#include "ImathEuler.h"
#include "ImathExport.h"
#include "ImathMatrix.h"
#include "ImathNamespace.h"
#include "ImathQuat.h"
#include "ImathVec.h"
#include <math.h>

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Functions
template<class T >
bool	extractScaling (const Matrix44< T > &mat, Vec3< T > &scl, bool exc=true)

template<class T >
Matrix44< T >	sansScaling (const Matrix44< T > &mat, bool exc=true)

template<class T >
bool	removeScaling (Matrix44< T > &mat, bool exc=true)

template<class T >
bool	extractScalingAndShear (const Matrix44< T > &mat, Vec3< T > &scl, Vec3< T > &shr, bool exc=true)

template<class T >
Matrix44< T >	sansScalingAndShear (const Matrix44< T > &mat, bool exc=true)

template<class T >
void	sansScalingAndShear (Matrix44< T > &result, const Matrix44< T > &mat, bool exc=true)

template<class T >
bool	removeScalingAndShear (Matrix44< T > &mat, bool exc=true)
	Remove scaling and shear from the given 4x4 matrix in place. More...

template<class T >
bool	extractAndRemoveScalingAndShear (Matrix44< T > &mat, Vec3< T > &scl, Vec3< T > &shr, bool exc=true)

template<class T >
void	extractEulerXYZ (const Matrix44< T > &mat, Vec3< T > &rot)

template<class T >
void	extractEulerZYX (const Matrix44< T > &mat, Vec3< T > &rot)

template<class T >
Quat< T >	extractQuat (const Matrix44< T > &mat)

template<class T >
bool	extractSHRT (const Matrix44< T > &mat, Vec3< T > &s, Vec3< T > &h, Vec3< T > &r, Vec3< T > &t, bool exc, typename Euler< T >::Order rOrder)

template<class T >
bool	extractSHRT (const Matrix44< T > &mat, Vec3< T > &s, Vec3< T > &h, Vec3< T > &r, Vec3< T > &t, bool exc=true)

template<class T >
bool	extractSHRT (const Matrix44< T > &mat, Vec3< T > &s, Vec3< T > &h, Euler< T > &r, Vec3< T > &t, bool exc=true)

template<class T >
bool	checkForZeroScaleInRow (const T &scl, const Vec3< T > &row, bool exc=true)

template<class T >
Matrix44< T >	outerProduct (const Vec4< T > &a, const Vec4< T > &b)
	Return the 4x4 outer product two 4-vectors. More...

template<class T >
Matrix44< T >	rotationMatrix (const Vec3< T > &fromDirection, const Vec3< T > &toDirection)

template<class T >
Matrix44< T >	rotationMatrixWithUpDir (const Vec3< T > &fromDir, const Vec3< T > &toDir, const Vec3< T > &upDir)

template<class T >
void	alignZAxisWithTargetDir (Matrix44< T > &result, Vec3< T > targetDir, Vec3< T > upDir)

template<class T >
Matrix44< T >	computeLocalFrame (const Vec3< T > &p, const Vec3< T > &xDir, const Vec3< T > &normal)

template<class T >
Matrix44< T >	addOffset (const Matrix44< T > &inMat, const Vec3< T > &tOffset, const Vec3< T > &rOffset, const Vec3< T > &sOffset, const Vec3< T > &ref)

template<class T >
Matrix44< T >	computeRSMatrix (bool keepRotateA, bool keepScaleA, const Matrix44< T > &A, const Matrix44< T > &B)

template<class T >
bool	extractScaling (const Matrix33< T > &mat, Vec2< T > &scl, bool exc=true)

template<class T >
Matrix33< T >	sansScaling (const Matrix33< T > &mat, bool exc=true)

template<class T >
bool	removeScaling (Matrix33< T > &mat, bool exc=true)

template<class T >
bool	extractScalingAndShear (const Matrix33< T > &mat, Vec2< T > &scl, T &shr, bool exc=true)

template<class T >
Matrix33< T >	sansScalingAndShear (const Matrix33< T > &mat, bool exc=true)

template<class T >
bool	removeScalingAndShear (Matrix33< T > &mat, bool exc=true)
	Remove scaling and shear from the given 3x3e matrix in place. More...

