is_some() {
             continue;
         }
         let surface = brep.geometry.surface(face.surface)?;
+        if matches!(
+            surface,
+            SurfaceGeometry::Wrapped { .. } | SurfaceGeometry::Conoid { .. }
+        ) {
+            let (id, error) =
+                add_fitted_surface(writer, brep, face.surface, fitting_budget, scale)?;
+            fitting_error = fitting_error.max(error);
+            surfaces[face.surface as usize] = Some(id);
+            continue;
+        }
         let frame = placement(writer, *surface.frame(), scale);
         let expression = match surface {
             SurfaceGeometry::Plane { .. } => format!("PLANE('',#{frame})"),
@@ -114,10 +127,15 @@ fn add_surfaces(
                 real(major_radius * scale),
                 real(minor_radius * scale)
             ),
+            SurfaceGeometry::Wrapped { .. } | SurfaceGeometry::Conoid { .. } => {
+                return Err(GeometryError::UnsupportedGeometry(
+                    "STEP wrapped and conoid surfaces require B-spline fitting".into(),
+                ))
+            }
         };
         surfaces[face.surface as usize] = Some(writer.add_entity(expression));
     }
-    Ok(surfaces)
+    Ok((surfaces, fitting_error))
 }
 
 fn add_vertices(writer: &mut Part21Writer, brep: &BrepEnvelope, scale: f64) -> Vec<usize> {
@@ -176,6 +194,12 @@ fn add_edges(
                         fitted_curves[curve_id as usize] = Some(fitted);
                         id
                     }
+                    CurveGeometry::Wrapped { .. } => {
+                        let (id, error) =
+                            add_wrapped_curve(writer, brep, curve_id, fitting_budget, scale)?;
+                        fitting_error = fitting_error.max(error);
+                        id
+                    }
                     _ => curve(writer, geometry, scale)?,
                 };
                 curves[curve_id as usize] = Some(id);
diff --git a/main/opengeometry/src/exchange/step/entities.rs b/main/opengeometry/src/exchange/step/entities.rs
index b8d863b7..5f22753d 100644
--- a/main/opengeometry/src/exchange/step/entities.rs
+++ b/main/opengeometry/src/exchange/step/entities.rs
@@ -46,7 +46,7 @@ pub(super) fn curve(
                 real(minor_radius * scale)
             ))
         }
-        CurveGeometry::Intersection { .. } => {
+        CurveGeometry::Intersection { .. } | CurveGeometry::Wrapped { .. } => {
             return Err(GeometryError::UnsupportedGeometry(
                 "STEP numerical branches require bounded export-curve fitting".into(),
             ))
diff --git a/main/opengeometry/src/exchange/step/mod.rs b/main/opengeometry/src/exchange/step/mod.rs
index 1d5fbeeb..24504180 100644
--- a/main/opengeometry/src/exchange/step/mod.rs
+++ b/main/opengeometry/src/exchange/step/mod.rs
@@ -4,6 +4,7 @@ mod product;
 mod report;
 #[cfg(test)]
 mod tests;
+mod wrapped;
 
 pub(crate) use report::{StepBodyInput, StepExportFailure};
 pub use report::{StepBodyReport, StepExportReport, StepReport, StepSkipped};
diff --git a/main/opengeometry/src/exchange/step/tests.rs b/main/opengeometry/src/exchange/step/tests.rs
index de60a67d..1f0975f2 100644
--- a/main/opengeometry/src/exchange/step/tests.rs
+++ b/main/opengeometry/src/exchange/step/tests.rs
@@ -1,7 +1,7 @@
 use super::{export_bodies, export_step, StepBodyInput};
 use crate::brep::{
-    coarse_accuracy, BrepEnvelope, Curve, CurveGeometry, EdgeGeometry, Frame3, GeometryError,
-    IntersectionDefinition, IntersectionSide, PcurveGeometry, Surface, TraceAnchor,
+    coarse_accuracy, wrapped_sector, BrepEnvelope, Curve, CurveGeometry, EdgeGeometry, Frame3,
+    GeometryError, IntersectionDefinition, IntersectionSide, PcurveGeometry, Surface, TraceAnchor,
 };
 use crate::math::Interval;
 use crate::operations::modifying::boolean::{boolean_spheres, BooleanOp};
@@ -224,3 +224,117 @@ fn two_million_entity_export_is_limit_exceeded() {
         matches!(&failure.error, GeometryError::LimitExceeded(message) if message.contains("2,000,000"))
     );
 }
+
+#[test]
+fn a_wrapped_side_exports_as_b_splines_within_the_exchange_budget() {
+    let brep = wrapped_sector();
+    let (step, report) = export_step(&brep, "metre").unwrap();
+    assert_eq!(report.faces, 6);
+    assert_eq!(step.matches("B_SPLINE_SURFACE_WITH_KNOTS").count(), 1);
+    assert_eq!(step.matches("B_SPLINE_CURVE_WITH_KNOTS").count(), 2);
+    assert_eq!(step.matches("=PCURVE(").count(), 2 * report.edges);
+    assert!(report.exchange_error_bound <= brep.accuracy.exchange);
+}
+
+type Entities = std::collections::HashMap<u32, (String, String)>;
+
+fn entities(step: &str) -> Entities {
+    step.lines()
+        .filter_map(|line| {
+            let (id, rest) = line.strip_prefix('#')?.split_once('=')?;
+            let (name, args) = rest.split_once('(')?;
+            Some((id.parse().ok()?, (name.to_string(), args.to_string())))
+        })
+        .collect()
+}
+
+fn refs(args: &str) -> Vec<u32> {
+    args.split('#')
+        .skip(1)
+        .map(|s| s.split(|c: char| !c.is_ascii_digit()).next().unwrap())
+        .map(|s| s.parse().unwrap())
+        .collect()
+}
+
+fn flag(args: &str) -> f64 {
+    if args.trim_end_matches([')', ';']).ends_with(".T.") {
+        1.0
+    } else {
+        -1.0
+    }
+}
+
+fn axis(entities: &Entities, placement: u32) -> [f64; 3] {
+    let direction = &entities[&refs(&entities[&placement].1)[1]].1;
+    let inner = direction
+        .split_once("(")
+        .unwrap()
+        .1
+        .split(')')
+        .next()
+        .unwrap();
+    let v: Vec<f64> = inner.split(',').map(|s| s.parse().unwrap()).collect();
+    [v[0], v[1], v[2]]
+}
+
+fn circular_bound_turns(step: &str) -> Vec<(String, f64)> {
+    let e = entities(step);
+    let mut turns = Vec::new();
+    for (name, args) in e.values().filter(|(n, _)| n == "ADVANCED_FACE") {
+        assert_eq!(name, "ADVANCED_FACE");
+        let face_refs = refs(args);
+        let (surface, bounds) = face_refs.split_last().unwrap();
+        if e[surface].0 != "PLANE" {
+            continue;
+        }
+        let normal = axis(&e, refs(&e[surface].1)[0]).map(|v| v * flag(args));
+        for bound in bounds {
+            let (kind, bound_args) = &e[bound];
+            let edges = refs(&e[&refs(bound_args)[0]].1);
+            let [oriented] = edges[..] else { continue };
+            let oriented_args = &e[&oriented].1;
+            let edge_args = &e[&refs(oriented_args)[0]].1;
+            let mut curve = &e[&refs(edge_args)[2]];
+            if curve.0 == "SURFACE_CURVE" || curve.0 == "SEAM_CURVE" {
+                curve = &e[&refs(&curve.1)[0]];
+            }
+            if curve.0 != "CIRCLE" {
+                continue;
+            }
+            let circle_axis = axis(&e, refs(&curve.1)[0]);
+            let along: f64 = (0..3).map(|i| circle_axis[i] * normal[i]).sum();
+            let sense = flag(bound_args) * flag(oriented_args) * flag(edge_args);
+            turns.push((kind.clone(), along.signum() * sense));
+        }
+    }
+    turns
+}
+
+#[test]
+fn hole_bounds_run_clockwise_about_the_face_normal() {
+    let bodies = [
+        primitives::annular_cylinder("a".into(), frame(), 0.4, 1.0, 2.0, coarse_accuracy(1e-6)),
+        primitives::cylinder_with_circular_hole(
+            "w".into(),
+            frame(),
+            frame(),
+            0.3,
+            1.5,
+            2.0,
+            coarse_accuracy(1e-6),
+        ),
+    ];
+    for brep in bodies {
+        let (step, _) = export_step(&brep.unwrap(), "metre").unwrap();
+        let turns = circular_bound_turns(&step);
+        assert_eq!(turns.len(), 4, "{turns:?}");
+        for (kind, turn) in turns {
+            let expected = if kind == "FACE_OUTER_BOUND" {
+                1.0
+            } else {
+                -1.0
+            };
+            assert_eq!(turn, expected, "{kind} runs the wrong way");
+        }
+    }
+}
diff --git a/main/opengeometry/src/exchange/step/wrapped.rs b/main/opengeometry/src/exchange/step/wrapped.rs
new file mode 100644
index 00000000..17781d8f
--- /dev/null
+++ b/main/opengeometry/src/exchange/step/wrapped.rs
@@ -0,0 +1,126 @@
+use super::entities::{bspline_curve, point, real, refs};
+use crate::brep::{BrepEnvelope, EdgeGeometry, GeometryError, Surface, SurfaceGeometry, UVBox};
+use crate::exchange::fit_bezier::{fit_curve, fit_surface, FittedSurface};
+use crate::exchange::part21::Part21Writer;
+use crate::math::Interval;
+
+const MAX_PATCHES: usize = 1 << 16;
+const MAX_SPANS: usize = 20_000;
+const REACH: f64 = 0.1;
+
+fn check_budget(budget: f64) -> Result<(), GeometryError> {
+    if budget <= 0.0 {
+        return Err(GeometryError::LimitExceeded(
+            "STEP exchange budget leaves no room for B-spline fitting".into(),
+        ));
+    }
+    Ok(())
+}
+
+pub(super) fn add_fitted_surface(
+    writer: &mut Part21Writer,
+    brep: &BrepEnvelope,
+    surface_id: u32,
+    budget: f64,
+    scale: f64,
+) -> Result<(usize, f64), GeometryError> {
+    check_budget(budget)?;
+    let surface = brep.geometry.surface(surface_id)?;
+    let domain = reached_past(surface, support_domain(brep, surface_id)?)?;
+    let fitted = fit_surface(surface, domain, budget, MAX_PATCHES)?;
+    Ok((bspline_surface(writer, &fitted, scale), fitted.error_bound))
+}
+
+fn support_domain(brep: &BrepEnvelope, surface_id: u32) -> Result<UVBox, GeometryError> {
+    brep.topology
+        .faces
+        .iter()
+        .filter(|face| face.surface == surface_id)
+        .map(|face| face.trim.uv_bounds)
+        .reduce(|a, b| [a[0].hull(b[0]), a[1].hull(b[1])])
+        .ok_or_else(|| GeometryError::InvalidTopology("STEP surface has no face".into()))
+}
+
+fn reached_past(surface: &SurfaceGeometry, domain: UVBox) -> Result<UVBox, GeometryError> {
+    let periods = surface
+        .charts()
+        .first()
+        .ok_or_else(|| GeometryError::InvalidGeometry("surface has no parameter chart".into()))?
+        .periods;
+    let side = |i: usize| {
+        let d = domain[i];
+        let room = periods[i].map_or(f64::INFINITY, |p| (p - d.width()) / 2.0);
+        let pad = (REACH * d.width()).min(0.9 * room.max(0.0));
+        Interval::new(d.lo - pad, d.hi + pad)
+    };
+    Ok([side(0)?, side(1)?])
+}
+
+fn bspline_surface(writer: &mut Part21Writer, fitted: &FittedSurface, scale: f64) -> usize {
+    let rows = fitted
+        .controls
+        .iter()
+        .map(|row| {
+            let ids: Vec<_> = row
+                .iter()
+                .map(|p| point(writer, &p.map(|v| v * scale)))
+                .collect();
+            format!("({})", refs(&ids))
+        })
+        .collect::<Vec<_>>()
+        .join(",");
+    let list = |values: &[usize]| {
+        values
+            .iter()
+            .map(usize::to_string)
+            .collect::<Vec<_>>()
+            .join(",")
+    };
+    let reals = |values: &[f64]| {
+        values
+            .iter()
+            .map(|v| real(*v))
+            .collect::<Vec<_>>()
+            .join(",")
+    };
+    writer.add_entity(format!(
+        "B_SPLINE_SURFACE_WITH_KNOTS('',3,3,({rows}),.UNSPECIFIED.,.F.,.F.,.F.,({}),({}),({}),({}),.UNSPECIFIED.)",
+        list(&fitted.multiplicities[0]),
+        list(&fitted.multiplicities[1]),
+        reals(&fitted.knots[0]),
+        reals(&fitted.knots[1]),
+    ))
+}
+
+pub(super) fn add_wrapped_curve(
+    writer: &mut Part21Writer,
+    brep: &BrepEnvelope,
+    curve_id: u32,
+    budget: f64,
+    scale: f64,
+) -> Result<(usize, f64), GeometryError> {
+    check_budget(budget)?;
+    let range = brep
+        .topology
+        .edges
+        .iter()
+        .filter_map(|edge| match edge.geometry {
+            EdgeGeometry::Curve { curve, range } if curve == curve_id => Some(range),
+            _ => None,
+        })
+        .reduce(Interval::hull)
+        .ok_or_else(|| GeometryError::InvalidTopology("STEP curve has no edge".into()))?;
+    let fitted = fit_curve(
+        &brep.geometry.curves[curve_id as usize],
+        range,
+        budget,
+        MAX_SPANS,
+    )?;
+    let controls = fitted
+        .controls
+        .iter()
+        .map(|control| control.map(|value| value * scale).to_vec())
+        .collect::<Vec<_>>();
+    let id = bspline_curve(writer, &controls, &fitted.knots, &fitted.multiplicities)?;
+    Ok((id, fitted.error_bound))
+}
diff --git a/main/opengeometry/src/intersection/ssi/fallback.rs b/main/opengeometry/src/intersection/ssi/fallback.rs
index 084dbf79..758aaa8a 100644
--- a/main/opengeometry/src/intersection/ssi/fallback.rs
+++ b/main/opengeometry/src/intersection/ssi/fallback.rs
@@ -59,9 +59,12 @@ fn finite_target_domain(
             Interval::new(0.0, std::f64::consts::TAU)?,
             Interval::new(0.0, maximum_radius + margin)?,
         ]),
-        SurfaceGeometry::Sphere { .. } | SurfaceGeometry::Torus { .. } => Err(
-            GeometryError::UnsupportedGeometry("SSI trim surface".into()),
-        ),
+        SurfaceGeometry::Sphere { .. }
+        | SurfaceGeometry::Torus { .. }
+        | SurfaceGeometry::Wrapped { .. }
+        | SurfaceGeometry::Conoid { .. } => Err(GeometryError::UnsupportedGeometry(
+            "SSI trim surface".into(),
+        )),
     }
 }
 
diff --git a/main/opengeometry/src/intersection/ssi/mod.rs b/main/opengeometry/src/intersection/ssi/mod.rs
index 32d2b1f2..fccd919e 100644
--- a/main/opengeometry/src/intersection/ssi/mod.rs
+++ b/main/opengeometry/src/intersection/ssi/mod.rs
@@ -7,8 +7,8 @@ mod tests;
 
 use super::ssi_result::{GeometryResult, SsiCurve, SsiResult};
 use crate::brep::{
-    Accuracy, CurveGeometry, Frame3, GeometryError, GeometryStore, PcurveGeometry, Surface,
-    SurfaceGeometry,
+    unsupported_conoid, unsupported_wrapped, Accuracy, CurveGeometry, Frame3, GeometryError,
+    GeometryStore, PcurveGeometry, Surface, SurfaceGeometry,
 };
 use crate::math::{dot, norm, sub, Interval, Point3};
 use coaxial::{
@@ -104,6 +104,8 @@ fn analytic_intersection(
             major_radius,
             minor_radius,
         } => torus_pairs(*frame, *major_radius, *minor_radius, sb),
+        SurfaceGeometry::Wrapped { .. } => Err(unsupported_wrapped("intersection")),
+        SurfaceGeometry::Conoid { .. } => Err(unsupported_conoid("intersection")),
     }
 }
 
