[3.0] Generic Matrix Math

[otac0n]

This PR is a continuation of #2459 and supersedes it.

Status snapshot:

Vector structs:

• Implements IEquatable with itself as the generic parameter
• Implements IReadOnlyList with the components as the list elements
• Implements ISpanFormattable
• Implements ISpanParsable
• Implements IUtf8SpanFormattable
• Implements IUtf8SpanParsable
• Implements IParsable
• Implemetns IFormattable
• The relevant number of properties to represent the mathematical vector's components (X and Y for Vector2) and relevant unit vectors
• Constructors which take either a single parameter and uses it for every component, a parameter for each component, or a ReadOnlySpan of values which has the same number of elements as our vector has components.
• Constructors for 3 dimensions and up must include lower dimension variants that use the lower dimensions for their specific components (vector2 -> X,Y).
• A ref indexer that takes a int index and returns the corresponding component value (0 -> x, 1 -> y, etc.)
• An AsSpan function which returns this vector as a Span of the generic type
• A LengthSquared extension property which returns the dot product of the vector with itself.
• A Length extension property which returns the magnitued of the vector when the type T defines IRootFunctions<T>.
• A Dot static method which takes another vector and returns the dot product with our original vector.
• For 3D Vectors, a Cross static method which takes another vector and returns the cross product with our original vector.
• +, -, *, /, and % operators defined between two vectors of the same type which returns a vector which has had each operation applied component-wise.
• +, -, *, /, and % operators defined between a vector and a scalar value that matches the generic type which returns a vector which has had each operation applied component-wise with the scalar value. Both vector first and scalar first should be implemented.
• A - unary operator which returns the negated vector.
• A + unary operator which returns the vector.
• Overrides ToString to show component values.
• CopyTo functions which copy to an array or span, with or without a starting index
• Explicit cast and checked cast operators to all builtin type parameter variants of the vector types of the same dimensionality. (e.g. from Vector2D<T> to Vector2D<float>)
• Explicit cast and checked cast to and from matching System.Numerics vector type
• Explicit cast cast to lower dimensional matching vector with matching generic type
• Static Unit Vectors for each component
• A Static Zero Vector with zero for all components
• A Static One Vector with one for all components
• A static AllBitsSet Vector with all bits set for all components
• A deconstruct method for detupling
• An implicit conversion from a value tuple of the same size.
• Define static CreateChecked, CreateSaturating, and CreateTruncating which converts other vector types to this type
  • Try variants of these methods should also be defined which out the resulting vector and return a bool representing success or failure of the operation.
• Define Transform static methods which take a Matrix of higher dimensionality assuming 1 in for the final missing component and 0 for the rest (Vector 2 can use Matrix2Xn, Matrix3Xn, and Matrix4Xn) and return a vector containing the output (type should match the outer type e.g. Vector2D.Transform(Matrix4X4) returns Vector2D)
• Define VectorND<T> * MatrixNXM operators where N is the same for both Vector and Matrix, but M is any number
  • These operators should function like Transform, but without needed assumptions
