SDL.Vect

Distance between two points in an affine space

Vector spaces have origins.

Compute the quadrance of the difference (the square of the distance)

An isomorphism between points and vectors, given a reference point.

Deserialize a linear type.

Serialize a linear type.

Matrix-scalar product

>>> V2 (V2 1 2) (V2 3 4) !!* 5V2 (V2 5 10) (V2 15 20)

Matrix-scalar division

Matrix * column vector

>>> V2 (V3 1 2 3) (V3 4 5 6) !* V3 7 8 9V2 50 122

Matrix product. This can compute any combination of sparse and dense multiplication.

>>> V2 (V3 1 2 3) (V3 4 5 6) !*! V3 (V2 1 2) (V2 3 4) (V2 4 5)V2 (V2 19 25) (V2 43 58)

>>> V2 (IntMap.fromList [(1,2)]) (IntMap.fromList [(2,3)]) !*! IntMap.fromList [(1,V3 0 0 1), (2, V3 0 0 5)]V2 (V3 0 0 2) (V3 0 0 15)

Entry-wise matrix addition.

>>> V2 (V3 1 2 3) (V3 4 5 6) !+! V2 (V3 7 8 9) (V3 1 2 3)V2 (V3 8 10 12) (V3 5 7 9)

Entry-wise matrix subtraction.

>>> V2 (V3 1 2 3) (V3 4 5 6) !-! V2 (V3 7 8 9) (V3 1 2 3)V2 (V3 (-6) (-6) (-6)) (V3 3 3 3)

Row vector * matrix

>>> V2 1 2 *! V2 (V3 3 4 5) (V3 6 7 8)V3 15 18 21

Scalar-matrix product

>>> 5 *!! V2 (V2 1 2) (V2 3 4)V2 (V2 5 10) (V2 15 20)

Extract a 2x2 matrix from a matrix of higher dimensions by dropping excess rows and columns.

Extract a 2x3 matrix from a matrix of higher dimensions by dropping excess rows and columns.

Extract a 2x4 matrix from a matrix of higher dimensions by dropping excess rows and columns.

Extract a 3x2 matrix from a matrix of higher dimensions by dropping excess rows and columns.

Extract a 3x3 matrix from a matrix of higher dimensions by dropping excess rows and columns.

Extract a 3x4 matrix from a matrix of higher dimensions by dropping excess rows and columns.

Extract a 4x2 matrix from a matrix of higher dimensions by dropping excess rows and columns.

Extract a 4x3 matrix from a matrix of higher dimensions by dropping excess rows and columns.

Extract a 4x4 matrix from a matrix of higher dimensions by dropping excess rows and columns.

Hermitian conjugate or conjugate transpose

>>> adjoint (V2 (V2 (1 :+ 2) (3 :+ 4)) (V2 (5 :+ 6) (7 :+ 8)))V2 (V2 (1.0 :+ (-2.0)) (5.0 :+ (-6.0))) (V2 (3.0 :+ (-4.0)) (7.0 :+ (-8.0)))

Solve a linear system with an upper-triangular matrix of coefficients with backwards substitution.

Solve a linear system with an upper-triangular matrix of coefficients with backwards substitution, using the vector's Finite instance to provide an index.

This is a generalization of inside to work over any corepresentable Functor.

column :: Representable f => Lens s t a b -> Lens (f s) (f t) (f a) (f b)

In practice it is used to access a column of a matrix.

>>> V2 (V3 1 2 3) (V3 4 5 6) ^._xV3 1 2 3

>>> V2 (V3 1 2 3) (V3 4 5 6) ^.column _xV2 1 4

2x2 matrix determinant.

>>> det22 (V2 (V2 a b) (V2 c d))a * d - b * c

3x3 matrix determinant.

>>> det33 (V3 (V3 a b c) (V3 d e f) (V3 g h i))a * (e * i - f * h) - d * (b * i - c * h) + g * (b * f - c * e)

4x4 matrix determinant.

Solve a linear system with a lower-triangular matrix of coefficients with forwards substitution.

Solve a linear system with a lower-triangular matrix of coefficients with forwards substitution, using the vector's Finite instance to provide an index.

Build a rotation matrix from a unit Quaternion.

The identity matrix for any dimension vector.

>>> identity :: M44 IntV4 (V4 1 0 0 0) (V4 0 1 0 0) (V4 0 0 1 0) (V4 0 0 0 1)>>> identity :: V3 (V3 Int)V3 (V3 1 0 0) (V3 0 1 0) (V3 0 0 1)

2x2 matrix inverse.

>>> inv22 $ V2 (V2 1 2) (V2 3 4)V2 (V2 (-2.0) 1.0) (V2 1.5 (-0.5))

3x3 matrix inverse.

>>> inv33 $ V3 (V3 1 2 4) (V3 4 2 2) (V3 1 1 1)V3 (V3 0.0 0.5 (-1.0)) (V3 (-0.5) (-0.75) 3.5) (V3 0.5 0.25 (-1.5))

4x4 matrix inverse.

Compute the (L, U) decomposition of a square matrix using Crout's algorithm. The Index of the vectors must be Integral.

Compute the determinant of a matrix using LU decomposition.

Compute the determinant of a matrix using LU decomposition, using the vector's Finite instance to provide an index.

Compute the (L, U) decomposition of a square matrix using Crout's algorithm, using the vector's Finite instance to provide an index.

Invert a matrix with LU decomposition.