template<class T >
bool	extractAndRemoveScalingAndShear (Matrix33< T > &mat, Vec2< T > &scl, T &shr, bool exc=true)

template<class T >
void	extractEuler (const Matrix22< T > &mat, T &rot)

template<class T >
void	extractEuler (const Matrix33< T > &mat, T &rot)

template<class T >
bool	extractSHRT (const Matrix33< T > &mat, Vec2< T > &s, T &h, T &r, Vec2< T > &t, bool exc=true)

template<class T >
bool	checkForZeroScaleInRow (const T &scl, const Vec2< T > &row, bool exc=true)

template<class T >
Matrix33< T >	outerProduct (const Vec3< T > &a, const Vec3< T > &b)
	Return the 3xe outer product two 3-vectors. More...

template<class T >
Matrix44< T >	addOffset (const Matrix44< T > &inMat, const Vec3< T > &tOffset, const Vec3< T > &rOffset, const Vec3< T > &sOffset, const Matrix44< T > &ref)

template<typename T >
M44d	procrustesRotationAndTranslation (const Vec3< T > A, const Vec3< T > B, const T *weights, const size_t numPoints, const bool doScaling=false)

template<typename T >
M44d	procrustesRotationAndTranslation (const Vec3< T > A, const Vec3< T > B, const size_t numPoints, const bool doScaling=false)

template<typename T >
void	jacobiSVD (const Matrix33< T > &A, Matrix33< T > &U, Vec3< T > &S, Matrix33< T > &V, const T tol=std::numeric_limits< T >::epsilon(), const bool forcePositiveDeterminant=false)

template<typename T >
void	jacobiSVD (const Matrix44< T > &A, Matrix44< T > &U, Vec4< T > &S, Matrix44< T > &V, const T tol=std::numeric_limits< T >::epsilon(), const bool forcePositiveDeterminant=false)

template<typename T >
void	jacobiEigenSolver (Matrix33< T > &A, Vec3< T > &S, Matrix33< T > &V, const T tol)

template<typename T >
void	jacobiEigenSolver (Matrix33< T > &A, Vec3< T > &S, Matrix33< T > &V)

template<typename T >
void	jacobiEigenSolver (Matrix44< T > &A, Vec4< T > &S, Matrix44< T > &V, const T tol)

template<typename T >
void	jacobiEigenSolver (Matrix44< T > &A, Vec4< T > &S, Matrix44< T > &V)

template<typename TM , typename TV >
void	maxEigenVector (TM &A, TV &S)

template<typename TM , typename TV >
void	minEigenVector (TM &A, TV &S)

Variables
IMATH_INTERNAL_NAMESPACE_HEADER_ENTER IMATH_EXPORT_CONST M22f	identity22f
	M22f identity matrix. More...

IMATH_EXPORT_CONST M33f	identity33f
	M33f identity matrix. More...

IMATH_EXPORT_CONST M44f	identity44f
	M44f identity matrix. More...

IMATH_EXPORT_CONST M22d	identity22d
	M22d identity matrix. More...

IMATH_EXPORT_CONST M33d	identity33d
	M33d identity matrix. More...

IMATH_EXPORT_CONST M44d	identity44d
	M44d identity matrix. More...

Function Documentation

template<class T >

Matrix44<T> addOffset	(	const Matrix44< T > &	inMat,
		const Vec3< T > &	tOffset,
		const Vec3< T > &	rOffset,
		const Vec3< T > &	sOffset,
		const Vec3< T > &	ref
	)

Add a translate/rotate/scale offset to a 4x4 input frame and put it in another frame of reference

Parameters

[in]	inMat	Input frame
[in]	tOffset	Translation offset
[in]	rOffset	Rotation offset in degrees
[in]	sOffset	Scale offset
[in]	ref	Frame of reference

Returns: The offsetted frame

template<class T >

Matrix44<T> addOffset	(	const Matrix44< T > &	inMat,
		const Vec3< T > &	tOffset,
		const Vec3< T > &	rOffset,
		const Vec3< T > &	sOffset,
		const Matrix44< T > &	ref
	)

Add a translate/rotate/scale offset to an input frame and put it in another frame of reference.