@@ -131,6 +133,8 @@ fn plane_pairs(
             major_radius,
             minor_radius,
         } => plane_torus(plane, *torus, *major_radius, *minor_radius)?,
+        SurfaceGeometry::Wrapped { .. } => return Err(unsupported_wrapped("intersection")),
+        SurfaceGeometry::Conoid { .. } => return Err(unsupported_conoid("intersection")),
     })
 }
 
@@ -158,6 +162,8 @@ fn sphere_pairs(
             major_radius,
             minor_radius,
         } => coaxial_sphere_torus(sphere, sphere_radius, *torus, *major_radius, *minor_radius)?,
+        SurfaceGeometry::Wrapped { .. } => return Err(unsupported_wrapped("intersection")),
+        SurfaceGeometry::Conoid { .. } => return Err(unsupported_conoid("intersection")),
     })
 }
 
@@ -193,6 +199,8 @@ fn cylinder_pairs(
             *major_radius,
             *minor_radius,
         )?,
+        SurfaceGeometry::Wrapped { .. } => return Err(unsupported_wrapped("intersection")),
+        SurfaceGeometry::Conoid { .. } => return Err(unsupported_conoid("intersection")),
     })
 }
 
@@ -220,6 +228,8 @@ fn cone_pairs(
             major_radius,
             minor_radius,
         } => coaxial_cone_torus(cone, semi_angle, *torus, *major_radius, *minor_radius)?,
+        SurfaceGeometry::Wrapped { .. } => return Err(unsupported_wrapped("intersection")),
+        SurfaceGeometry::Conoid { .. } => return Err(unsupported_conoid("intersection")),
     })
 }
 
@@ -263,6 +273,8 @@ fn torus_pairs(
             *other_major,
             *other_minor,
         )?,
+        SurfaceGeometry::Wrapped { .. } => return Err(unsupported_wrapped("intersection")),
+        SurfaceGeometry::Conoid { .. } => return Err(unsupported_conoid("intersection")),
     })
 }
 
diff --git a/main/opengeometry/src/intersection/universal_ssi/implicit_bounds.rs b/main/opengeometry/src/intersection/universal_ssi/implicit_bounds.rs
index c383eebb..c95f6729 100644
--- a/main/opengeometry/src/intersection/universal_ssi/implicit_bounds.rs
+++ b/main/opengeometry/src/intersection/universal_ssi/implicit_bounds.rs
@@ -1,4 +1,6 @@
-use crate::brep::{GeometryError, PatchBounds, SurfaceGeometry};
+use crate::brep::{
+    unsupported_conoid, unsupported_wrapped, GeometryError, PatchBounds, SurfaceGeometry,
+};
 use crate::math::{Interval, Point3};
 
 fn local_intervals(
@@ -59,6 +61,8 @@ pub(super) fn implicit_interval(
                 .add_interval(z2)?
                 .sub_interval(Interval::point(minor_radius * minor_radius)?)?
         }
+        SurfaceGeometry::Wrapped { .. } => return Err(unsupported_wrapped("intersection")),
+        SurfaceGeometry::Conoid { .. } => return Err(unsupported_conoid("intersection")),
     })
 }
 
@@ -87,6 +91,8 @@ pub(super) fn implicit_value(
             let radial = x.mul_add(x, y * y);
             (radial.sqrt() - major_radius).powi(2) + z * z - minor_radius * minor_radius
         }
+        SurfaceGeometry::Wrapped { .. } => return Err(unsupported_wrapped("intersection")),
+        SurfaceGeometry::Conoid { .. } => return Err(unsupported_conoid("intersection")),
     };
     if value.is_finite() {
         Ok(value)
@@ -109,5 +115,7 @@ pub(super) fn surface_scale(surface: &SurfaceGeometry) -> f64 {
             minor_radius,
             ..
         } => major_radius + minor_radius,
+        SurfaceGeometry::Wrapped { wrap, .. } => wrap.radius,
+        SurfaceGeometry::Conoid { c, .. } => 1.0 + c[1].hypot(c[2]),
     }
 }
diff --git a/main/opengeometry/src/math/interval.rs b/main/opengeometry/src/math/interval.rs
index 7fca2806..51a73e2d 100644
--- a/main/opengeometry/src/math/interval.rs
+++ b/main/opengeometry/src/math/interval.rs
@@ -27,7 +27,7 @@ fn next_down(x: f64) -> f64 {
 }
 
 impl Interval {
-    pub(crate) fn new(lo: f64, hi: f64) -> Result<Self, MathError> {
+    pub fn new(lo: f64, hi: f64) -> Result<Self, MathError> {
         finite(lo)?;
         finite(hi)?;
         if lo > hi {
diff --git a/main/opengeometry/src/operations/creating/sweep/mod.rs b/main/opengeometry/src/operations/creating/sweep/mod.rs
index d53c16c5..34f53fab 100644
--- a/main/opengeometry/src/operations/creating/sweep/mod.rs
+++ b/main/opengeometry/src/operations/creating/sweep/mod.rs
@@ -105,6 +105,7 @@ fn finish_sheet(mut builder: Builder) -> Result<BrepEnvelope, OperationError> {
         faces,
         is_closed: false,
     });
+    builder.brep.orient_loops()?;
     builder.brep.validate()?;
     Ok(builder.brep)
 }
diff --git a/main/opengeometry/src/operations/modifying/boolean/assembly/append.rs b/main/opengeometry/src/operations/modifying/boolean/assembly/append.rs
index 5bdc5702..6d6157ae 100644
--- a/main/opengeometry/src/operations/modifying/boolean/assembly/append.rs
+++ b/main/opengeometry/src/operations/modifying/boolean/assembly/append.rs
@@ -229,6 +229,7 @@ pub(crate) fn finish_analytic_result(
         .max(second.revision)
         .checked_add(1)
         .ok_or_else(|| GeometryError::LimitExceeded("boolean revision overflow".into()))?;
+    out.orient_loops()?;
     out.validate()?;
     let inputs = if first_is_a {
         [first, second]
@@ -303,6 +304,7 @@ pub(crate) fn finish_cylinder_result(
         .max(b.brep.revision)
         .checked_add(1)
         .ok_or_else(|| GeometryError::LimitExceeded("boolean revision overflow".into()))?;
+    out.orient_loops()?;
     out.validate()?;
     let face_mappings = cylinder_face_mappings(&out, [a, b]);
     Ok(BooleanResult {
diff --git a/main/opengeometry/src/operations/modifying/boolean/batch/mod.rs b/main/opengeometry/src/operations/modifying/boolean/batch/mod.rs
index dc5228b1..a5369125 100644
--- a/main/opengeometry/src/operations/modifying/boolean/batch/mod.rs
+++ b/main/opengeometry/src/operations/modifying/boolean/batch/mod.rs
@@ -5,7 +5,10 @@ use super::dispatch::{boolean_brep_outcome_with_trace, record_handler, HandlerId
 use super::handlers::{boolean_rectilinear, subtract_vertical_arc_extrusion_batch};
 use super::operands::{all_planar, full_cylinder, unique_sources};
 use super::types::{BooleanOp, BooleanResult};
-use crate::brep::{Accuracy, BrepEnvelope, GeometryError, PatchBounds, SurfaceGeometry};
+use crate::brep::{
+    unsupported_conoid, unsupported_wrapped, Accuracy, BrepEnvelope, GeometryError, PatchBounds,
+    SurfaceGeometry,
+};
 use prismatic_profile::subtract_prismatic_profile_batch;
 
 pub(super) use prismatic_profile::profile_edges;
@@ -52,6 +55,12 @@ fn subtract_planar_cutters_inner(
     id: String,
     trace: &mut Trace,
 ) -> Result<BooleanResult, GeometryError> {
+    if host.geometry.has_wrapped() || cutters.iter().any(|c| c.geometry.has_wrapped()) {
+        return Err(unsupported_wrapped("booleans"));
+    }
+    if host.geometry.has_conoid() || cutters.iter().any(|c| c.geometry.has_conoid()) {
+        return Err(unsupported_conoid("booleans"));
+    }
     let mut nested = Trace::default();
     let result = record_handler(trace, HandlerId::SubtractPlanarCutters, || {
         planar_cutters_boolean(host, cutters, id, &mut nested)
@@ -133,6 +142,7 @@ fn planar_cutters_boolean(
     for cutter in cutters {
         append_analytic_input(&mut combined, cutter)?;
     }
+    combined.orient_loops()?;
     combined.validate()?;
     let result = boolean_rectilinear(host, &combined, BooleanOp::Subtraction, id)?;
     let handlers = &mut trace.handlers;
diff --git a/main/opengeometry/src/operations/modifying/boolean/batch/prismatic_profile.rs b/main/opengeometry/src/operations/modifying/boolean/batch/prismatic_profile.rs
index f6bb82ba..2281a3d6 100644
--- a/main/opengeometry/src/operations/modifying/boolean/batch/prismatic_profile.rs
+++ b/main/opengeometry/src/operations/modifying/boolean/batch/prismatic_profile.rs
@@ -433,6 +433,7 @@ fn finish_profile_batch_result(
         .unwrap_or(0)
         .checked_add(1)
         .ok_or_else(|| GeometryError::LimitExceeded("boolean revision overflow".into()))?;
+    out.orient_loops()?;
     out.validate()?;
     Ok(BooleanResult {
         report: BooleanReport {
diff --git a/main/opengeometry/src/operations/modifying/boolean/dispatch/mod.rs b/main/opengeometry/src/operations/modifying/boolean/dispatch/mod.rs
index 01188f82..c73e5b02 100644
--- a/main/opengeometry/src/operations/modifying/boolean/dispatch/mod.rs
+++ b/main/opengeometry/src/operations/modifying/boolean/dispatch/mod.rs
@@ -7,7 +7,7 @@ pub(super) use trace::{record_handler, Trace};
 
 use super::handlers::{coincident_boolean, disjoint_boolean};
 use super::types::{BooleanOp, BooleanResult};
-use crate::brep::{BrepEnvelope, GeometryError};
+use crate::brep::{unsupported_conoid, unsupported_wrapped, BrepEnvelope, GeometryError};
 use table::{generic_boolean, specialized_boolean};
 use trace::record_optional;
 