Invert a matrix with LU decomposition, using the vector's Finite instance to provide an index.

Solve a linear system with LU decomposition.

Solve a linear system with LU decomposition, using the vector's Finite instance to provide an index.

Convert a 3x3 matrix to a 4x4 matrix extending it with 0's in the new row and column.

Convert from a 4x3 matrix to a 4x4 matrix, extending it with the [ 0 0 0 1 ] column vector

Build a transformation matrix from a rotation expressed as a Quaternion and a translation vector.

Build a transformation matrix from a rotation matrix and a translation vector.

Extract the translation vector (first three entries of the last column) from a 3x4 or 4x4 matrix.

transpose is just an alias for distribute

transpose (V3 (V2 1 2) (V2 3 4) (V2 5 6))

V2 (V3 1 3 5) (V3 2 4 6)

Normalize a Metric functor to have unit norm. This function does not change the functor if its norm is 0 or 1.

project u v computes the projection of v onto u.

Build a perspective matrix per the classic glFrustum arguments.

Build a matrix for a symmetric perspective-view frustum with a far plane at infinite

Build an inverse orthographic perspective matrix from 6 clipping planes

Build an inverse perspective matrix

Build a look at view matrix

Build an orthographic perspective matrix from 6 clipping planes. This matrix takes the region delimited by these planes and maps it to normalized device coordinates between [-1,1]

This call is designed to mimic the parameters to the OpenGL glOrtho call, so it has a slightly strange convention: Notably: the near and far planes are negated.

Consequently:

ortho l r b t n f !* V4 l b (-n) 1 = V4 (-1) (-1) (-1) 1
ortho l r b t n f !* V4 r t (-f) 1 = V4 1 1 1 1

Examples:

>>> ortho 1 2 3 4 5 6 !* V4 1 3 (-5) 1V4 (-1.0) (-1.0) (-1.0) 1.0

>>> ortho 1 2 3 4 5 6 !* V4 2 4 (-6) 1V4 1.0 1.0 1.0 1.0

Build a matrix for a symmetric perspective-view frustum

norm of the imaginary component

acosh with a specified branch cut.

acos with a specified branch cut.

asinh with a specified branch cut.

asin with a specified branch cut.

atanh with a specified branch cut.

atan with a specified branch cut.

axisAngle axis theta builds a Quaternion representing a rotation of theta radians about axis.

raise a Quaternion to a scalar power

Apply a rotation to a vector.

Spherical linear interpolation between two quaternions.

Compute the Frobenius norm of a matrix.

The Z-component of the cross product of two vectors in the XY-plane.

>>> crossZ (V2 1 0) (V2 0 1)1

the counter-clockwise perpendicular vector

>>> perp $ V2 10 20V2 (-20) 10

cross product

scalar triple product

Convert 4-dimensional projective coordinates to a 3-dimensional point. This operation may be denoted, euclidean [x:y:z:w] = (x/w, y/w, z/w) where the projective, homogenous, coordinate [x:y:z:w] is one of many associated with a single point (x/w, y/w, z/w).

Convert a 3-dimensional affine point into a 4-dimensional homogeneous vector, i.e. sets the w coordinate to 1.

Convert a 3-dimensional affine vector into a 4-dimensional homogeneous vector, i.e. sets the w coordinate to 0.

Compute the left scalar product

>>> 2 *^ V2 3 4V2 6 8

Compute the right scalar product

>>> V2 3 4 ^* 2V2 6 8

Compute division by a scalar on the right.

Produce a default basis for a vector space. If the dimensionality of the vector space is not statically known, see basisFor.

Produce a default basis for a vector space from which the argument is drawn.

Compute the negation of a vector

>>> negated (V2 2 4)V2 (-2) (-4)

Outer (tensor) product of two vectors

Produce a diagonal (scale) matrix from a vector.

>>> scaled (V2 2 3)V2 (V2 2 0) (V2 0 3)

Sum over multiple vectors

>>> sumV [V2 1 1, V2 3 4]V2 4 5

Create a unit vector.

>>> unit _x :: V2 IntV2 1 0

An affine space is roughly a vector space in which we have forgotten or at least pretend to have forgotten the origin.

a .+^ (b .-. a)  =  b@
(a .+^ u) .+^ v  =  a .+^ (u ^+^ v)@
(a .-. b) ^+^ v  =  (a .+^ v) .-. q@

A handy wrapper to help distinguish points from vectors at the type level

An associative unital algebra over a ring

A coassociative counital coalgebra over a ring

An involutive ring

Requires and provides a default definition such that

conjugate = id

Linear functionals from elements of an (infinite) free module to a scalar

Provides a fairly subjective test to see if a quantity is near zero.

>>> nearZero (1e-11 :: Double)False

>>> nearZero (1e-17 :: Double)True

>>> nearZero (1e-5 :: Float)False

>>> nearZero (1e-7 :: Float)True

A 2x2 matrix with row-major representation

A 2x3 matrix with row-major representation

A 2x4 matrix with row-major representation

A 3x2 matrix with row-major representation

A 3x3 matrix with row-major representation

A 3x4 matrix with row-major representation

A 4x2 matrix with row-major representation

A 4x3 matrix with row-major representation

A 4x4 matrix with row-major representation

Free and sparse inner product/metric spaces.

A vector space that includes the basis elements _e and _i

A vector space that includes the basis elements _e, _i, _j and _k

Quaternions

A 0-dimensional vector

>>> pure 1 :: V0 IntV0

>>> V0 + V0V0