Parameters

inMat	input frame
tOffset	translate offset
rOffset	rotate offset in degrees
sOffset	scale offset
ref	Frame of reference

Returns: The offsetted frame

Definition at line 1027 of file ImathMatrixAlgo.h.

template<class T >

void alignZAxisWithTargetDir	(	Matrix44< T > &	result,
		Vec3< T >	targetDir,
		Vec3< T >	upDir
	)

Construct a 4x4 matrix that rotates the z-axis so that it points towards targetDir. You must also specify that you want the up vector to be pointing in a certain direction upDir.

Notes: The following degenerate cases are handled: (a) when the directions given by toDir and upDir are parallel or opposite (the direction vectors must have a non-zero cross product); (b) when any of the given direction vectors have zero length

Parameters

[out]	result	The output matrix
[in]	targetDir	The target direction vector
[in]	upDir	The up direction vector

Definition at line 908 of file ImathMatrixAlgo.h.

template<class T >

bool checkForZeroScaleInRow	(	const T &	scl,
		const Vec3< T > &	row,
		bool	exc = `true`
	)

Return true if the given scale can be removed from the given row matrix, false if scl is small enough that the operation would overflow. If exc is true, throw an exception on overflow.

Definition at line 828 of file ImathMatrixAlgo.h.

template<class T >

bool checkForZeroScaleInRow	(	const T &	scl,
		const Vec2< T > &	row,
		bool	exc = `true`
	)

Return true if the given scale can be removed from the given row matrix, false if scl is small enough that the operation would overflow. If exc is true, throw an exception on overflow.

template<class T >

Matrix44< T > computeLocalFrame	(	const Vec3< T > &	p,
		const Vec3< T > &	xDir,
		const Vec3< T > &	normal
	)

Compute an orthonormal direct 4x4 frame from a position, an x axis direction and a normal to the y axis. If the x axis and normal are perpendicular, then the normal will have the same direction as the z axis.

Parameters

[in]	p	The position of the frame
[in]	xDir	The x axis direction of the frame
[in]	normal	A normal to the y axis of the frame

Returns: The orthonormal frame

Definition at line 985 of file ImathMatrixAlgo.h.

template<class T >

Matrix44< T > computeRSMatrix	(	bool	keepRotateA,
		bool	keepScaleA,
		const Matrix44< T > &	A,
		const Matrix44< T > &	B
	)

Compute 4x4 translate/rotate/scale matrix from A with the rotate/scale of B.

Parameters

[in]	keepRotateA	If true, keep rotate from matrix `A`, use `B` otherwise
[in]	keepScaleA	If true, keep scale from matrix `A`, use `B` otherwise
[in]	A	Matrix A
[in]	B	Matrix B

Returns: Matrix A with tweaked rotation/scale

Definition at line 1060 of file ImathMatrixAlgo.h.

template<class T >

bool extractAndRemoveScalingAndShear	(	Matrix44< T > &	mat,
		Vec3< T > &	scl,
		Vec3< T > &	shr,
		bool	exc = `true`
	)

Remove scaling and shear from the given 4x4 matrix in place, returning the extracted values.

Parameters

[in,out]	mat	The matrix to operate on
[out]	scl	The extracted scale
[out]	shr	The extracted shear
[in]	exc	If true, throw an exception if the scaling in `mat` is very close to zero.

Returns: True if the scale could be extracted, false if the matrix is degenerate.

Definition at line 523 of file ImathMatrixAlgo.h.

template<class T >

bool extractAndRemoveScalingAndShear	(	Matrix33< T > &	mat,
		Vec2< T > &	scl,
		T &	shr,
		bool	exc = `true`
	)

Remove scaling and shear from the given 3x3 matrix in place, returning the extracted values.

Parameters

[in,out]	mat	The matrix to operate on
[out]	scl	The extracted scale
[out]	shr	The extracted shear
[in]	exc	If true, throw an exception if the scaling in `mat` is very close to zero.

Returns: True if the scale could be extracted, false if the matrix is degenerate.