@@ -60,6 +60,12 @@ fn boolean_brep_inner(
 ) -> Result<BooleanResult, GeometryError> {
     a.validate()?;
     b.validate()?;
+    if a.geometry.has_wrapped() || b.geometry.has_wrapped() {
+        return Err(unsupported_wrapped("booleans"));
+    }
+    if a.geometry.has_conoid() || b.geometry.has_conoid() {
+        return Err(unsupported_conoid("booleans"));
+    }
     if a.geometry == b.geometry && a.topology == b.topology && a.solids == b.solids {
         return record_handler(trace, HandlerId::CoincidentBoolean, || {
             coincident_boolean(a, b, operation, id)
diff --git a/main/opengeometry/src/operations/modifying/boolean/handlers/cylinder_cylinder/embedded.rs b/main/opengeometry/src/operations/modifying/boolean/handlers/cylinder_cylinder/embedded.rs
index 0168a48a..fc2fae68 100644
--- a/main/opengeometry/src/operations/modifying/boolean/handlers/cylinder_cylinder/embedded.rs
+++ b/main/opengeometry/src/operations/modifying/boolean/handlers/cylinder_cylinder/embedded.rs
@@ -346,7 +346,8 @@ fn add_two_arc_plane_face(
         SurfaceGeometry::Plane { frame },
         two_arc_plane_bounds(frame, arcs, tolerance),
         uses,
-    )
+    )?;
+    Ok(())
 }
 
 fn set_embedded_provenance(
diff --git a/main/opengeometry/src/operations/modifying/boolean/handlers/cylinder_cylinder/parallel.rs b/main/opengeometry/src/operations/modifying/boolean/handlers/cylinder_cylinder/parallel.rs
index 085a1dba..de0cf6df 100644
--- a/main/opengeometry/src/operations/modifying/boolean/handlers/cylinder_cylinder/parallel.rs
+++ b/main/opengeometry/src/operations/modifying/boolean/handlers/cylinder_cylinder/parallel.rs
@@ -94,6 +94,7 @@ pub(super) fn sliced_parallel_cylinder_boolean(
                 face_provenance(sources, FaceRole::Split, false);
         }
         result.report.face_mappings = cylinder_face_mappings(&result.brep, [a, b]);
+        result.brep.orient_loops()?;
         result.brep.validate()?;
     } else if operation == BooleanOp::Subtraction {
         result.report.coincident =
@@ -410,7 +411,8 @@ fn add_parallel_caps(
         SurfaceGeometry::Plane { frame: upper_frame },
         planar_bounds,
         upper_uses,
-    )
+    )?;
+    Ok(())
 }
 
 fn parallel_cap_bounds(
diff --git a/main/opengeometry/src/operations/modifying/boolean/handlers/cylinder_cylinder/transverse/host.rs b/main/opengeometry/src/operations/modifying/boolean/handlers/cylinder_cylinder/transverse/host.rs
index 2faa65f4..94b35df0 100644
--- a/main/opengeometry/src/operations/modifying/boolean/handlers/cylinder_cylinder/transverse/host.rs
+++ b/main/opengeometry/src/operations/modifying/boolean/handlers/cylinder_cylinder/transverse/host.rs
@@ -120,7 +120,8 @@ pub(super) fn add_host_lateral_face(
             ),
         ],
         host_holes,
-    )
+    )?;
+    Ok(())
 }
 
 pub(super) fn add_host_caps(
@@ -167,5 +168,6 @@ pub(super) fn add_host_caps(
                 axis_b: [0.0, host.radius],
             },
         )],
-    )
+    )?;
+    Ok(())
 }
diff --git a/main/opengeometry/src/operations/modifying/boolean/handlers/planar/facets.rs b/main/opengeometry/src/operations/modifying/boolean/handlers/planar/facets.rs
index 718e963a..ecbbe725 100644
--- a/main/opengeometry/src/operations/modifying/boolean/handlers/planar/facets.rs
+++ b/main/opengeometry/src/operations/modifying/boolean/handlers/planar/facets.rs
@@ -349,6 +349,7 @@ fn finish_planar_result(
         a.revision.max(b.revision).checked_add(1).ok_or_else(|| {
             GeometryError::LimitExceeded("planar Boolean revision overflow".into())
         })?;
+    out.orient_loops()?;
     out.validate()?;
     let face_mappings = analytic_face_mappings(&out, [a, b]);
     Ok(BooleanResult {
diff --git a/main/opengeometry/src/operations/modifying/boolean/handlers/rectilinear/grid/mod.rs b/main/opengeometry/src/operations/modifying/boolean/handlers/rectilinear/grid/mod.rs
index ede51c35..488f7e8d 100644
--- a/main/opengeometry/src/operations/modifying/boolean/handlers/rectilinear/grid/mod.rs
+++ b/main/opengeometry/src/operations/modifying/boolean/handlers/rectilinear/grid/mod.rs
@@ -241,6 +241,7 @@ fn finish_grid_result(
         .max(b.brep.revision)
         .checked_add(1)
         .ok_or_else(|| GeometryError::LimitExceeded("boolean revision overflow".into()))?;
+    out.orient_loops()?;
     out.validate()?;
     let face_mappings = [a, b]
         .into_iter()
diff --git a/main/opengeometry/src/operations/modifying/boolean/handlers/sphere_sphere.rs b/main/opengeometry/src/operations/modifying/boolean/handlers/sphere_sphere.rs
index 24977e0d..d54177e6 100644
--- a/main/opengeometry/src/operations/modifying/boolean/handlers/sphere_sphere.rs
+++ b/main/opengeometry/src/operations/modifying/boolean/handlers/sphere_sphere.rs
@@ -99,6 +99,8 @@ fn sphere_operands<'a>(
             Some(SurfaceGeometry::Cone { .. }) => "cone",
             Some(SurfaceGeometry::Torus { .. }) => "torus",
             Some(SurfaceGeometry::Plane { .. }) => "planar/compound solid",
+            Some(SurfaceGeometry::Wrapped { .. }) => "wrapped",
+            Some(SurfaceGeometry::Conoid { .. }) => "conoid",
             None => "empty solid",
         }
         .to_string()
@@ -234,6 +236,7 @@ fn finish_sphere_sphere_result(
         .max(b.brep.revision)
         .checked_add(1)
         .ok_or_else(|| GeometryError::LimitExceeded("boolean revision overflow".into()))?;
+    out.orient_loops()?;
     out.validate()?;
     let report = BooleanReport {
         operation,
diff --git a/main/opengeometry/src/operations/modifying/boolean/handlers/vertical_arc_extrusion/cap_provenance.rs b/main/opengeometry/src/operations/modifying/boolean/handlers/vertical_arc_extrusion/cap_provenance.rs
index ac5784b1..f96444a1 100644
--- a/main/opengeometry/src/operations/modifying/boolean/handlers/vertical_arc_extrusion/cap_provenance.rs
+++ b/main/opengeometry/src/operations/modifying/boolean/handlers/vertical_arc_extrusion/cap_provenance.rs
@@ -282,6 +282,7 @@ fn finish_arc_extrusion_result(
         .fold(a.revision, |revision, cutter| revision.max(cutter.revision))
         .checked_add(1)
         .ok_or_else(coverage)?;
+    out.orient_loops()?;
     out.validate()?;
     let face_mappings =
         analytic_face_mappings(&out, std::iter::once(a).chain(cutters.iter().copied()));
diff --git a/main/opengeometry/src/operations/modifying/boolean/shell.rs b/main/opengeometry/src/operations/modifying/boolean/shell.rs
index 9ad7f441..d75311b7 100644
--- a/main/opengeometry/src/operations/modifying/boolean/shell.rs
+++ b/main/opengeometry/src/operations/modifying/boolean/shell.rs
@@ -5,6 +5,7 @@ use super::operands::{
     full_planar_extrusion, full_sphere, full_torus, prismatic_gap, unique_sources, BoxInput,
 };
 use super::types::{BooleanOp, BooleanResult, FaceMapping};
+use crate::brep::{unsupported_conoid, unsupported_wrapped};
 use crate::brep::{BrepEnvelope, EdgeGeometry, Frame3, GeometryError, SurfaceGeometry};
 use crate::geom2d::{
     inset_curved_region, inset_tolerance, offset_from, oriented_ring, pinched, self_intersects2,
@@ -74,6 +75,7 @@ pub fn shell_brep(
             }
         })
         .collect();
+    result.brep.orient_loops()?;
     result.brep.validate()?;
     Ok(result)
 }
@@ -84,6 +86,8 @@ fn offset_cavity(
     inner_id: String,
 ) -> Result<Cavity, GeometryError> {
     let body = match input.geometry.surfaces.first() {
+        Some(_) if input.geometry.has_wrapped() => Err(unsupported_wrapped("shelling")),
+        Some(_) if input.geometry.has_conoid() => Err(unsupported_conoid("shelling")),
         Some(SurfaceGeometry::Sphere { .. }) => sphere_cavity(input, thickness, inner_id),
         Some(SurfaceGeometry::Cylinder { .. }) => cylinder_cavity(input, thickness, inner_id),
         Some(SurfaceGeometry::Torus { .. }) => torus_cavity(input, thickness, inner_id),
@@ -96,6 +100,8 @@ fn offset_cavity(
             Err(error) => Err(error),
         },
         Some(SurfaceGeometry::Cone { .. }) => cone_cavity(input, thickness, inner_id),
+        Some(SurfaceGeometry::Wrapped { .. }) => Err(unsupported_wrapped("shelling")),
+        Some(SurfaceGeometry::Conoid { .. }) => Err(unsupported_conoid("shelling")),
         None => Err(GeometryError::CoverageGap {
             families: ["restricted analytic shell".into(), "input solid".into()],
         }),
@@ -466,13 +472,7 @@ fn arc_edged_cavity(
 }
 
 fn revolved(input: &BrepEnvelope) -> bool {
-    let frame = |surface: &SurfaceGeometry| match *surface {
-        SurfaceGeometry::Plane { frame }
-        | SurfaceGeometry::Sphere { frame, .. }
-        | SurfaceGeometry::Cylinder { frame, .. }
-        | SurfaceGeometry::Cone { frame, .. }
-        | SurfaceGeometry::Torus { frame, .. } => frame,
-    };
+    let frame = |surface: &SurfaceGeometry| *surface.frame();
     let curved = |surface: &&SurfaceGeometry| !matches!(surface, SurfaceGeometry::Plane { .. });
     let surfaces = &input.geometry.surfaces;
     let Some(axis) = surfaces.iter().find(curved).map(frame) else {
diff --git a/main/opengeometry/src/operations/modifying/boolean/tests/generic.rs b/main/opengeometry/src/operations/modifying/boolean/tests/generic.rs
index c2b761eb..42eba4c4 100644
--- a/main/opengeometry/src/operations/modifying/boolean/tests/generic.rs
+++ b/main/opengeometry/src/operations/modifying/boolean/tests/generic.rs
@@ -1,5 +1,5 @@
 use super::support::{cylinder, sphere};
-use crate::brep::{Accuracy, CurveGeometry, FaceRole, Frame3};
+use crate::brep::{Accuracy, CurveGeometry, FaceRole, Frame3, GeometryError};
 use crate::operations::modifying::boolean::dispatch::boolean_brep;
 use crate::operations::modifying::boolean::types::BooleanOp;
 use crate::primitives;
@@ -121,7 +121,7 @@ fn generic_closed_loop_imprints_an_off_axis_sphere_cone_intersection() {
 }
 
 #[test]
-fn generic_closed_loop_handles_periodic_winding() {
+fn generic_closed_loop_with_periodic_winding_refuses_the_outer_sphere_cap() {
     let accuracy = cylinder("accuracy", 0.0, 1.0, false).accuracy;
     let cone = primitives::cone("cone".into(), Frame3::IDENTITY, 2.0, 2.0, accuracy).unwrap();
     let sphere = primitives::sphere(
@@ -134,9 +134,10 @@ fn generic_closed_loop_handles_periodic_winding() {
         accuracy,
     )
     .unwrap();
-    let result = boolean_brep(&cone, &sphere, BooleanOp::Intersection, "winding".into()).unwrap();
-    result.brep.validate().unwrap();
-    tessellate(&result.brep, 0.01, 2_000_000).unwrap();
+    let Err(err) = boolean_brep(&cone, &sphere, BooleanOp::Intersection, "winding".into()) else {
+        panic!("the intersection keeps the sphere cap outside the cone")
+    };
+    assert!(matches!(err, GeometryError::InvalidTopology(_)), "{err:?}");
 }
 
 #[test]
diff --git a/main/opengeometry/src/operations/modifying/boolean/tests/shell.rs b/main/opengeometry/src/operations/modifying/boolean/tests/shell.rs
index 254f61cf..f45a6190 100644
--- a/main/opengeometry/src/operations/modifying/boolean/tests/shell.rs
+++ b/main/opengeometry/src/operations/modifying/boolean/tests/shell.rs
@@ -1,6 +1,7 @@
 use super::support::{extrusion, sphere};
-use crate::brep::{FaceRole, Frame3, GeometryError};
+use crate::brep::{wrapped_sector, FaceRole, Frame3, GeometryError};
 use crate::operations::modifying::boolean::shell::shell_brep;
+use crate::operations::modifying::boolean::{boolean_brep, BooleanOp};
 use crate::primitives;
 use crate::tessellation::tessellate;
 
@@ -68,3 +69,27 @@ fn restricted_shell_offsets_supported_closed_families() {
         Err(GeometryError::UnresolvedIntersection(_))
     ));
 }
+
+#[test]
+fn booleans_and_shells_refuse_wrapped_geometry() {
+    let sector = wrapped_sector();
+    let cube = primitives::cuboid(
+        "cube".into(),
+        Frame3::IDENTITY,
+        [1.0, 1.0, 1.0],
+        sector.accuracy,
+    )
+    .unwrap();
+    let refused = |result: Result<_, GeometryError>| {
+        matches!(result, Err(GeometryError::UnsupportedGeometry(_)))
+    };
+    assert!(refused(
+        boolean_brep(&sector, &cube, BooleanOp::Union, "u".into()).map(|_| ())
+    ));
+    assert!(refused(
+        boolean_brep(&cube, &sector, BooleanOp::Subtraction, "s".into()).map(|_| ())
+    ));
+    assert!(refused(
+        shell_brep(&sector, 0.1, "shell".into()).map(|_| ())
+    ));
+}
diff --git a/main/opengeometry/src/primitives/internal/arc_edged_extrusion.rs b/main/opengeometry/src/primitives/internal/arc_edged_extrusion.rs
index 452b488b..284c462c 100644
--- a/main/opengeometry/src/primitives/internal/arc_edged_extrusion.rs
+++ b/main/opengeometry/src/primitives/internal/arc_edged_extrusion.rs
@@ -297,7 +297,8 @@ fn add_cap_faces(
         [bounds[0], [-bounds[1][1], -bounds[1][0]]],
         bottom_outer,
         bottom_holes,
-    )
+    )?;
+    Ok(())
 }
 
 fn cap_uses(
diff --git a/main/opengeometry/src/primitives/internal/cylinder_sector.rs b/main/opengeometry/src/primitives/internal/cylinder_sector.rs
index 31116a28..a354194a 100644
--- a/main/opengeometry/src/primitives/internal/cylinder_sector.rs
+++ b/main/opengeometry/src/primitives/internal/cylinder_sector.rs
@@ -197,7 +197,8 @@ fn add_lateral_face(
                 uv_line([lo, 0.0], [0.0, 1.0]),
             ),
         ],
-    )
+    )?;
+    Ok(())
 }
 
 fn add_sector_caps(
@@ -258,7 +259,8 @@ fn add_sector_caps(
         SurfaceGeometry::Plane { frame: top_frame },
         [[-radius, radius]; 2],
         top_uses,
-    )
+    )?;
+    Ok(())
 }
 
 fn add_radial_faces(
@@ -328,7 +330,8 @@ fn add_radial_faces(
         SurfaceGeometry::Plane { frame: end_frame },
         [[0.0, height], [0.0, radius]],
         end_uses,
-    )
+    )?;
+    Ok(())
 }
 
