diff --git a/main/opengeometry/src/tessellation/tests/edge_on_triangles.rs b/main/opengeometry/src/tessellation/tests/edge_on_triangles.rs
new file mode 100644
index 0000000..bedd70c
--- /dev/null
+++ b/main/opengeometry/src/tessellation/tests/edge_on_triangles.rs
@@ -0,0 +1,55 @@
+use super::super::mesh::Tessellation;
+use super::super::trimmed_face::general_curved_trimmed_face;
+use crate::brep::{fine_accuracy, Frame3, Surface, SurfaceGeometry};
+use crate::primitives;
+use std::collections::BTreeSet;
+
+const SLIVER: [[f64; 2]; 6] = [
+    [0.0, 0.4315522348698053],
+    [-2.4066236448435014e-16, 0.0],
+    [0.0970789271307487, 0.0],
+    [0.1941578542615101, 2.2648549702353193e-14],
+    [0.12598686852943008, 0.13848252444618137],
+    [0.061224770648327764, 0.2824284296054027],
+];
+
+#[test]
+fn an_ear_of_end_samples_rounded_off_their_line_turns_like_its_face() {
+    let body =
+        primitives::cylinder("c".into(), Frame3::IDENTITY, 3.35, 1.0, fine_accuracy()).unwrap();
+    let face = body
+        .topology
+        .faces
+        .iter()
+        .find(|face| {
+            matches!(
+                body.geometry.surfaces[face.surface as usize],
+                SurfaceGeometry::Cylinder { .. }
+            )
+        })
+        .unwrap();
+    let surface = body.geometry.surface(face.surface).unwrap();
+    let ring = SLIVER
+        .into_iter()
+        .map(|uv| {
+            let on_end = [uv[0], if uv[1].abs() < 1e-12 { 0.0 } else { uv[1] }];
+            (uv, surface.point_at(on_end).unwrap())
+        })
+        .collect::<Vec<_>>();
+    let mut mesh = Tessellation::new(0, 0.02);
+    general_curved_trimmed_face(&body, face, [0, 0], &[ring], 0.02, &mut mesh, 1000).unwrap();
+    let mut directed = BTreeSet::new();
+    for triangle in mesh.indices.chunks_exact(3) {
+        for edge in [
+            (triangle[0], triangle[1]),
+            (triangle[1], triangle[2]),
+            (triangle[2], triangle[0]),
+        ] {
+            assert!(
+                directed.insert(edge),
+                "edge {edge:?} runs the same way twice"
+            );
+        }
+    }
+    assert!(mesh.indices.len() >= 12);
+}
diff --git a/main/opengeometry/src/tessellation/tests/mod.rs b/main/opengeometry/src/tessellation/tests/mod.rs
index 404d549..a205550 100644
--- a/main/opengeometry/src/tessellation/tests/mod.rs
+++ b/main/opengeometry/src/tessellation/tests/mod.rs
@@ -1,4 +1,5 @@
 mod display;
+mod edge_on_triangles;
 mod seams;
 mod short_loops;
 mod smooth_edges;
diff --git a/main/opengeometry/src/tessellation/trimmed_face.rs b/main/opengeometry/src/tessellation/trimmed_face.rs
index acb8528..553edfb 100644
--- a/main/opengeometry/src/tessellation/trimmed_face.rs
+++ b/main/opengeometry/src/tessellation/trimmed_face.rs
@@ -5,7 +5,7 @@ use super::trim::{
 };
 use crate::brep::{BrepEnvelope, Face, GeometryError, Surface, SurfaceGeometry};
 use crate::geom2d::{point_in_ring2, Pt2};
-use crate::math::{scale, Interval, Point3};
+use crate::math::{cross, dot, norm, scale, sub, Interval, Point3};
 use std::collections::{BTreeSet, HashMap};
 
 const COUNT_ROUNDING: f64 = 1e-9;
@@ -194,7 +194,7 @@ fn shift_periodic_holes(
     Ok(())
 }
 
-fn general_curved_trimmed_face(
+pub(super) fn general_curved_trimmed_face(
     brep: &BrepEnvelope,
     face: &Face,
     _desired: [usize; 2],
@@ -252,17 +252,47 @@ fn general_curved_trimmed_face(
         let normal = trim_normal(brep, face, surface, *coordinate)?;
         mesh.vertex(point, normal)?;
     }
-    for triangle in triangles {
-        let center = std::array::from_fn(|axis| {
-            (uv[triangle[0]][axis] + uv[triangle[1]][axis] + uv[triangle[2]][axis]) / 3.0
-        });
+    let triangles = triangles
+        .into_iter()
+        .map(|triangle| {
+            (
+                triangle.map(|index| (start + index) as u32),
+                triangle.map(|index| uv[index]),
+            )
+        })
+        .collect::<Vec<_>>();
+    push_triangles_with_one_winding(brep, face, surface, &triangles, mesh, max_triangles)
+}
+
+fn push_triangles_with_one_winding(
+    brep: &BrepEnvelope,
+    face: &Face,
+    surface: &SurfaceGeometry,
+    triangles: &[([u32; 3], [[f64; 2]; 3])],
+    mesh: &mut Tessellation,
+    max_triangles: usize,
+) -> Result<(), GeometryError> {
+    let mut oriented = Vec::with_capacity(triangles.len());
+    let mut flip = None;
+    for (ids, coordinates) in triangles {
+        let center =
+            std::array::from_fn(|axis| coordinates.iter().map(|uv| uv[axis]).sum::<f64>() / 3.0);
         let normal = trim_normal(brep, face, surface, center)?;
-        mesh.triangle(
-            triangle.map(|index| (start + index) as u32),
-            face.id,
-            normal,
-            max_triangles,
-        )?;
+        let [a, b, c] = ids.map(|id| mesh.point(id));
+        let direction = cross(sub(b, a), sub(c, a));
+        let along = dot(direction, normal);
+        let edge_on = along.abs() <= 1e-6 * norm(direction);
+        if !edge_on && flip.is_none() {
+            flip = Some(along < 0.0);
+        }
+        oriented.push((*ids, normal, direction, edge_on));
+    }
+    for (ids, normal, direction, edge_on) in oriented {
+        let normal = match (edge_on, flip) {
+            (true, Some(flip)) => scale(direction, if flip { -1.0 } else { 1.0 }),
+            _ => normal,
+        };
+        mesh.triangle(ids, face.id, normal, max_triangles)?;
     }
     Ok(())
 }