Definition at line 1182 of file ImathMatrixAlgo.h.

template<class T >

void extractEuler	(	const Matrix22< T > &	mat,
		T &	rot
	)

Extract the rotation from the given 2x2 matrix

Parameters

[in]	mat	The input matrix
[out]	rot	The extracted rotation value

Definition at line 1269 of file ImathMatrixAlgo.h.

template<class T >

void extractEuler	(	const Matrix33< T > &	mat,
		T &	rot
	)

Extract the rotation from the given 3x3 matrix

Parameters

[in]	mat	The input matrix
[out]	rot	The extracted rotation value

Definition at line 1290 of file ImathMatrixAlgo.h.

template<class T >

void extractEulerXYZ	(	const Matrix44< T > &	mat,
		Vec3< T > &	rot
	)

Extract the rotation from the given 4x4 matrix in the form of XYZ euler angles.

Parameters

[in]	mat	The input matrix
[out]	rot	The extracted XYZ euler angle vector

Definition at line 636 of file ImathMatrixAlgo.h.

template<class T >

void extractEulerZYX	(	const Matrix44< T > &	mat,
		Vec3< T > &	rot
	)

Extract the rotation from the given 4x4 matrix in the form of ZYX euler angles.

Parameters

[in]	mat	The input matrix
[out]	rot	The extracted ZYX euler angle vector

Definition at line 679 of file ImathMatrixAlgo.h.

template<class T >

Quat< T > extractQuat ( const Matrix44< T > & mat )

Extract the rotation from the given 4x4 matrix in the form of a quaternion.

Parameters

[in] mat The input matrix

Returns: The extracted quaternion

Definition at line 722 of file ImathMatrixAlgo.h.

template<class T >

bool extractScaling	(	const Matrix44< T > &	mat,
		Vec3< T > &	scl,
		bool	exc = `true`
	)

Extract the scaling component of the given 4x4 matrix.

Parameters

[in]	mat	The input matrix
[out]	scl	The extracted scale, i.e. the output value
[in]	exc	If true, throw an exception if the scaling in `mat` is very close to zero.

Returns: True if the scale could be extracted, false if the matrix is degenerate.

Definition at line 419 of file ImathMatrixAlgo.h.

template<class T >

bool extractScaling	(	const Matrix33< T > &	mat,
		Vec2< T > &	scl,
		bool	exc = `true`
	)

Extract the scaling component of the given 3x3 matrix.

Parameters

[in]	mat	The input matrix
[out]	scl	The extracted scale, i.e. the output value
[in]	exc	If true, throw an exception if the scaling in `mat` is very close to zero.

Returns: True if the scale could be extracted, false if the matrix is degenerate.

Definition at line 1089 of file ImathMatrixAlgo.h.

template<class T >

bool extractScalingAndShear	(	const Matrix44< T > &	mat,
		Vec3< T > &	scl,
		Vec3< T > &	shr,
		bool	exc = `true`
	)

Extract the scaling and shear components of the given 4x4 matrix. Return true if the scale could be successfully extracted, false if the matrix is degenerate.

Parameters

[in]	mat	The input matrix
[out]	scl	The extracted scale
[out]	shr	The extracted shear
[in]	exc	If true, throw an exception if the scaling in `mat` is very close to zero.

Returns: True if the scale could be extracted, false if the matrix is degenerate.

Definition at line 473 of file ImathMatrixAlgo.h.

template<class T >

bool extractScalingAndShear	(	const Matrix33< T > &	mat,
		Vec2< T > &	scl,
		T &	shr,
		bool	exc = `true`
	)

Extract the scaling and shear components of the given 3x3 matrix. Return true if the scale could be successfully extracted, false if the matrix is degenerate.

Parameters

[in]	mat	The input matrix
[out]	scl	The extracted scale
[out]	shr	The extracted shear
[in]	exc	If true, throw an exception if the scaling in `mat` is very close to zero.

Returns: True if the scale could be extracted, false if the matrix is degenerate.

Definition at line 1143 of file ImathMatrixAlgo.h.

template<class T >

bool extractSHRT	(	const Matrix44< T > &	mat,
		Vec3< T > &	s,
		Vec3< T > &	h,
		Vec3< T > &	r,
		Vec3< T > &	t,
		bool	exc,
		typename Euler< T >::Order	rOrder
	)

Extract the scaling, shear, rotation, and translation components of the given 4x4 matrix. The values are such that:

M = S * H * R * T

Parameters

[in]	mat	The input matrix
[out]	s	The extracted scale
[out]	h	The extracted shear
[out]	r	The extracted rotation
[out]	t	The extracted translation
[in]	exc	If true, throw an exception if the scaling in `mat` is very close to zero.
[in]	rOrder	The order with which to extract the rotation

Returns: True if the values could be extracted, false if the matrix is degenerate.