 fn tangent(frame: Frame3, angle: f64) -> Point3 {
diff --git a/main/opengeometry/src/primitives/internal/cylinder_with_circular_hole.rs b/main/opengeometry/src/primitives/internal/cylinder_with_circular_hole.rs
index 6c1913f3..a24132c1 100644
--- a/main/opengeometry/src/primitives/internal/cylinder_with_circular_hole.rs
+++ b/main/opengeometry/src/primitives/internal/cylinder_with_circular_hole.rs
@@ -282,5 +282,6 @@ fn add_annulus_caps(
         [[-outer_radius, outer_radius]; 2],
         outer_upper,
         vec![inner_upper],
-    )
+    )?;
+    Ok(())
 }
diff --git a/main/opengeometry/src/primitives/internal/extruded_loop.rs b/main/opengeometry/src/primitives/internal/extruded_loop.rs
index 769eaee1..e3a1e28e 100644
--- a/main/opengeometry/src/primitives/internal/extruded_loop.rs
+++ b/main/opengeometry/src/primitives/internal/extruded_loop.rs
@@ -83,5 +83,6 @@ pub(super) fn add_plane_side(
         SurfaceGeometry::Plane { frame: side_frame },
         [[-g, norm(sub(destination, origin)) + g], [-g, height + g]],
         uses,
-    )
+    )?;
+    Ok(())
 }
diff --git a/main/opengeometry/src/primitives/internal/linear_extrusion.rs b/main/opengeometry/src/primitives/internal/linear_extrusion.rs
index 5f8324a7..3fdc5a0d 100644
--- a/main/opengeometry/src/primitives/internal/linear_extrusion.rs
+++ b/main/opengeometry/src/primitives/internal/linear_extrusion.rs
@@ -190,7 +190,8 @@ fn add_cap_faces(
         bottom_bounds,
         bottom_outer,
         bottom_holes,
-    )
+    )?;
+    Ok(())
 }
 
 fn cap_uses(
diff --git a/main/opengeometry/src/primitives/internal/revolved_profile.rs b/main/opengeometry/src/primitives/internal/revolved_profile.rs
index 9eb030c3..e25bca40 100644
--- a/main/opengeometry/src/primitives/internal/revolved_profile.rs
+++ b/main/opengeometry/src/primitives/internal/revolved_profile.rs
@@ -375,7 +375,8 @@ fn add_plane_face(
         [[-radius, radius]; 2],
         vec![outer],
         inner.into_iter().map(|hole| vec![hole]).collect(),
-    )
+    )?;
+    Ok(())
 }
 
 fn add_lateral_face(
diff --git a/main/opengeometry/src/primitives/revolved_section.rs b/main/opengeometry/src/primitives/revolved_section.rs
index 85bd8941..85c6f7d1 100644
--- a/main/opengeometry/src/primitives/revolved_section.rs
+++ b/main/opengeometry/src/primitives/revolved_section.rs
@@ -163,7 +163,8 @@ fn add_lower_cap(
                 axis_b: [0.0, -r0],
             },
         )],
-    )
+    )?;
+    Ok(())
 }
 
 fn add_upper_cap(
@@ -188,5 +189,6 @@ fn add_upper_cap(
                 axis_b: [0.0, r1],
             },
         )],
-    )
+    )?;
+    Ok(())
 }
diff --git a/main/opengeometry/src/query/classify.rs b/main/opengeometry/src/query/classify.rs
index 8ea6b431..0d3f6492 100644
--- a/main/opengeometry/src/query/classify.rs
+++ b/main/opengeometry/src/query/classify.rs
@@ -1,6 +1,7 @@
 use super::classification::PointClassification;
 use super::face_trim::face_contains_uv;
 use super::ray_roots::{ray_box_domain, support_roots};
+use super::wrapped_roots::{wrapped_ray_roots, Ray};
 use crate::brep::{
     unit, BrepEnvelope, Face, GeometryError, PatchBounds, SolidRegion, Surface, SurfaceGeometry, UV,
 };
@@ -45,16 +46,21 @@ fn classify_with_ray<'a>(
         return Ok(PointClassification::Outside);
     };
     let mut hits: Vec<(f64, Point3)> = Vec::new();
+    let ray = Ray {
+        origin: point,
+        direction,
+        domain,
+        tolerance: brep.accuracy.intersection,
+    };
     for face in faces {
         let surface = brep.geometry.surface(face.surface)?;
-        let Some(roots) = support_roots(
-            surface,
-            point,
-            direction,
-            domain,
-            brep.accuracy.intersection,
-        )?
-        else {
+        let roots = match surface {
+            SurfaceGeometry::Wrapped { .. } => {
+                wrapped_ray_roots(surface, face.trim.uv_bounds, &ray)?
+            }
+            _ => support_roots(surface, point, direction, domain, ray.tolerance)?,
+        };
+        let Some(roots) = roots else {
             return Ok(PointClassification::Unknown);
         };
         for parameter in roots {
diff --git a/main/opengeometry/src/query/mod.rs b/main/opengeometry/src/query/mod.rs
index 3ae440d4..fc0fc3c2 100644
--- a/main/opengeometry/src/query/mod.rs
+++ b/main/opengeometry/src/query/mod.rs
@@ -5,6 +5,7 @@ mod measure;
 mod ray_roots;
 #[cfg(test)]
 mod tests;
+mod wrapped_roots;
 
 pub use classification::PointClassification;
 pub use classify::classify_point;
diff --git a/main/opengeometry/src/query/ray_roots.rs b/main/opengeometry/src/query/ray_roots.rs
index 942d533f..33a7b977 100644
--- a/main/opengeometry/src/query/ray_roots.rs
+++ b/main/opengeometry/src/query/ray_roots.rs
@@ -68,6 +68,8 @@ pub(super) fn support_roots(
             domain,
             tolerance,
         )?),
+        SurfaceGeometry::Wrapped { .. } => None,
+        SurfaceGeometry::Conoid { c, .. } => conoid_roots(o, d, c, domain, tolerance),
     };
     Ok(result.map(|roots| {
         roots
@@ -77,6 +79,45 @@ pub(super) fn support_roots(
     }))
 }
 
+fn conoid_roots(
+    o: Point3,
+    d: Point3,
+    c: &[f64; 3],
+    domain: Interval,
+    tolerance: f64,
+) -> Option<Vec<f64>> {
+    let w = [o[2] - c[0], d[2]];
+    let l = [c[1] * o[0] + c[2] * o[1], c[1] * d[0] + c[2] * d[1]];
+    let square = |p: [f64; 2]| [p[0] * p[0], 2.0 * p[0] * p[1], p[1] * p[1]];
+    let radial = [
+        o[0].mul_add(o[0], o[1] * o[1]),
+        2.0 * o[0].mul_add(d[0], o[1] * d[1]),
+        d[0].mul_add(d[0], d[1] * d[1]),
+    ];
+    let (w2, l2) = (square(w), square(l));
+    let mut coefficients = [0.0; 5];
+    for i in 0..3 {
+        coefficients[i] -= l2[i];
+        for j in 0..3 {
+            coefficients[i + j] += w2[i] * radial[j];
+        }
+    }
+    let candidates = isolate_candidates(&coefficients, domain, tolerance, 100_000).ok()?;
+    let on_sheet = |t: f64| {
+        let rho = (o[0] + t * d[0]).hypot(o[1] + t * d[1]);
+        let (height, lean) = ((w[0] + t * w[1]) * rho, l[0] + t * l[1]);
+        rho > tolerance && (height - lean).abs() <= (height + lean).abs()
+    };
+    Some(
+        candidates
+            .intervals
+            .into_iter()
+            .map(|candidate| refine_polynomial_root(&coefficients, candidate))
+            .filter(|t| on_sheet(*t))
+            .collect(),
+    )
+}
+
 fn torus_roots(
     o: Point3,
     d: Point3,
diff --git a/main/opengeometry/src/query/tests/mod.rs b/main/opengeometry/src/query/tests/mod.rs
index 9465023a..0bb3fdf7 100644
--- a/main/opengeometry/src/query/tests/mod.rs
+++ b/main/opengeometry/src/query/tests/mod.rs
@@ -1,6 +1,6 @@
 use super::classification::PointClassification;
 use super::classify::{classify_point, classify_point_validated};
-use crate::brep::{coarse_accuracy, unit, Accuracy, BrepEnvelope, Frame3};
+use crate::brep::{coarse_accuracy, unit, wrapped_sector, Accuracy, BrepEnvelope, Frame3, SECTOR};
 use crate::math::{add, cross, dot, scale, Point3};
 use crate::primitives;
 
@@ -433,3 +433,28 @@ fn cuboid_corners_edges_and_face_points_are_boundary() {
             .collect(),
     );
 }
+
+#[test]
+fn points_on_either_side_of_a_wrapped_wall_classify_correctly() {
+    let brep = wrapped_sector();
+    let w = SECTOR.wrap();
+    let mid = SECTOR.inner + SECTOR.thickness / 2.0;
+    for x in [0.5, 2.0, 3.5] {
+        let y = x * SECTOR.rise / SECTOR.length;
+        let inside = w.point([x, y + 0.01, mid]);
+        let outside = w.point([x, y - 0.01, mid]);
+        assert_eq!(
+            classify_point(&brep, inside).unwrap(),
+            PointClassification::Inside
+        );
+        assert_eq!(
+            classify_point(&brep, outside).unwrap(),
+            PointClassification::Outside
+        );
+    }
+    let middle = w.point([SECTOR.length / 2.0, SECTOR.width / 2.0, mid]);
+    assert_eq!(
+        classify_point(&brep, middle).unwrap(),
+        PointClassification::Inside
+    );
+}
diff --git a/main/opengeometry/src/query/wrapped_roots.rs b/main/opengeometry/src/query/wrapped_roots.rs
new file mode 100644
index 00000000..21666f80
--- /dev/null
+++ b/main/opengeometry/src/query/wrapped_roots.rs
@@ -0,0 +1,123 @@
+use crate::brep::{GeometryError, PatchBounds, Surface, SurfaceGeometry, UVBox};
+use crate::math::{add, cross, dot, norm, scale, sub, Interval, Point3};
+
+const MAX_VISITS: usize = 50_000;
+const MAX_DEPTH: usize = 24;
+const LEAF_FRACTION: f64 = 1.0 / 64.0;
+const NEWTON_STEPS: usize = 32;
+
+pub(super) struct Ray {
+    pub(super) origin: Point3,
+    pub(super) direction: Point3,
+    pub(super) domain: Interval,
+    pub(super) tolerance: f64,
+}
+
+fn ray_in_box(bounds: PatchBounds, ray: &Ray) -> Option<Interval> {
+    let (o, d) = (ray.origin, ray.direction);
+    let (mut lo, mut hi) = (ray.domain.lo, ray.domain.hi);
+    for axis in 0..3 {
+        let (a, b) = (bounds.axes[axis].lo, bounds.axes[axis].hi);
+        if d[axis] == 0.0 {
+            if o[axis] < a || o[axis] > b {
+                return None;
+            }
+            continue;
+        }
+        let (mut t0, mut t1) = ((a - o[axis]) / d[axis], (b - o[axis]) / d[axis]);
+        if t0 > t1 {
+            std::mem::swap(&mut t0, &mut t1);
+        }
+        lo = lo.max(t0);
+        hi = hi.min(t1);
+    }
+    Interval::new(lo, hi).ok().filter(|_| lo <= hi)
+}
+
+fn diagonal(bounds: PatchBounds) -> f64 {
+    norm(std::array::from_fn(|i| bounds.axes[i].width()))
+}
+
+pub(super) fn wrapped_ray_roots(
+    surface: &SurfaceGeometry,
+    uv: UVBox,
+    ray: &Ray,
+) -> Result<Option<Vec<f64>>, GeometryError> {
+    let leaf = diagonal(surface.enclose(uv)?) * LEAF_FRACTION;
+    let mut roots: Vec<f64> = Vec::new();
+    let mut stack = vec![(uv, 0usize)];
+    let mut visits = 0usize;
+    while let Some((b, depth)) = stack.pop() {
+        visits += 1;
+        if visits > MAX_VISITS {
+            return Ok(None);
+        }
+        let bounds = surface.enclose(b)?;
+        let Some(span) = ray_in_box(bounds, ray) else {
+            continue;
+        };
+        if diagonal(bounds) <= leaf || depth >= MAX_DEPTH {
+            let start = [b[0].midpoint(), b[1].midpoint(), span.midpoint()];
+            if let Some([u, v, t]) = ray_newton(surface, start, ray)? {
+                if uv[0].contains(u)
+                    && uv[1].contains(v)
+                    && ray.domain.contains(t)
+                    && roots.iter().all(|r| (r - t).abs() > ray.tolerance)
+                {
+                    roots.push(t);
+                }
+            }
+            continue;
+        }
+        for child in halves(surface, b)? {
+            stack.push((child, depth + 1));
+        }
+    }
+    Ok(Some(roots))
+}
+
+fn halves(surface: &SurfaceGeometry, b: UVBox) -> Result<[UVBox; 2], GeometryError> {
+    let jet = surface.derivatives([b[0].midpoint(), b[1].midpoint()])?;
+    let axis = usize::from(norm(jet.dv) * b[1].width() > norm(jet.du) * b[0].width());
+    let m = b[axis].midpoint();
+    let (mut low, mut high) = (b, b);
+    low[axis] = Interval::new(b[axis].lo, m)?;
+    high[axis] = Interval::new(m, b[axis].hi)?;
+    Ok([low, high])
+}
+
+fn ray_newton(
+    surface: &SurfaceGeometry,
+    start: [f64; 3],
+    ray: &Ray,
+) -> Result<Option<[f64; 3]>, GeometryError> {
+    let mut x = start;
+    for _ in 0..NEWTON_STEPS {
+        let uv = [x[0], x[1]];
+        let (point, jet) = match surface
+            .point_at(uv)
+            .and_then(|p| Ok((p, surface.derivatives(uv)?)))
+        {
+            Ok(value) => value,
+            Err(GeometryError::SingularParameterization) => return Ok(None),
+            Err(error) => return Err(error),
+        };
+        let r = sub(point, add(ray.origin, scale(ray.direction, x[2])));
+        if norm(r) <= ray.tolerance {
+            return Ok(Some(x));
+        }
+        let m = [jet.du, jet.dv, scale(ray.direction, -1.0)];
+        let det = dot(m[0], cross(m[1], m[2]));
+        if det.abs() <= f64::EPSILON * norm(m[0]) * norm(m[1]) {
+            return Ok(None);
+        }
+        let rhs = scale(r, -1.0);
+        x[0] += dot(rhs, cross(m[1], m[2])) / det;
+        x[1] += dot(m[0], cross(rhs, m[2])) / det;
+        x[2] += dot(m[0], cross(m[1], rhs)) / det;
+        if x.iter().any(|v| !v.is_finite()) {
+            return Ok(None);
+        }
+    }
+    Ok(None)
+}
diff --git a/main/opengeometry/src/tessellation/area_bound.rs b/main/opengeometry/src/tessellation/area_bound.rs
index 4dd3c848..af7a6d2d 100644
--- a/main/opengeometry/src/tessellation/area_bound.rs
+++ b/main/opengeometry/src/tessellation/area_bound.rs
@@ -132,6 +132,7 @@ fn deviation(
             sense,
             deflection,
         ),
+        SurfaceGeometry::Wrapped { .. } | SurfaceGeometry::Conoid { .. } => None,
     }
 }
 