• Define TransformNormal static methods which take a Matrix of higher dimensionality assuming 0 in for all missing components (Vector 2 can use Matrix2Xn, Matrix3Xn, and Matrix4Xn) and return a vector containing the output (type should match the outer type e.g. Vector2D.Transform(Matrix4X4) returns Vector2D)
• For types implementing IBinaryNumber
  • BitwiseAnd, BitwiseOr, and BitwiseXor static methods defined between two vectors which returns a vector which has had these operators applied on a component-wise basis.
  • BitwiseAnd, BitwiseOr, and BitwiseXor static methods operators defined between a vectors and a scalar value that matches the generic type which returns a vector which has had these operators applied on a component-wise basis with the scalar.
  • BitwiseNot static method defined which negates the bits of the vector components. (BitwiseComplement?)
• A Normalize extension method which divides all components by the length of the vector, when T implements IRootFunctions<T>
• A Reflect static method which takes a normal vector and reflects the vector over the normal
• The following static Vector properties which have the given value for all components
  • PositiveInfinity
  • NegativeInfinity
  • NaN
  • Epsilon
  • NegativeZero
  • Pi
  • Tau
  • E
• Define the following static methods for these types which calls the specified function and returns a new vector with the specified multiplicity:
  • INumber<>.Sign, (Memberwise)
    • Returns VectorND<int>, where N matches the dimensionality of the vector
  • INumber<>.Max, (Memberwise, Memberwise)
  • INumber<>.Max, (Memberwise, Scalar)
  • INumber<>.MaxNumber, (Memberwise, Memberwise)
  • INumber<>.MaxNumber, (Memberwise, Scalar)
  • INumber<>.Min, (Memberwise, Memberwise)
  • INumber<>.Min, (Memberwise, Scalar)
  • INumber<>.MinNumber, (Memberwise, Memberwise)
  • INumber<>.MinNumber, (Memberwise, Scalar)
  • INumber<>.Clamp, (Memberwise, Memberwise, Memberwise)
  • INumber<>.Clamp, (Memberwise, Scalar, Scalar)
  • INumber<>.CopySign, (Memberwise, Memberwise)
  • INumber<>.CopySign, (Memberwise, Scalar)
  • INumberBase<>.Abs, (Memberwise)
  • INumberBase<>.MaxMagnitude, (Memberwise, Memberwise)
  • INumberBase<>.MaxMagnitudeNumber, (Memberwise, Memberwise)
  • INumberBase<>.MinMagnitude, (Memberwise, Memberwise)
  • INumberBase<>.MinMagnitudeNumber, (Memberwise, Memberwise)
  • INumberBase<>.MultiplyAddEstimate, (Memberwise, Memberwise, Memberwise)
  • INumberBase<>.MultiplyAddEstimate, (Memberwise, Memberwise, Scalar)
  • INumberBase<>.MultiplyAddEstimate, (Memberwise, Scalar, Memberwise)
  • INumberBase<>.MultiplyAddEstimate, (Memberwise, Scalar, Scalar)
  • Log2, (Memberwise)
  • IBinaryInteger<>.DivRem, (Memberwise)
    • Returns tuple of 2 Vectors (Vector Quotient, Vector Remainder)
  • IBinaryInteger<>.PopCount, (Memberwise)
  • IBinaryInteger<>.TrailingZeroCount, (Memberwise)
  • IFloatingPoint<>.Ceiling, (Memberwise)
  • IFloatingPoint<>.Floor, (Memberwise)
  • IFloatingPoint<>.Round, (Memberwise)
  • IFloatingPoint<>.Round, (Memberwise, Scalar) *int digits
  • IFloatingPoint<>.Round, (Memberwise, Scalar) *MidpointRounding mode
  • IFloatingPoint<>.Round, (Memberwise, Scalar, Scalar) *int digits, MidpointRounding mode
  • IFloatingPoint<>.Truncate, (Memberwise)
  • IFloatingPointIeee754<>.Atan2, (Memberwise, Memberwise)
  • IFloatingPointIeee754<>.Atan2Pi, (Memberwise, Memberwise)
  • IFloatingPointIeee754<>.Lerp, (Memberwise, Memberwise, Memberwise)
  • IFloatingPointIeee754<>.Lerp, (Memberwise, Memberwise, Scalar)
  • IFloatingPointIeee754<>.BitDecrement, (Memberwise)
  • IFloatingPointIeee754<>.BitIncrement, (Memberwise)
  • IFloatingPointIeee754<>.FusedMultiplyAdd, (Memberwise, Memberwise, Memberwise)
  • IFloatingPointIeee754<>.FusedMultiplyAdd, (Memberwise, Memberwise, Scalar)