Definition at line 777 of file ImathMatrixAlgo.h.

template<class T >

bool extractSHRT	(	const Matrix44< T > &	mat,
		Vec3< T > &	s,
		Vec3< T > &	h,
		Vec3< T > &	r,
		Vec3< T > &	t,
		bool	exc = `true`
	)

Extract the scaling, shear, rotation, and translation components of the given 4x4 matrix.

Parameters

[in]	mat	The input matrix
[out]	s	The extracted scale
[out]	h	The extracted shear
[out]	r	The extracted rotation, in XYZ euler angles
[out]	t	The extracted translation
[in]	exc	If true, throw an exception if the scaling in `mat` is very close to zero.

Returns: True if the values could be extracted, false if the matrix is degenerate.

Definition at line 809 of file ImathMatrixAlgo.h.

template<class T >

bool extractSHRT	(	const Matrix44< T > &	mat,
		Vec3< T > &	s,
		Vec3< T > &	h,
		Euler< T > &	r,
		Vec3< T > &	t,
		bool	exc = `true`
	)

Extract the scaling, shear, rotation, and translation components of the given 4x4 matrix.

Parameters

[in]	mat	The input matrix
[out]	s	The extracted scale
[out]	h	The extracted shear
[out]	r	The extracted rotation, in Euler angles
[out]	t	The extracted translation
[in]	exc	If true, throw an exception if the scaling in `mat` is very close to zero.

Returns: True if the values could be extracted, false if the matrix is degenerate.

Definition at line 816 of file ImathMatrixAlgo.h.

template<class T >

bool extractSHRT	(	const Matrix33< T > &	mat,
		Vec2< T > &	s,
		T &	h,
		T &	r,
		Vec2< T > &	t,
		bool	exc = `true`
	)

Extract the scaling, shear, rotation, and translation components of the given 3x3 matrix. The values are such that:

M = S * H * R * T

Parameters

[in]	mat	The input matrix
[out]	s	The extracted scale
[out]	h	The extracted shear
[out]	r	The extracted rotation
[out]	t	The extracted translation
[in]	exc	If true, throw an exception if the scaling in `mat` is very close to zero.

Returns: True if the values could be extracted, false if the matrix is degenerate.

Definition at line 1311 of file ImathMatrixAlgo.h.

template<typename T >

void jacobiEigenSolver	(	Matrix33< T > &	A,
		Vec3< T > &	S,
		Matrix33< T > &	V,
		const T	tol
	)

Compute the eigenvalues (S) and the eigenvectors (V) of a real symmetric matrix using Jacobi transformation, using a given tolerance tol.

Jacobi transformation of a 3x3/4x4 matrix A outputs S and V: A = V * S * V^T where V is orthonormal and S is the diagonal matrix of eigenvalues. Input matrix A must be symmetric. A is also modified during the computation so that upper diagonal entries of A become zero.

template<typename T >

void jacobiEigenSolver	(	Matrix33< T > &	A,
		Vec3< T > &	S,
		Matrix33< T > &	V
	)

inline

Compute the eigenvalues (S) and the eigenvectors (V) of a real symmetric matrix using Jacobi transformation.

Jacobi transformation of a 3x3/4x4 matrix A outputs S and V: A = V * S * V^T where V is orthonormal and S is the diagonal matrix of eigenvalues. Input matrix A must be symmetric. A is also modified during the computation so that upper diagonal entries of A become zero.

Definition at line 1474 of file ImathMatrixAlgo.h.

template<typename T >

void jacobiEigenSolver	(	Matrix44< T > &	A,
		Vec4< T > &	S,
		Matrix44< T > &	V,
		const T	tol
	)

Compute the eigenvalues (S) and the eigenvectors (V) of a real symmetric matrix using Jacobi transformation, using a given tolerance tol.