diff --git a/main/opengeometry/src/tessellation/deviation.rs b/main/opengeometry/src/tessellation/deviation.rs
index b0c2878d..126a223b 100644
--- a/main/opengeometry/src/tessellation/deviation.rs
+++ b/main/opengeometry/src/tessellation/deviation.rs
@@ -184,6 +184,18 @@ pub(super) fn edge_count(
             CurveGeometry::Intersection { definition } => {
                 bounded_intersection_count(brep, *definition, curve, range, error, max_triangles)
             }
+            geometry @ CurveGeometry::Wrapped { curve: flat, .. } => {
+                let arc = matches!(
+                    **flat,
+                    CurveGeometry::Circle { .. } | CurveGeometry::Ellipse { .. }
+                );
+                turn_count(
+                    range.width() * (geometry.derivative_bounds(range)?.1 / (8.0 * error)).sqrt(),
+                    if arc { range.width() } else { 0.0 },
+                    1,
+                    max_triangles,
+                )
+            }
         },
     }
 }
diff --git a/main/opengeometry/src/tessellation/grid.rs b/main/opengeometry/src/tessellation/grid.rs
index d0e50772..03312ba8 100644
--- a/main/opengeometry/src/tessellation/grid.rs
+++ b/main/opengeometry/src/tessellation/grid.rs
@@ -1,3 +1,4 @@
+use super::wrapped::second_derivative_bounds;
 use crate::brep::{
     period_lifted_uv, BrepEnvelope, Face, GeometryError, PcurveGeometry, Surface, SurfaceGeometry,
 };
@@ -123,6 +124,9 @@ pub(super) fn grid_size(
             turn_count(u * ((r + a) / (4.0 * error)).sqrt(), u, 3, max_triangles)?,
             turn_count(v * (a / (4.0 * error)).sqrt(), v, 3, max_triangles)?,
         ),
+        SurfaceGeometry::Wrapped { .. } | SurfaceGeometry::Conoid { .. } => {
+            sampled_grid(surface, bounds, error, max_triangles)?
+        }
         SurfaceGeometry::Plane { .. } => {
             return Err(GeometryError::InvalidGeometry(
                 "plane does not use a curved grid".into(),
@@ -141,6 +145,31 @@ pub(super) fn grid_size(
     Ok([nu, nv])
 }
 
+fn sampled_grid(
+    surface: &SurfaceGeometry,
+    bounds: [Interval; 2],
+    error: f64,
+    max_triangles: usize,
+) -> Result<(usize, usize), GeometryError> {
+    let second = second_derivative_bounds(surface, bounds)?;
+    let periods = surface.charts()[0].periods;
+    let side = |i: usize, minimum: usize| {
+        let width = bounds[i].width();
+        let value = width * (second[i] / (8.0 * error)).sqrt();
+        match periods[i] {
+            Some(_) => turn_count(value, width, minimum, max_triangles),
+            None => count(value, minimum, max_triangles),
+        }
+    };
+    let nu = side(0, 3)?;
+    let nv = side(1, 1)?;
+    if !matches!(surface, SurfaceGeometry::Conoid { .. }) {
+        return Ok((nu, nv));
+    }
+    let twist = bounds[1].width() * second[2] * bounds[0].width() / (4.0 * error * nu as f64);
+    Ok((nu, nv.max(count(twist, 1, max_triangles)?)))
+}
+
 pub(super) fn turn_count(
     value: f64,
     sweep: f64,
diff --git a/main/opengeometry/src/tessellation/mod.rs b/main/opengeometry/src/tessellation/mod.rs
index 358b45fa..17126e9c 100644
--- a/main/opengeometry/src/tessellation/mod.rs
+++ b/main/opengeometry/src/tessellation/mod.rs
@@ -17,6 +17,7 @@ mod tessellate;
 mod tests;
 mod trim;
 mod trimmed_face;
+mod wrapped;
 
 pub use measure::{measure, Measure};
 pub use mesh::Tessellation;
diff --git a/main/opengeometry/src/tessellation/tests/mod.rs b/main/opengeometry/src/tessellation/tests/mod.rs
index 404d5492..5fcfc05f 100644
--- a/main/opengeometry/src/tessellation/tests/mod.rs
+++ b/main/opengeometry/src/tessellation/tests/mod.rs
@@ -3,6 +3,7 @@ mod seams;
 mod short_loops;
 mod smooth_edges;
 mod support;
+mod wrapped;
 
 use super::cache::TessellationCache;
 use super::tessellate::tessellate;
diff --git a/main/opengeometry/src/tessellation/tests/wrapped.rs b/main/opengeometry/src/tessellation/tests/wrapped.rs
new file mode 100644
index 00000000..3a443cf7
--- /dev/null
+++ b/main/opengeometry/src/tessellation/tests/wrapped.rs
@@ -0,0 +1,24 @@
+use crate::brep::{wrapped_sector, SECTOR};
+use crate::math::Point3;
+use crate::tessellation::tessellate::tessellate;
+
+#[test]
+fn a_wrapped_side_meshes_to_the_exact_volume() {
+    let mesh = tessellate(&wrapped_sector(), 1e-3, 1_000_000).unwrap();
+    let p = |i: u32| -> Point3 { std::array::from_fn(|j| mesh.positions[i as usize * 3 + j]) };
+    let volume: f64 = mesh
+        .indices
+        .chunks_exact(3)
+        .map(|t| {
+            let (a, b, c) = (p(t[0]), p(t[1]), p(t[2]));
+            (a[0] * (b[1] * c[2] - b[2] * c[1]) - a[1] * (b[0] * c[2] - b[2] * c[0])
+                + a[2] * (b[0] * c[1] - b[1] * c[0]))
+                / 6.0
+        })
+        .sum();
+    let exact = SECTOR.volume();
+    assert!(
+        ((volume - exact) / exact).abs() < 1e-4,
+        "{volume} vs {exact}"
+    );
+}
diff --git a/main/opengeometry/src/tessellation/trim.rs b/main/opengeometry/src/tessellation/trim.rs
index 75e87714..54aaa77d 100644
--- a/main/opengeometry/src/tessellation/trim.rs
+++ b/main/opengeometry/src/tessellation/trim.rs
@@ -1,3 +1,4 @@
+use super::wrapped::chord_deviation;
 use crate::brep::{
     period_lifted_uv, BrepEnvelope, EdgeGeometry, Face, GeometryError, HalfEdge, Orientation,
     Surface, SurfaceGeometry,
@@ -262,6 +263,9 @@ pub(super) fn trimmed_surface_edge_deviation(
         }
     };
     let acceleration = match surface {
+        SurfaceGeometry::Wrapped { .. } | SurfaceGeometry::Conoid { .. } => {
+            return chord_deviation(surface, uv[a], uv[b])
+        }
         SurfaceGeometry::Plane { .. } => 0.0,
         SurfaceGeometry::Sphere { radius, .. } => {
             let cos_v = maximum_trig(0.0, f64::cos);
diff --git a/main/opengeometry/src/tessellation/trimmed_face.rs b/main/opengeometry/src/tessellation/trimmed_face.rs
index acb8528a..488b7733 100644
--- a/main/opengeometry/src/tessellation/trimmed_face.rs
+++ b/main/opengeometry/src/tessellation/trimmed_face.rs
@@ -54,7 +54,7 @@ pub(super) fn curved_trimmed_face(
     if rect.is_some() && !face.trim.holes.is_empty() {
         shift_periodic_holes(face, surface, periods, &mut loops)?;
     }
-    if !axis_aligned {
+    if !axis_aligned || matches!(surface, SurfaceGeometry::Conoid { .. }) {
         return general_curved_trimmed_face(brep, face, size, &loops, error, mesh, max_triangles);
     }
     let all_boundary = loops.iter().flatten().collect::<Vec<_>>();
diff --git a/main/opengeometry/src/tessellation/wrapped.rs b/main/opengeometry/src/tessellation/wrapped.rs
new file mode 100644
index 00000000..ef085b2f
--- /dev/null
+++ b/main/opengeometry/src/tessellation/wrapped.rs
@@ -0,0 +1,41 @@
+use crate::brep::{GeometryError, Surface, SurfaceGeometry};
+use crate::math::{norm, Interval, Point3};
+
+const MARGIN: f64 = 1.5;
+const GRID: usize = 8;
+
+pub(super) fn second_derivative_bounds(
+    surface: &SurfaceGeometry,
+    bounds: [Interval; 2],
+) -> Result<[f64; 3], GeometryError> {
+    let mut maximum = [0.0_f64; 3];
+    for i in 0..=GRID {
+        for j in 0..=GRID {
+            let uv = [
+                bounds[0].lo + bounds[0].width() * i as f64 / GRID as f64,
+                bounds[1].lo + bounds[1].width() * j as f64 / GRID as f64,
+            ];
+            let jet = surface.derivatives(uv)?;
+            maximum[0] = maximum[0].max(norm(jet.duu));
+            maximum[1] = maximum[1].max(norm(jet.dvv));
+            maximum[2] = maximum[2].max(norm(jet.duv));
+        }
+    }
+    Ok(maximum.map(|m| m * MARGIN))
+}
+
+pub(super) fn chord_deviation(surface: &SurfaceGeometry, a: [f64; 2], b: [f64; 2]) -> f64 {
+    let (du, dv) = (b[0] - a[0], b[1] - a[1]);
+    let mut maximum = 0.0_f64;
+    for k in 0..=2 {
+        let t = k as f64 / 2.0;
+        let Ok(jet) = surface.derivatives([a[0] + t * du, a[1] + t * dv]) else {
+            return f64::INFINITY;
+        };
+        let acceleration: Point3 = std::array::from_fn(|i| {
+            jet.duu[i] * du * du + 2.0 * jet.duv[i] * du * dv + jet.dvv[i] * dv * dv
+        });
+        maximum = maximum.max(norm(acceleration));
+    }
+    maximum * MARGIN / 8.0
+}
diff --git a/main/opengeometry/test-support/src/part21/document.rs b/main/opengeometry/test-support/src/part21/document.rs
index 31447a7c..197d4e3e 100644
--- a/main/opengeometry/test-support/src/part21/document.rs
+++ b/main/opengeometry/test-support/src/part21/document.rs
@@ -188,6 +188,9 @@ impl Document {
 