  • IFloatingPointIeee754<>.FusedMultiplyAdd, (Memberwise, Scalar, Memberwise)
  • IFloatingPointIeee754<>.FusedMultiplyAdd, (Memberwise, Scalar, Scalar)
  • IFloatingPointIeee754<>.Ieee754Remainder, (Memberwise, Memberwise)
  • IFloatingPointIeee754<>.Ieee754Remainder, (Memberwise, Scalar)
  • IFloatingPointIeee754<>.ILogB, (Memberwise)
    • Returns VectorND<int>, where N matches the dimensionality of the vector
  • IFloatingPointIeee754<>.ReciprocalEstimate, (Memberwise)
  • IFloatingPointIeee754<>.ReciprocalSqrtEstimate, (Memberwise)
  • IFloatingPointIeee754<>.ScaleB, (Memberwise, Memberwise)
  • IFloatingPointIeee754<>.ScaleB, (Memberwise, Scalar)
  • IPowerFunctions<>.Pow, (Memberwise, Memberwise)
  • IPowerFunctions<>.Pow, (Memberwise, Scalar)
  • IRootFunctions<>.Cbrt, (Memberwise)
  • IRootFunctions<>.Sqrt, (Memberwise)
  • IRootFunctions<>.RootN, (Memberwise, Memberwise)
  • IRootFunctions<>.RootN, (Memberwise, Scalar)
  • IRootFunctions<>.Hypot, (Memberwise, Memberwise)
  • IRootFunctions<>.Hypot, (Memberwise, Scalar)
  • ILogarithmicFunctions<>.Log, (Memberwise)
  • ILogarithmicFunctions<>.Log, (Memberwise, Memberwise)
  • ILogarithmicFunctions<>.Log, (Memberwise, Scalar)
  • ILogarithmicFunctions<>.LogP1, (Memberwise)
  • ILogarithmicFunctions<>.Log2P1, (Memberwise)
  • ILogarithmicFunctions<>.Log10, (Memberwise)
  • ILogarithmicFunctions<>.Log10P1, (Memberwise)
  • IExponentialFunctions<>.Exp, (Memberwise)
  • IExponentialFunctions<>.ExpM1, (Memberwise)
  • IExponentialFunctions<>.Exp2, (Memberwise)
  • IExponentialFunctions<>.Exp2M1, (Memberwise)
  • IExponentialFunctions<>.Exp10, (Memberwise)
  • IExponentialFunctions<>.Exp10M1, (Memberwise)
  • ITrigonometricFunctions<>.Acos, (Memberwise)
  • ITrigonometricFunctions<>.AcosPi, (Memberwise)
  • ITrigonometricFunctions<>.Asin, (Memberwise)
  • ITrigonometricFunctions<>.AsinPi, (Memberwise)
  • ITrigonometricFunctions<>.Atan, (Memberwise)
  • ITrigonometricFunctions<>.AtanPi, (Memberwise)
  • ITrigonometricFunctions<>.Cos, (Memberwise)
  • ITrigonometricFunctions<>.CosPi, (Memberwise)
  • ITrigonometricFunctions<>.Sin, (Memberwise)
  • ITrigonometricFunctions<>.SinPi, (Memberwise)
  • ITrigonometricFunctions<>.SinCos, (Memberwise)
    • Returns a tuple of 2 Vectors (Sin, Cos)
  • ITrigonometricFunctions<>.SinCosPi, (Memberwise)
    • Returns a tuple of 2 Vectors (Sin, Cos)
  • ITrigonometricFunctions<>.Tan, (Memberwise)
  • ITrigonometricFunctions<>.TanPi, (Memberwise)
  • ITrigonometricFunctions<>.DegreesToRadians, (Memberwise)
  • ITrigonometricFunctions<>.RadiansToDegrees, (Memberwise)
  • IHyperbolicFunctions<>.Acosh, (Memberwise)
  • IHyperbolicFunctions<>.Asinh, (Memberwise)
  • IHyperbolicFunctions<>.Atanh, (Memberwise)
  • IHyperbolicFunctions<>.Cosh, (Memberwise)
  • IHyperbolicFunctions<>.Sinh, (Memberwise)
  • IHyperbolicFunctions<>.Tanh, (Memberwise)
  • RoundToInt(Vector x)
    • Returns VectorND<int>, where N matches the dimensionality of the vector
    • INFORMATIVE This may require multiple methods depending on implementation
  • FloorToInt(Vector x)
    • Returns VectorND<int>, where N matches the dimensionality of the vector
    • INFORMATIVE This may require multiple methods depending on implementation
  • CeilingToInt(Vector x)
    • Returns VectorND<int>, where N matches the dimensionality of the vector
    • INFORMATIVE This may require multiple methods depending on implementation
  • ToVector64(Vector x)
    • Returns System.Runtime.Intrinsics.Vector64<TScalar>
  • ToVector128(Vector x)
    • Returns System.Runtime.Intrinsics.Vector128<TScalar>
  • ToVector256(Vector x)
    • Returns System.Runtime.Intrinsics.Vector256<TScalar>
  • ToVector512(Vector x)
    • Returns System.Runtime.Intrinsics.Vector512<TScalar>