Jacobi transformation of a 3x3/4x4 matrix A outputs S and V: A = V * S * V^T where V is orthonormal and S is the diagonal matrix of eigenvalues. Input matrix A must be symmetric. A is also modified during the computation so that upper diagonal entries of A become zero.

template<typename T >

void jacobiEigenSolver	(	Matrix44< T > &	A,
		Vec4< T > &	S,
		Matrix44< T > &	V
	)

inline

Compute the eigenvalues (S) and the eigenvectors (V) of a real symmetric matrix using Jacobi transformation.

Jacobi transformation of a 3x3/4x4 matrix A outputs S and V: A = V * S * V^T where V is orthonormal and S is the diagonal matrix of eigenvalues. Input matrix A must be symmetric. A is also modified during the computation so that upper diagonal entries of A become zero.

Definition at line 1501 of file ImathMatrixAlgo.h.

template<typename T >

void jacobiSVD	(	const Matrix33< T > &	A,
		Matrix33< T > &	U,
		Vec3< T > &	S,
		Matrix33< T > &	V,
		const T	tol = `std::numeric_limits< T >::epsilon()`,
		const bool	forcePositiveDeterminant = `false`
	)

Compute the SVD of a 3x3 matrix using Jacobi transformations. This method should be quite accurate (competitive with LAPACK) even for poorly conditioned matrices, and because it has been written specifically for the 3x3/4x4 case it is much faster than calling out to LAPACK.

The SVD of a 3x3/4x4 matrix A is defined as follows: A = U * S * V^T where S is the diagonal matrix of singular values and both U and V are orthonormal. By convention, the entries S are all positive and sorted from the largest to the smallest. However, some uses of this function may require that the matrix U*V^T have positive determinant; in this case, we may make the smallest singular value negative to ensure that this is satisfied.

Currently only available for single- and double-precision matrices.

template<typename T >

void jacobiSVD	(	const Matrix44< T > &	A,
		Matrix44< T > &	U,
		Vec4< T > &	S,
		Matrix44< T > &	V,
		const T	tol = `std::numeric_limits< T >::epsilon()`,
		const bool	forcePositiveDeterminant = `false`
	)

Compute the SVD of a 3x3 matrix using Jacobi transformations. This method should be quite accurate (competitive with LAPACK) even for poorly conditioned matrices, and because it has been written specifically for the 3x3/4x4 case it is much faster than calling out to LAPACK.

The SVD of a 3x3/4x4 matrix A is defined as follows: A = U * S * V^T where S is the diagonal matrix of singular values and both U and V are orthonormal. By convention, the entries S are all positive and sorted from the largest to the smallest. However, some uses of this function may require that the matrix U*V^T have positive determinant; in this case, we may make the smallest singular value negative to ensure that this is satisfied.

Currently only available for single- and double-precision matrices.

template<typename TM , typename TV >

void maxEigenVector	(	TM &	A,
		TV &	S
	)

Compute a eigenvector corresponding to the abs max eigenvalue of a real symmetric matrix using Jacobi transformation.

template<typename TM , typename TV >

void minEigenVector	(	TM &	A,
		TV &	S
	)

Compute a eigenvector corresponding to the abs min eigenvalue of a real symmetric matrix using Jacobi transformation.

template<class T >

Matrix44< T > outerProduct	(	const Vec4< T > &	a,
		const Vec4< T > &	b
	)

Return the 4x4 outer product two 4-vectors.

Definition at line 847 of file ImathMatrixAlgo.h.

template<class T >

Matrix33< T > outerProduct	(	const Vec3< T > &	a,
		const Vec3< T > &	b
	)

Return the 3xe outer product two 3-vectors.

Definition at line 1349 of file ImathMatrixAlgo.h.

template<typename T >

M44d procrustesRotationAndTranslation	(	const Vec3< T > *	A,
		const Vec3< T > *	B,
		const T *	weights,
		const size_t	numPoints,
		const bool	doScaling = `false`
	)

Computes the translation and rotation that brings the 'from' points as close as possible to the 'to' points under the Frobenius norm. To be more specific, let x be the matrix of 'from' points and y be the matrix of 'to' points, we want to find the matrix A of the form [ R t ] [ 0 1 ] that minimizes || (A*x - y)^T * W * (A*x - y) ||_F If doScaling is true, then a uniform scale is allowed also.