     pub(super) fn surface_point(&self, id: usize, uv: [f64; 2]) -> Result<[f64; 3], String> {
         let expression = self.entity(id)?;
+        if expression.starts_with("B_SPLINE_SURFACE_WITH_KNOTS(") {
+            return self.spline_surface_point(expression, uv);
+        }
         let args = fields(expression)?;
         let (origin, z, x) = self.axis(reference(&args[1])?)?;
         let y = [
diff --git a/main/opengeometry/test-support/src/part21/mod.rs b/main/opengeometry/test-support/src/part21/mod.rs
index aded8271..0caba1c2 100644
--- a/main/opengeometry/test-support/src/part21/mod.rs
+++ b/main/opengeometry/test-support/src/part21/mod.rs
@@ -8,6 +8,7 @@ mod report;
 mod report_match;
 mod rules;
 mod solids;
+mod surfaces;
 #[cfg(test)]
 mod tests;
 mod values;
diff --git a/main/opengeometry/test-support/src/part21/rules.rs b/main/opengeometry/test-support/src/part21/rules.rs
index bf85a019..3b0a26c0 100644
--- a/main/opengeometry/test-support/src/part21/rules.rs
+++ b/main/opengeometry/test-support/src/part21/rules.rs
@@ -13,6 +13,7 @@ fn allowed(keyword: &str) -> bool {
             | "AXIS2_PLACEMENT_3D"
             | "BREP_WITH_VOIDS"
             | "B_SPLINE_CURVE_WITH_KNOTS"
+            | "B_SPLINE_SURFACE_WITH_KNOTS"
             | "CARTESIAN_POINT"
             | "CIRCLE"
             | "CLOSED_SHELL"
diff --git a/main/opengeometry/test-support/src/part21/surfaces.rs b/main/opengeometry/test-support/src/part21/surfaces.rs
new file mode 100644
index 00000000..c0fb48aa
--- /dev/null
+++ b/main/opengeometry/test-support/src/part21/surfaces.rs
@@ -0,0 +1,63 @@
+use super::document::Document;
+use super::lexer::{fields, references};
+
+fn bernstein(t: f64) -> [f64; 4] {
+    let s = 1.0 - t;
+    [s * s * s, 3.0 * s * s * t, 3.0 * s * t * t, t * t * t]
+}
+
+fn span(knots: &[f64], multiplicities: &[f64], value: f64) -> Result<(usize, f64), String> {
+    let last = knots.len().checked_sub(1).ok_or("empty knot vector")?;
+    let piecewise = last > 0
+        && knots.len() == multiplicities.len()
+        && multiplicities
+            .iter()
+            .enumerate()
+            .all(|(i, &m)| m == if i == 0 || i == last { 4.0 } else { 3.0 });
+    if !piecewise {
+        return Err("B-spline surface is not piecewise cubic Bezier".into());
+    }
+    let index = knots[1..last]
+        .iter()
+        .take_while(|&&knot| knot <= value)
+        .count();
+    let (lo, hi) = (knots[index], knots[index + 1]);
+    Ok((index, (value - lo) / (hi - lo)))
+}
+
+impl Document {
+    pub(super) fn spline_surface_point(
+        &self,
+        expression: &str,
+        uv: [f64; 2],
+    ) -> Result<[f64; 3], String> {
+        let args = fields(expression)?;
+        if args[1] != "3" || args[2] != "3" {
+            return Err("B-spline surface is not bicubic".into());
+        }
+        let rows = fields(&args[3])?
+            .iter()
+            .map(|row| references(row))
+            .collect::<Vec<_>>();
+        let (i, s) = span(&self.tuple(&args[10])?, &self.tuple(&args[8])?, uv[0])?;
+        let (j, t) = span(&self.tuple(&args[11])?, &self.tuple(&args[9])?, uv[1])?;
+        let (wu, wv) = (bernstein(s), bernstein(t));
+        let mut point = [0.0; 3];
+        for (a, weight_u) in wu.iter().enumerate() {
+            for (b, weight_v) in wv.iter().enumerate() {
+                let id = *rows
+                    .get(3 * i + a)
+                    .and_then(|row| row.get(3 * j + b))
+                    .ok_or("B-spline surface control net is too small")?;
+                let control = self.tuple(&fields(self.entity(id)?)?[1])?;
+                if control.len() != 3 {
+                    return Err("B-spline surface control is not three-dimensional".into());
+                }
+                for axis in 0..3 {
+                    point[axis] += weight_u * weight_v * control[axis];
+                }
+            }
+        }
+        Ok(point)
+    }
+}
diff --git a/main/opengeometry/tests/conoid_bend.rs b/main/opengeometry/tests/conoid_bend.rs
new file mode 100644
index 00000000..9fba7d72
--- /dev/null
+++ b/main/opengeometry/tests/conoid_bend.rs
@@ -0,0 +1,375 @@
+use opengeometry::brep::{
+    conoid_ellipse, mirrored, placed, plane_boundary, projected_boundary, uv_line, BrepEnvelope,
+    Builder, CurveGeometry, Frame3, GeometryError, Orientation, SurfaceGeometry, Use,
+};
+use opengeometry::exchange::export_step;
+use opengeometry::math::{Interval, Point3};
+use opengeometry::operations::modifying::boolean::{boolean_brep, BooleanOp};
+use opengeometry::primitives::cuboid;
+use opengeometry::query::{classify_point, PointClassification};
+use opengeometry_test_support::accuracy::accuracy;
+use opengeometry_test_support::part21::Document;
+use opengeometry_test_support::volume::estimate;
+use std::f64::consts::FRAC_PI_2;
+
+const RADII: [f64; 2] = [2.0, 4.0];
+const ENDS: [f64; 2] = [0.0, FRAC_PI_2];
+const PAD: f64 = 1e-3;
+
+struct Shell {
+    frame: Frame3,
+    c: [f64; 3],
+}
+
+fn dot(a: Point3, b: Point3) -> f64 {
+    a[0] * b[0] + a[1] * b[1] + a[2] * b[2]
+}
+
+fn combine(a: Point3, s: f64, b: Point3, t: f64) -> Point3 {
+    std::array::from_fn(|i| a[i] * s + b[i] * t)
+}
+
+impl Shell {
+    fn new() -> Self {
+        let frame = Frame3::from_axis([1.0, 2.0, 3.0], [1.0, 0.2, 0.1], [0.0, 1.0, 0.3]).unwrap();
+        let normal = [1.0, -1.0, 0.0];
+        let local = [frame.x, frame.y, frame.z].map(|axis| dot(normal, axis));
+        let k = dot(normal, frame.origin) + 5.0 * local[2];
+        let c = [
+            (k - dot(normal, frame.origin)) / local[2],
+            -RADII[0] * local[0] / local[2],
+            -RADII[0] * local[1] / local[2],
+        ];
+        Self { frame, c }
+    }
+
+    fn height(&self, u: f64) -> f64 {
+        self.c[0] + self.c[1] * u.cos() + self.c[2] * u.sin()
+    }
+
+    fn at(&self, u: f64, radius: f64, z: f64) -> Point3 {
+        self.frame.point([radius * u.cos(), radius * u.sin(), z])
+    }
+
+    fn radial(&self, u: f64) -> Point3 {
+        combine(self.frame.x, u.cos(), self.frame.y, u.sin())
+    }
+
+    fn tangent(&self, u: f64) -> Point3 {
+        combine(self.frame.x, -u.sin(), self.frame.y, u.cos())
+    }
+
+    fn volume(&self) -> f64 {
+        let n = 64;
+        let h = FRAC_PI_2 / n as f64;
+        let ring = (RADII[1] * RADII[1] - RADII[0] * RADII[0]) / 2.0;
+        let sum: f64 = (0..=n)
+            .map(|i| {
+                let weight = if i == 0 || i == n {
+                    1.0
+                } else {
+                    2.0 + 2.0 * (i % 2) as f64
+                };
+                weight * self.height(i as f64 * h)
+            })
+            .sum();
+        ring * sum * h / 3.0
+    }
+}
+
+struct Edges {
+    vertices: [[[u32; 2]; 2]; 2],
+    arcs: [u32; 2],
+    ellipses: [(u32, f64); 2],
+    radials: [u32; 2],
+    rulings: [u32; 2],
+    axials: [[u32; 2]; 2],
+}
+
+fn line(b: &mut Builder, origin: Point3, direction: Point3, length: f64) -> u32 {
+    let curve = CurveGeometry::Line { origin, direction };
+    b.edge(curve, Interval::new(0.0, length).unwrap(), false)
+}
+
+fn edges(b: &mut Builder, s: &Shell) -> Edges {
+    let mut vertices = [[[0; 2]; 2]; 2];
+    for (e, &u) in ENDS.iter().enumerate() {
+        for (r, &radius) in RADII.iter().enumerate() {
+            vertices[e][r] = [0.0, s.height(u)].map(|z| b.vertex(s.at(u, radius, z)));
+        }
+    }
+    let quarter = Interval::new(0.0, FRAC_PI_2).unwrap();
+    let arcs = RADII.map(|radius| {
+        let circle = CurveGeometry::Circle {
+            frame: s.frame,
+            radius,
+        };
+        b.edge(circle, quarter, false)
+    });
+    let ellipses = RADII.map(|radius| {
+        let (ellipse, phase) = conoid_ellipse(s.frame, s.c, radius).unwrap();
+        let range = Interval::new(-phase, FRAC_PI_2 - phase).unwrap();
+        (b.edge(ellipse, range, false), phase)
+    });
+    let thickness = RADII[1] - RADII[0];
+    let radials = ENDS.map(|u| line(b, s.at(u, RADII[0], 0.0), s.radial(u), thickness));
+    let rulings = ENDS.map(|u| line(b, s.at(u, RADII[0], s.height(u)), s.radial(u), thickness));
+    let axials =
+        ENDS.map(|u| RADII.map(|radius| line(b, s.at(u, radius, 0.0), s.frame.z, s.height(u))));
+    Edges {
+        vertices,
+        arcs,
+        ellipses,
+        radials,
+        rulings,
+        axials,
+    }
+}
+
+type Step = (u32, u32, u32, Orientation);
+
+fn plane(b: &mut Builder, key: &str, frame: Frame3, steps: [Step; 4]) {
+    let uses = steps
+        .map(|(edge, from, to, sense)| plane_boundary(b, frame, edge, from, to, sense).unwrap());
+    let big = 100.0;
+    let surface = SurfaceGeometry::Plane { frame };
+    b.face(key, surface, [[-big, big], [-big, big]], uses.into())
+        .unwrap();
+}
+
+fn planes(b: &mut Builder, s: &Shell, e: &Edges) {
+    let (fwd, rev) = (Orientation::Forward, Orientation::Reverse);
+    let v = |end: usize, r: usize, level: usize| e.vertices[end][r][level];
+    let f = s.frame;
+    let bottom = Frame3::from_axis(f.origin, f.z.map(|x| -x), f.x).unwrap();
+    plane(
+        b,
+        "bottom",
+        bottom,
+        [
+            (e.arcs[0], v(0, 0, 0), v(1, 0, 0), fwd),
+            (e.radials[1], v(1, 0, 0), v(1, 1, 0), fwd),
+            (e.arcs[1], v(1, 1, 0), v(0, 1, 0), rev),
+            (e.radials[0], v(0, 1, 0), v(0, 0, 0), rev),
+        ],
+    );
+    for (end, &u) in ENDS.iter().enumerate() {
+        let outward = s.tangent(u).map(|x| if end == 0 { -x } else { x });
+        let frame = Frame3::from_axis(f.origin, outward, s.radial(u)).unwrap();
+        plane(
+            b,
+            ["start", "end"][end],
+            frame,
+            [
+                (e.radials[end], v(end, 0, 0), v(end, 1, 0), fwd),
+                (e.axials[end][1], v(end, 1, 0), v(end, 1, 1), fwd),
+                (e.rulings[end], v(end, 1, 1), v(end, 0, 1), rev),
+                (e.axials[end][0], v(end, 0, 1), v(end, 0, 0), rev),
+            ],
+        );
+    }
+}
+
+fn cylinders(b: &mut Builder, s: &Shell, e: &Edges) {
+    let (fwd, rev) = (Orientation::Forward, Orientation::Reverse);
+    let top = s.c[0] + s.c[1].hypot(s.c[2]);
+    for (r, &radius) in RADII.iter().enumerate() {
+        let v = |end: usize, level: usize| e.vertices[end][r][level];
+        let surface = SurfaceGeometry::Cylinder {
+            frame: s.frame,
+            radius,
+        };
+        let ellipse = projected_boundary(
+            b,
+            &surface,
+            e.ellipses[r].0,
+            v(1, 1),
+            v(0, 1),
+            rev,
+            Some([0.0, s.height(0.0)]),
+        )
+        .unwrap();
+        let uses: Vec<Use> = vec![
+            uv_use(e.arcs[r], v(0, 0), v(1, 0), fwd, [0.0, 0.0], [1.0, 0.0]),
+            uv_use(
+                e.axials[1][r],
+                v(1, 0),
+                v(1, 1),
+                fwd,
+                [ENDS[1], 0.0],
+                [0.0, 1.0],
+            ),
+            ellipse,
+            uv_use(
+                e.axials[0][r],
+                v(0, 1),
+                v(0, 0),
+                rev,
+                [0.0, 0.0],
+                [0.0, 1.0],
+            ),
+        ];
+        let bounds = [[-PAD, FRAC_PI_2 + PAD], [-PAD, top + PAD]];
+        let face = b
+            .face(["inner", "outer"][r], surface, bounds, uses)
+            .unwrap();
+        if r == 0 {
+            b.set_face_sense(face, rev).unwrap();
+        }
+    }
+}
+
+fn uv_use(
+    edge: u32,
+    from: u32,
+    to: u32,
+    sense: Orientation,
+    origin: [f64; 2],
+    dir: [f64; 2],
+) -> Use {
+    opengeometry::brep::boundary(edge, from, to, sense, uv_line(origin, dir))
+}
+
+fn conoid_face(b: &mut Builder, s: &Shell, e: &Edges, bounds: [[f64; 2]; 2]) {
+    let (fwd, rev) = (Orientation::Forward, Orientation::Reverse);