Matrix structs must fulfill the following requirements:

• Fulfills IEquatable<T> where T is the same matrix class
• Stored in row major format
• Both row vectors and individual values (M11, etc.) accessible via properties
• A ref indexer that takes row and column indicies and outputs the value
• Add, subtract, and multiply operators defined with Matricies of the same size
• Multiply operators defined with compatible matricies, if the output matrix type already exists (AxB * BxC = AxC)
• Negate Operator defined
• Implicit conversion to and from the System.Numerics matrix type, if available
• Invert extension method for square matricies
• GetDeterminant extension method for square matricies and Matrix3X2, Matrix4X3, and Matrix5X4
• Transpose extension method
• Lerp static method
• static identity property for square matrices
• For Matrix3X2, Matrix3X3, Matrix4X3, and Matrix4X4 include the following static functions
  • CreateBillboardRH
  • CreateBillboardLH
  • CreateRotation
    • 3x3, 4x3, and 4x4 Matricies get X, Y, and Z variants for this function instead
  • CreateTranslation
  • CreateScale
  • Decompose (separate out any transformations)
• For Matrix3X2 include a CreateSkew static function
• For Matrix3X3, Matrix4X3, and Matrix4X4 include the following static functions
  • CreateFromAxisAngle
  • CreateFromQuaternion
  • Transform
    • from a Quaternion
  • CreateFromYawPitchRoll
• For Matrix4X3 and Matrix4X4 include the following static functions
  • CreateConstrainedBillboardLH
  • CreateConstrainedBillboardRH
  • CreateLookAtLH
  • CreateLookToLH
  • CreateOrthographicLH
  • CreateOrthographicOffCenterLH
  • CreatePerspectiveLH
  • CreatePerspectiveFieldOfViewLH
  • CreatePerspectiveOffCenterLH
  • CreateLookAtRH
  • CreateLookToRH
  • CreateOrthographicRH
  • CreateOrthographicOffCenterRH
  • CreatePerspectiveRH
  • CreatePerspectiveFieldOfViewRH
  • CreatePerspectiveOffCenterRH
  • CreateReflection
  • CreateWorld
  • CreateViewport

Several of these functions exist, but not for all types or not for both LH/RH versions.

A Quaternion struct:

• A generic struct with a type parameter T which is constrained by IBinaryFloatingPointIeee754<T> representing the scalar type
• Implements IEquatable with itself
• Contain 4 scalar properties (X, Y, Z, W)
• Define a Constructor taking 4 scalar values matching the properties
• Define a Constructor taking a Vector3D<T> and a Scalar, with the vector 3 mapping to X, Y, Z and the Scalar to the W
• Define a Constructor taking a Vector4D<T>
• A Vector3D<T> Axis property mapping to (X, Y, Z)
• A T Angle property mapping to 2 * Acos(W)
• A ref Indexer which takes an int and returns the components in order
• An AsSpan function which returns this quaternion as a Span of the generic type
• An IsIdentity property which returns if this Quaternion matches an Identity Quaternion
• Define +, -, *, and / between two Quaternions
• Define * with T multiplying each component by the scalar value returning a new quaternion
• Define unary ~
• A Dot function which takes another Quaternion and returns its the dotproduct between them
  • A static implementation of this function must be available
• A LengthSquared property which returns the dot product of the quaternion with itself
• A Length property which returns the Square Root of LengthSquared
• An Invert function inverts the Quaternion
  • a static Inverse function must be available but it returns the inverse rather than affecting the original
• A Normalize function which normalizes the Quaternion
  • A static implemenation of this function must be available but returns the normalized Quaternion rather than affecting the original
• A Concatenate function which takes another Quaternion and concatenates it with this quaternion
  • A static implementation of this function must be available but it returns a new Quaternion rather than affecting the originals
• A Conjugate function which returns the conjugate of this quaternion
  • A static implementation of this function must be available
• A static CreateFromAxisAngle function which takes in a Vector3D<T> and an angle and returns a Quaternion representing that rotation
• A static CreateFromRotationMatrix function which takes either a Matrix3X3 or Matrix4X4 and returns a Quaternion representing that rotation
• A static CreateFromYawPitchRoll which takes either each components separately or in a Vector3D<T> and outputs a Quaternion representing that rotation
• A static Lerp function which takes 2 Quaternions and a Scalar matching the generic type which linearly interpolates between the 2 Quaternions with scalar used as the amount to lerp
• A static SLerp function which takes 2 Quaternions and a Scalar matching the generic type which Spherical linearly interpolates between the 2 Quaternions with scalar used as the amount to lerp
• A static Zero Quaternion Property
• A static Identity Quaternion Property

Geometric Types:

• Box2
• Box3
• Circle
• Cube
• Plane
• Ray2
• Ray3
• Rectangle
• Sphere

Geometric Types details:

• Intersect functions with both another instance of the type and a point
• GetDistanceToNearest(Point,Edge,etc) functions must be available for a given point
• For all but the rays and planes, GetInflated function that takes a point and returns the scaled object that is closest to the original and contains the given point
• Include Scale and Translation transformation functions
• For Box and Rectangle the following Vector properties must be defined
  • Min
  • Max
  • Center
  • Size
• For Planes and Rays, Normalize functions
• For Planes include the following static functions
  • CreateFromVerticies
  • CreateFromPointNormal
  • Dot
    • with a Vector4
  • DotCoordinate and DotNormal
    • with a Vector3
  • Transform
    • With a Matrix4X4 or Quaternion, if relevant