+    let v = |end: usize, r: usize| e.vertices[end][r][1];
+    let [(inner, inner_phase), (outer, outer_phase)] = e.ellipses;
+    let uses = vec![
+        uv_use(
+            inner,
+            v(0, 0),
+            v(1, 0),
+            fwd,
+            [inner_phase, RADII[0]],
+            [1.0, 0.0],
+        ),
+        uv_use(
+            e.rulings[1],
+            v(1, 0),
+            v(1, 1),
+            fwd,
+            [ENDS[1], RADII[0]],
+            [0.0, 1.0],
+        ),
+        uv_use(
+            outer,
+            v(1, 1),
+            v(0, 1),
+            rev,
+            [outer_phase, RADII[1]],
+            [1.0, 0.0],
+        ),
+        uv_use(
+            e.rulings[0],
+            v(0, 1),
+            v(0, 0),
+            rev,
+            [ENDS[0], RADII[0]],
+            [0.0, 1.0],
+        ),
+    ];
+    let surface = SurfaceGeometry::Conoid {
+        frame: s.frame,
+        c: s.c,
+    };
+    let face = b.face("cut", surface, bounds, uses).unwrap();
+    b.set_face_sense(face, rev).unwrap();
+}
+
+fn build(s: &Shell, conoid_bounds: [[f64; 2]; 2]) -> Result<BrepEnvelope, GeometryError> {
+    let mut b = Builder::new("conoid-cut".into(), accuracy(1e-5))?;
+    let e = edges(&mut b, s);
+    planes(&mut b, s, &e);
+    cylinders(&mut b, s, &e);
+    conoid_face(&mut b, s, &e, conoid_bounds);
+    b.finish_solid()
+}
+
+fn padded() -> [[f64; 2]; 2] {
+    [[-PAD, FRAC_PI_2 + PAD], [RADII[0] - PAD, RADII[1] + PAD]]
+}
+
+fn exact() -> [[f64; 2]; 2] {
+    [ENDS, RADII]
+}
+
+fn check_volume(brep: &BrepEnvelope, exact: f64, deflection: f64, relative: f64) {
+    let volume = estimate(brep, deflection);
+    let error = (volume.value - exact).abs();
+    assert!(error <= volume.error_bound, "{} vs {exact}", volume.value);
+    assert!(error <= relative * exact, "{} vs {exact}", volume.value);
+}
+
+#[test]
+fn a_conoid_cut_shell_is_a_valid_solid_with_its_exact_volume() {
+    let s = Shell::new();
+    let exact_volume = s.volume();
+    let closed_form = 6.0 * (s.c[0] * FRAC_PI_2 + s.c[1] + s.c[2]);
+    assert!((exact_volume - closed_form).abs() < 1e-9 * closed_form);
+    for (bounds, relative) in [(exact(), 2e-4), (padded(), 2e-4)] {
+        let brep = build(&s, bounds).unwrap();
+        brep.validate().unwrap();
+        assert_eq!(brep.topology.faces.len(), 6);
+        assert_eq!(brep.topology.edges.len(), 12);
+        check_volume(&brep, exact_volume, 1e-3, relative);
+    }
+}
+
+#[test]
+fn points_on_either_side_of_the_conoid_classify_correctly() {
+    let s = Shell::new();
+    let brep = build(&s, exact()).unwrap();
+    for u in [0.2, FRAC_PI_2 / 2.0, 1.3] {
+        let h = s.height(u);
+        let classify = |radius: f64, z: f64| classify_point(&brep, s.at(u, radius, z)).unwrap();
+        assert_eq!(classify(3.0, h / 2.0), PointClassification::Inside);
+        assert_eq!(classify(3.5, h - 0.05), PointClassification::Inside);
+        assert_eq!(classify(2.5, h + 0.05), PointClassification::Outside);
+        assert_eq!(classify(4.5, h / 2.0), PointClassification::Outside);
+    }
+}
+
+#[test]
+fn a_conoid_cut_shell_exports_to_step_and_round_trips_through_json() {
+    let s = Shell::new();
+    let brep = build(&s, exact()).unwrap();
+    let (step, report) = export_step(&brep, "metre").unwrap();
+    assert!(report.exchange_error_bound <= brep.accuracy.exchange);
+    let document = Document::parse(&step).unwrap();
+    assert_eq!(document.count("ADVANCED_FACE"), 6);
+    assert_eq!(document.count("B_SPLINE_SURFACE_WITH_KNOTS"), 1);
+    assert_eq!(document.count("CYLINDRICAL_SURFACE"), 2);
+    assert_eq!(document.count("SURFACE_CURVE"), 10);
+    assert_eq!(document.solids().unwrap().len(), 1);
+    let (millimetres, _) = export_step(&brep, "millimetre").unwrap();
+    Document::parse(&millimetres).unwrap();
+    let json = brep.to_json().unwrap();
+    assert!(json.contains("\"kind\":\"Conoid\""));
+    let back = BrepEnvelope::from_json(&json).unwrap();
+    back.validate().unwrap();
+    assert_eq!(back.geometry, brep.geometry);
+    assert_eq!(back.to_json().unwrap(), json);
+}
+
+#[test]
+fn a_conoid_cut_shell_moves_scales_and_mirrors_but_refuses_booleans() {
+    let s = Shell::new();
+    let brep = build(&s, exact()).unwrap();
+    let volume = s.volume();
+    let placement = Frame3::from_axis([-4.0, 1.0, 2.0], [0.3, -1.0, 2.0], [1.0, 0.0, 0.0]).unwrap();
+    check_volume(
+        &placed(&brep, placement, 2.0).unwrap(),
+        8.0 * volume,
+        2e-3,
+        2e-4,
+    );
+    let mirror = mirrored(&brep, [0.5, 0.0, 1.0], [1.0, 2.0, -0.5]).unwrap();
+    check_volume(&mirror, volume, 1e-3, 2e-4);
+    let cube = cuboid("cube".into(), Frame3::IDENTITY, [1.0; 3], brep.accuracy).unwrap();
+    for (a, b) in [(&brep, &cube), (&cube, &brep)] {
+        assert!(matches!(
+            boolean_brep(a, b, BooleanOp::Union, "u".into()),
+            Err(GeometryError::UnsupportedGeometry(_))
+        ));
+    }
+}
diff --git a/main/opengeometry/tests/creating/sweep_bends.rs b/main/opengeometry/tests/creating/sweep_bends.rs
index af26e7b3..a15ef9dc 100644
--- a/main/opengeometry/tests/creating/sweep_bends.rs
+++ b/main/opengeometry/tests/creating/sweep_bends.rs
@@ -99,6 +99,8 @@ fn surface_kinds(body: &BrepEnvelope) -> Vec<&'static str> {
             SurfaceGeometry::Cone { .. } => "cone",
             SurfaceGeometry::Torus { .. } => "torus",
             SurfaceGeometry::Sphere { .. } => "sphere",
+            SurfaceGeometry::Wrapped { .. } => "wrapped",
+            SurfaceGeometry::Conoid { .. } => "conoid",
         })
         .collect::<Vec<_>>();
     kinds.sort_unstable();
diff --git a/main/opengeometry/tests/fixtures/cases/annular-cylinder.brep.json b/main/opengeometry/tests/fixtures/cases/annular-cylinder.brep.json
index 6755c583..dca1644f 100644
--- a/main/opengeometry/tests/fixtures/cases/annular-cylinder.brep.json
+++ b/main/opengeometry/tests/fixtures/cases/annular-cylinder.brep.json
@@ -1 +1 @@
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\ No newline at end of file
diff --git a/main/opengeometry/tests/fixtures/cases/batch-matrix/prismatic.single-round.fallback.json b/main/opengeometry/tests/fixtures/cases/batch-matrix/prismatic.single-round.fallback.json
index d398bf70..1bc0b661 100644
--- a/main/opengeometry/tests/fixtures/cases/batch-matrix/prismatic.single-round.fallback.json
+++ b/main/opengeometry/tests/fixtures/cases/batch-matrix/prismatic.single-round.fallback.json
@@ -1 +1 @@
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\ No newline at end of file
diff --git a/main/opengeometry/tests/fixtures/cases/validity/builder-annular-cylinder.json b/main/opengeometry/tests/fixtures/cases/validity/builder-annular-cylinder.json
index 3cb65889..77f38b8b 100644
--- a/main/opengeometry/tests/fixtures/cases/validity/builder-annular-cylinder.json
+++ b/main/opengeometry/tests/fixtures/cases/validity/builder-annular-cylinder.json
@@ -1 +1 @@
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diff --git a/main/opengeometry/tests/fixtures/cases/validity/builder-cylinder-with-hole.json b/main/opengeometry/tests/fixtures/cases/validity/builder-cylinder-with-hole.json
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+++ b/main/opengeometry/tests/fixtures/cases/validity/builder-cylinder-with-hole.json
@@ -1 +1 @@
-{"accuracy":{"exchange":1e-6,"geometric":1e-8,"intersection":1e-9,"tessellation":0.01},"geometry":{"curves":[{"frame":{"origin":[0.0,0.0,0.0],"x":[1.0,0.0,0.0],"y":[0.0,0.0,-1.0],"z":[0.0,1.0,0.0]},"kind":"Circle","radius":1.0},{"frame":{"origin":[0.0,2.0,0.0],"x":[1.0,0.0,0.0],"y":[0.0,0.0,-1.0],"z":[0.0,1.0,0.0]},"kind":"Circle","radius":1.0},{"frame":{"origin":[0.0,0.0,0.0],"x":[1.0,0.0,0.0],"y":[0.0,0.0,-1.0],"z":[0.0,1.0,0.0]},"kind":"Circle","radius":0.4},{"frame":{"origin":[0.0,2.0,0.0],"x":[1.0,0.0,0.0],"y":[0.0,0.0,-1.0],"z":[0.0,1.0,0.0]},"kind":"Circle","radius":0.4},{"direction":[0.0,1.0,0.0],"kind":"Line","origin":[1.0,0.0,0.0]},{"direction":[0.0,1.0,0.0],"kind":"Line","origin":[0.4,0.0,0.0]}],"intersections":[],"pcurves":[{"direction":[1.0,0.0],"kind":"Line2","origin":[0.0,0.0]},{"direction":[0.0,1.0],"kind":"Line2","origin":[6.283185307179586,0.0]},{"direction":[1.0,0.0],"kind":"Line2","origin":[0.0,2.0]},{"direction":[0.0,1.0],"kind":"Line2","origin":[0.0,0.0]},{"direction":[1.0,0.0],"kind":"Line2","origin":[0.0,0.0]},{"direction":[0.0,1.0],"kind":"Line2","origin":[6.283185307179586,0.0]},{"direction":[1.0,0.0],"kind":"Line2","origin":[0.0,2.0]},{"direction":[0.0,1.0],"kind":"Line2","origin":[0.0,0.0]},{"axis_a":[1.0,0.0],"axis_b":[0.0,-1.0],"kind":"Conic2","origin":[0.0,0.0]},{"axis_a":[0.4,0.0],"axis_b":[0.0,-0.4],"kind":"Conic2","origin":[0.0,0.0]},{"axis_a":[1.0,0.0],"axis_b":[0.0,1.0],"kind":"Conic2","origin":[0.0,0.0]},{"axis_a":[0.4,0.0],"axis_b":[0.0,0.4],"kind":"Conic2","origin":[0.0,0.0]}],"surfaces":[{"frame":{"origin":[0.0,0.0,0.0],"x":[1.0,0.0,0.0],"y":[0.0,0.0,-1.0],"z":[0.0,1.0,0.0]},"kind":"Cylinder","radius":1.0},{"frame":{"origin":[0.0,0.0,0.0],"x":[1.0,0.0,0.0],"y":[0.0,0.0,-1.0],"z":[0.0,1.0,0.0]},"kind":"Cylinder","radius":0.4},{"frame":{"origin":[0.0,0.0,0.0],"x":[1.0,0.0,0.0],"y":[-0.0,-0.0,1.0],"z":[-0.0,-1.0,-0.0]},"kind":"Plane"},{"frame":{"origin":[0.0,2.0,0.0],"x":[1.0,0.0,0.0],"y":[0.0,0.0,-1.0],"z":[0.0,1.0,0.0]},"kind":"Plane"}]},"id":"cylinder-with-hole","quality":{"kind":"Analytic"},"revision":0,"schema_version":2,"solids":[{"cavity_shells":[],"outer_shell":0}],"topology":{"edges":[{"chart_seam":false,"geometry":{"curve":0,"kind":"Curve","range":{"hi":6.283185307179586,"lo":0.0}},"halfedge":0,"id":0,"tolerance":1e-8,"twin_halfedge":8},{"chart_seam":false,"geometry":{"curve":1,"kind":"Curve","range":{"hi":6.283185307179586,"lo":0.0}},"halfedge":2,"id":1,"tolerance":1e-8,"twin_halfedge":10},{"chart_seam":false,"geometry":{"curve":2,"kind":"Curve","range":{"hi":6.283185307179586,"lo":0.0}},"halfedge":4,"id":2,"tolerance":1e-8,"twin_halfedge":9},{"chart_seam":false,"geometry":{"curve":3,"kind":"Curve","range":{"hi":6.283185307179586,"lo":0.0}},"halfedge":6,"id":3,"tolerance":1e-8,"twin_halfedge":11},{"chart_seam":true,"geometry":{"curve":4,"kind":"Curve","range":{"hi":2.0,"lo":0.0}},"halfedge":1,"id":4,"tolerance":1e-8,"twin_halfedge":3},{"chart_seam":true,"geometry":{"curve":5,"kind":"Curve","range":{"hi":2.0,"lo":0.0}},"halfedge":5,"id":5,"tolerance":1e-8,"twin_halfedge":7}],"faces":[{"id":0,"key":"outer","provenance":{"reversed":false,"role":"Authored","sources":[{"body":"cylinder-with-hole","entity":"cylinder-with-hole","face":0,"key":"outer"}]},"sense":"Forward","shell_ref":0,"surface":0,"trim":{"chart":0,"holes":[],"outer":0,"uv_bounds":[{"hi":6.283185307179586,"lo":0.0},{"hi":2.0,"lo":0.0}]}},{"id":1,"key":"inner","provenance":{"reversed":false,"role":"Authored","sources":[{"body":"cylinder-with-hole","entity":"cylinder-with-hole","face":1,"key":"inner"}]},"sense":"Reverse","shell_ref":0,"surface":1,"trim":{"chart":0,"holes":[],"outer":1,"uv_bounds":[{"hi":6.283185307179586,"lo":0.0},{"hi":2.0,"lo":0.0}]}},{"id":2,"key":"lower_cap","provenance":{"reversed":false,"role":"Authored","sources":[{"body":"cylinder-with-hole","entity":"cylinder-with-hole","face":2,"key":"lower_cap"}]},"sense":"Forward","shell_ref":0,"surface":2,"trim":{"chart":0,"holes":[3],"outer":2,"uv_bounds":[{"hi":1.0,"lo":-1.0},{"hi":1.0,"lo":-1.0}]}},{"id":3,"key":"upper_cap","provenance":{"reversed":false,"role":"Authored","sources":[{"body":"cylinder-with-hole","entity":"cylinder-with-hole","face":3,"key":"upper_cap"}]},"sense":"Forward","shell_ref":0,"surface":3,"trim":{"chart":0,"holes":[5],"outer":4,"uv_bounds":[{"hi":1.0,"lo":-1.0},{"hi":1.0,"lo":-1.0}]}}],"halfedges":[{"edge":0,"face":0,"from":0,"geometry_use":{"pcurve":0,"periodic_lift":[0,0],"sense":"Forward"},"id":0,"loop_ref":0,"next":1,"prev":3,"to":0,"twin":8,"wire_ref":null},{"edge":4,"face":0,"from":0,"geometry_use":{"pcurve":1,"periodic_lift":[0,0],"sense":"Forward"},"id":1,"loop_ref":0,"next":2,"prev":0,"to":1,"twin":3,"wire_ref":null},{"edge":1,"face":0,"from":1,"geometry_use":{"pcurve":2,"periodic_lift":[0,0],"sense":"Reverse"},"id":2,"loop_ref":0,"next":3,"prev":1,"to":1,"twin":10,"wire_ref":null},{"edge":4,"face":0,"from":1,"geometry_use":{"pcurve":3,"periodic_lift":[0,0],"sense":"Reverse"},"id":3,"loop_ref":0,"next":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\ No newline at end of file
+{"accuracy":{"exchange":1e-6,"geometric":1e-8,"intersection":1e-9,"tessellation":0.01},"geometry":{"curves":[{"frame":{"origin":[0.0,0.0,0.0],"x":[1.0,0.0,0.0],"y":[0.0,0.0,-1.0],"z":[0.0,1.0,0.0]},"kind":"Circle","radius":1.0},{"frame":{"origin":[0.0,2.0,0.0],"x":[1.0,0.0,0.0],"y":[0.0,0.0,-1.0],"z":[0.0,1.0,0.0]},"kind":"Circle","radius":1.0},{"frame":{"origin":[0.0,0.0,0.0],"x":[1.0,0.0,0.0],"y":[0.0,0.0,-1.0],"z":[0.0,1.0,0.0]},"kind":"Circle","radius":0.4},{"frame":{"origin":[0.0,2.0,0.0],"x":[1.0,0.0,0.0],"y":[0.0,0.0,-1.0],"z":[0.0,1.0,0.0]},"kind":"Circle","radius":0.4},{"direction":[0.0,1.0,0.0],"kind":"Line","origin":[1.0,0.0,0.0]},{"direction":[0.0,1.0,0.0],"kind":"Line","origin":[0.4,0.0,0.0]}],"intersections":[],"pcurves":[{"direction":[1.0,0.0],"kind":"Line2","origin":[0.0,0.0]},{"direction":[0.0,1.0],"kind":"Line2","origin":[6.283185307179586,0.0]},{"direction":[1.0,0.0],"kind":"Line2","origin":[0.0,2.0]},{"direction":[0.0,1.0],"kind":"Line2","origin":[0.0,0.0]},{"direction":[1.0,0.0],"kind":"Line2","origin":[0.0,0.0]},{"direction":[0.0,1.0],"kind":"Line2","origin":[6.283185307179586,0.0]},{"direction":[1.0,0.0],"kind":"Line2","origin":[0.0,2.0]},{"direction":[0.0,1.0],"kind":"Line2","origin":[0.0,0.0]},{"axis_a":[1.0,0.0],"axis_b":[0.0,-1.0],"kind":"Conic2","origin":[0.0,0.0]},{"axis_a":[0.4,0.0],"axis_b":[0.0,-0.4],"kind":"Conic2","origin":[0.0,0.0]},{"axis_a":[1.0,0.0],"axis_b":[0.0,1.0],"kind":"Conic2","origin":[0.0,0.0]},{"axis_a":[0.4,0.0],"axis_b":[0.0,0.4],"kind":"Conic2","origin":[0.0,0.0]}],"surfaces":[{"frame":{"origin":[0.0,0.0,0.0],"x":[1.0,0.0,0.0],"y":[0.0,0.0,-1.0],"z":[0.0,1.0,0.0]},"kind":"Cylinder","radius":1.0},{"frame":{"origin":[0.0,0.0,0.0],"x":[1.0,0.0,0.0],"y":[0.0,0.0,-1.0],"z":[0.0,1.0,0.0]},"kind":"Cylinder","radius":0.4},{"frame":{"origin":[0.0,0.0,0.0],"x":[1.0,0.0,0.0],"y":[-0.0,-0.0,1.0],"z":[-0.0,-1.0,-0.0]},"kind":"Plane"},{"frame":{"origin":[0.0,2.0,0.0],"x":[1.0,0.0,0.0],"y":[0.0,0.0,-1.0],"z":[0.0,1.0,0.0]},"kind":"Plane"}]},"id":"cylinder-with-hole","quality":{"kind":"Analytic"},"revision":0,"schema_version":2,"solids":[{"cavity_shells":[],"outer_shell":0}],"topology":{"edges":[{"chart_seam":false,"geometry":{"curve":0,"kind":"Curve","range":{"hi":6.283185307179586,"lo":0.0}},"halfedge":0,"id":0,"tolerance":1e-8,"twin_halfedge":8},{"chart_seam":false,"geometry":{"curve":1,"kind":"Curve","range":{"hi":6.283185307179586,"lo":0.0}},"halfedge":2,"id":1,"tolerance":1e-8,"twin_halfedge":10},{"chart_seam":false,"geometry":{"curve":2,"kind":"Curve","range":{"hi":6.283185307179586,"lo":0.0}},"halfedge":4,"id":2,"tolerance":1e-8,"twin_halfedge":9},{"chart_seam":false,"geometry":{"curve":3,"kind":"Curve","range":{"hi":6.283185307179586,"lo":0.0}},"halfedge":6,"id":3,"tolerance":1e-8,"twin_halfedge":11},{"chart_seam":true,"geometry":{"curve":4,"kind":"Curve","range":{"hi":2.0,"lo":0.0}},"halfedge":1,"id":4,"tolerance":1e-8,"twin_halfedge":3},{"chart_seam":true,"geometry":{"curve":5,"kind":"Curve","range":{"hi":2.0,"lo":0.0}},"halfedge":5,"id":5,"tolerance":1e-8,"twin_halfedge":7}],"faces":[{"id":0,"key":"outer","provenance":{"reversed":false,"role":"Authored","sources":[{"body":"cylinder-with-hole","entity":"cylinder-with-hole","face":0,"key":"outer"}]},"sense":"Forward","shell_ref":0,"surface":0,"trim":{"chart":0,"holes":[],"outer":0,"uv_bounds":[{"hi":6.283185307179586,"lo":0.0},{"hi":2.0,"lo":0.0}]}},{"id":1,"key":"inner","provenance":{"reversed":false,"role":"Authored","sources":[{"body":"cylinder-with-hole","entity":"cylinder-with-hole","face":1,"key":"inner"}]},"sense":"Reverse","shell_ref":0,"surface":1,"trim":{"chart":0,"holes":[],"outer":1,"uv_bounds":[{"hi":6.283185307179586,"lo":0.0},{"hi":2.0,"lo":0.0}]}},{"id":2,"key":"lower_cap","provenance":{"reversed":false,"role":"Authored","sources":[{"body":"cylinder-with-hole","entity":"cylinder-with-hole","face":2,"key":"lower_cap"}]},"sense":"Forward","shell_ref":0,"surface":2,"trim":{"chart":0,"holes":[3],"outer":2,"uv_bounds":[{"hi":1.0,"lo":-1.0},{"hi":1.0,"lo":-1.0}]}},{"id":3,"key":"upper_cap","provenance":{"reversed":false,"role":"Authored","sources":[{"body":"cylinder-with-hole","entity":"cylinder-with-hole","face":3,"key":"upper_cap"}]},"sense":"Forward","shell_ref":0,"surface":3,"trim":{"chart":0,"holes":[5],"outer":4,"uv_bounds":[{"hi":1.0,"lo":-1.0},{"hi":1.0,"lo":-1.0}]}}],"halfedges":[{"edge":0,"face":0,"from":0,"geometry_use":{"pcurve":0,"periodic_lift":[0,0],"sense":"Forward"},"id":0,"loop_ref":0,"next":1,"prev":3,"to":0,"twin":8,"wire_ref":null},{"edge":4,"face":0,"from":0,"geometry_use":{"pcurve":1,"periodic_lift":[0,0],"sense":"Forward"},"id":1,"loop_ref":0,"next":2,"prev":0,"to":1,"twin":3,"wire_ref":null},{"edge":1,"face":0,"from":1,"geometry_use":{"pcurve":2,"periodic_lift":[0,0],"sense":"Reverse"},"id":2,"loop_ref":0,"next":3,"prev":1,"to":1,"twin":10,"wire_ref":null},{"edge":4,"face":0,"from":1,"geometry_use":{"pcurve":3,"periodic_lift":[0,0],"sense":"Reverse"},"id":3,"loop_ref":0,"next":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\ No newline at end of file
diff --git a/main/opengeometry/tests/primitives.rs b/main/opengeometry/tests/primitives.rs
index e6a03995..09203007 100644
--- a/main/opengeometry/tests/primitives.rs
+++ b/main/opengeometry/tests/primitives.rs
@@ -279,6 +279,8 @@ fn surface_kinds(body: &BrepEnvelope) -> Vec<&'static str> {
             SurfaceGeometry::Cylinder { .. } => "cylinder",
             SurfaceGeometry::Cone { .. } => "cone",
             SurfaceGeometry::Torus { .. } => "torus",
+            SurfaceGeometry::Wrapped { .. } => "wrapped",
+            SurfaceGeometry::Conoid { .. } => "conoid",
         })
         .collect::<Vec<_>>();
     kinds.sort();
diff --git a/main/opengeometry/tests/public_brep_builder.rs b/main/opengeometry/tests/public_brep_builder.rs
new file mode 100644
index 00000000..a8f8e15a
--- /dev/null
+++ b/main/opengeometry/tests/public_brep_builder.rs
@@ -0,0 +1,142 @@
+use opengeometry::brep::{
+    plane_boundary, Builder, CurveGeometry, Frame3, GeometryError, Orientation, SurfaceGeometry,
+};
+use opengeometry::exchange::export_step;
+use opengeometry::math::{Interval, Point3};
+use opengeometry::query::{classify_point, PointClassification};
+use opengeometry_test_support::accuracy::accuracy;
+use opengeometry_test_support::volume::estimate;
+
+const FACES: [(&str, [usize; 4], Point3); 6] = [
+    ("bottom", [0, 2, 3, 1], [0.0, 0.0, -1.0]),
+    ("top", [4, 5, 7, 6], [0.0, 0.0, 1.0]),
+    ("left", [0, 4, 6, 2], [-1.0, 0.0, 0.0]),
+    ("right", [1, 3, 7, 5], [1.0, 0.0, 0.0]),
+    ("front", [0, 1, 5, 4], [0.0, -1.0, 0.0]),
+    ("back", [2, 6, 7, 3], [0.0, 1.0, 0.0]),
+];
+
+fn sub(a: Point3, b: Point3) -> Point3 {
+    [a[0] - b[0], a[1] - b[1], a[2] - b[2]]
+}
+
+fn dot(a: Point3, b: Point3) -> f64 {
+    a[0] * b[0] + a[1] * b[1] + a[2] * b[2]
+}
+
+fn corners(size: Point3) -> Vec<Point3> {
+    (0..8)
+        .map(|i| {
+            std::array::from_fn(|axis| {
+                if i & (1 << axis) == 0 {
+                    0.0
+                } else {
+                    size[axis]
+                }
+            })
+        })
+        .collect()
+}
+
+fn box_edges(builder: &mut Builder, points: &[Point3], size: Point3) -> [[u32; 3]; 8] {
+    let axes = [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]];
+    let mut edges = [[u32::MAX; 3]; 8];
+    for i in 0..8 {
+        for axis in (0..3).filter(|axis| i & (1 << axis) == 0) {
+            let line = CurveGeometry::Line {
+                origin: points[i],
+                direction: axes[axis],
+            };
+            edges[i][axis] = builder.edge(line, Interval::new(0.0, size[axis]).unwrap(), false);
+        }
+    }
+    edges
+}
+
+fn box_builder(size: Point3) -> Result<(Builder, Vec<u32>), GeometryError> {
+    let mut builder = Builder::new("box".into(), accuracy(1e-6))?;
+    let points = corners(size);
+    for point in &points {
+        builder.vertex(*point);
+    }
+    let edges = box_edges(&mut builder, &points, size);
+    let mut faces = Vec::new();
+    for (key, corner, normal) in FACES {
+        let x = sub(points[corner[1]], points[corner[0]]);
+        let frame = Frame3::from_axis(points[corner[0]], normal, x)?;
+        let mut uses = Vec::new();
+        let mut uv = Vec::new();
+        for i in 0..4 {
+            let (from, to) = (corner[i], corner[(i + 1) % 4]);
+            let axis = (from ^ to).trailing_zeros() as usize;
+            let sense = if from < to {
+                Orientation::Forward
+            } else {
+                Orientation::Reverse
+            };
+            let edge = edges[from.min(to)][axis];
+            uses.push(plane_boundary(
+                &builder,
+                frame,
+                edge,
+                from as u32,
+                to as u32,
+                sense,
+            )?);
+            let local = sub(points[from], frame.origin);
+            uv.push([dot(local, frame.x), dot(local, frame.y)]);
+        }
+        let bounds = std::array::from_fn(|axis| {
+            let values = uv.iter().map(|point| point[axis]);
+            let lo = values.clone().fold(f64::INFINITY, f64::min);
+            [lo - 1e-7, values.fold(f64::NEG_INFINITY, f64::max) + 1e-7]
+        });
+        faces.push(builder.face(key, SurfaceGeometry::Plane { frame }, bounds, uses)?);
+    }
+    Ok((builder, faces))
+}
+
+#[test]
+fn a_box_built_through_the_public_builder_is_a_valid_exportable_solid() {
+    let (builder, faces) = box_builder([2.0, 3.0, 4.0]).unwrap();
+    assert_eq!(faces, [0, 1, 2, 3, 4, 5]);
+    assert_eq!(builder.brep().topology.faces.len(), 6);
+    let brep = builder.finish_solid().unwrap();
+    brep.validate().unwrap();
+    assert_eq!(brep.topology.edges.len(), 12);
+    assert_eq!(brep.topology.halfedges.len(), 24);
+    let volume = estimate(&brep, 1e-3);
+    assert!((volume.value - 24.0).abs() <= volume.error_bound.max(1e-9));
+    let inside = classify_point(&brep, [1.0, 1.5, 2.0]).unwrap();
+    assert_eq!(inside, PointClassification::Inside);
+    let outside = classify_point(&brep, [3.0, 1.5, 2.0]).unwrap();
+    assert_eq!(outside, PointClassification::Outside);
+    let (step, _) = export_step(&brep, "millimetre").unwrap();
+    assert_eq!(step.matches("=ADVANCED_FACE(").count(), 6);
+}
+
+#[test]
+fn a_face_sense_is_set_by_face_id() {
+    let (mut builder, faces) = box_builder([1.0, 1.0, 1.0]).unwrap();
+    let top = [0.5, 0.5, 1.0];
+    let normal = |builder: &Builder| builder.brep().face_normal_at(faces[1], top).unwrap();
+    assert!((normal(&builder)[2] - 1.0).abs() < 1e-12);
+    builder
+        .set_face_sense(faces[1], Orientation::Reverse)
+        .unwrap();
+    assert!((normal(&builder)[2] + 1.0).abs() < 1e-12);
+    assert!(matches!(
+        builder.set_face_sense(6, Orientation::Reverse),
+        Err(GeometryError::MissingReference { .. })
+    ));
+}
+
+#[test]
+fn an_edge_used_by_a_third_loop_is_rejected() {
+    let (mut builder, _) = box_builder([1.0, 1.0, 1.0]).unwrap();
+    let frame = Frame3::from_axis([0.0; 3], [0.0, 0.0, -1.0], [0.0, 1.0, 0.0]).unwrap();
+    let reuse = plane_boundary(&builder, frame, 0, 0, 1, Orientation::Forward).unwrap();
+    let plane = SurfaceGeometry::Plane { frame };
+    let result = builder.face("extra", plane, [[-1.0, 1.0], [-1.0, 1.0]], vec![reuse]);
+    assert!(matches!(result, Err(GeometryError::InvalidTopology(_))));
+}