Parameters

A	From points
B	To points
weights	Per-point weights
numPoints	The number of points in `A`, `B`, and `weights` (must be equal)
doScaling	If true, include a scaling transformation

Returns: The procrustes transformation

template<typename T >

M44d procrustesRotationAndTranslation	(	const Vec3< T > *	A,
		const Vec3< T > *	B,
		const size_t	numPoints,
		const bool	doScaling = `false`
	)

Computes the translation and rotation that brings the 'from' points as close as possible to the 'to' points under the Frobenius norm. To be more specific, let x be the matrix of 'from' points and y be the matrix of 'to' points, we want to find the matrix A of the form [ R t ] [ 0 1 ] that minimizes || (A*x - y)^T * W * (A*x - y) ||_F If doScaling is true, then a uniform scale is allowed also.

Parameters

A	From points
B	To points
numPoints	The number of points in `A` and `B` (must be equal)
doScaling	If true, include a scaling transformation

Returns: The procrustes transformation

template<class T >

bool removeScaling	(	Matrix44< T > &	mat,
		bool	exc = `true`
	)

Remove scaling from the given 4x4 matrix in place. Return true if the scale could be successfully extracted, false if the matrix is degenerate.

Parameters

[in]	mat	The matrix to operate on
[in]	exc	If true, throw an exception if the scaling in `mat` is very close to zero.

Returns: True if the scale could be extracted, false if the matrix is degenerate.

Definition at line 453 of file ImathMatrixAlgo.h.

template<class T >

bool removeScaling	(	Matrix33< T > &	mat,
		bool	exc = `true`
	)

Remove scaling from the given 3x3 matrix in place. Return true if the scale could be successfully extracted, false if the matrix is degenerate.

Parameters

[in]	mat	The matrix to operate on
[in]	exc	If true, throw an exception if the scaling in `mat` is very close to zero.

Returns: True if the scale could be extracted, false if the matrix is degenerate.

Definition at line 1123 of file ImathMatrixAlgo.h.

template<class T >

bool removeScalingAndShear	(	Matrix44< T > &	mat,
		bool	exc = `true`
	)

Remove scaling and shear from the given 4x4 matrix in place.

Parameters

[in,out]	mat	The matrix to operate on
[in]	exc	If true, throw an exception if the scaling in `mat` is very close to zero.

Returns: True if the scale could be extracted, false if the matrix is degenerate.

Definition at line 510 of file ImathMatrixAlgo.h.

template<class T >

bool removeScalingAndShear	(	Matrix33< T > &	mat,
		bool	exc = `true`
	)

Remove scaling and shear from the given 3x3e matrix in place.

Parameters

[in,out]	mat	The matrix to operate on
[in]	exc	If true, throw an exception if the scaling in `mat` is very close to zero.

Returns: True if the scale could be extracted, false if the matrix is degenerate.

Definition at line 1169 of file ImathMatrixAlgo.h.

template<class T >

Matrix44< T > rotationMatrix	(	const Vec3< T > &	fromDirection,
		const Vec3< T > &	toDirection
	)

Return a 4x4 matrix that rotates the vector fromDirection to toDirection

Definition at line 869 of file ImathMatrixAlgo.h.

template<class T >

Matrix44< T > rotationMatrixWithUpDir	(	const Vec3< T > &	fromDir,
		const Vec3< T > &	toDir,
		const Vec3< T > &	upDir
	)

Return a 4x4 matrix that rotates the fromDir vector so that it points towards `toDir1. You may also specify that you want the up vector to be pointing in a certain direction 1upDir`.

Definition at line 878 of file ImathMatrixAlgo.h.

template<class T >

Matrix44< T > sansScaling	(	const Matrix44< T > &	mat,
		bool	exc = `true`
	)

Return the given 4x4 matrix with scaling removed.

Parameters

[in]	mat	The input matrix
[in]	exc	If true, throw an exception if the scaling in `mat`

Definition at line 432 of file ImathMatrixAlgo.h.

template<class T >

Matrix33< T > sansScaling	(	const Matrix33< T > &	mat,
		bool	exc = `true`
	)

Return the given 3x3 matrix with scaling removed.

Parameters

[in]	mat	The input matrix
[in]	exc	If true, throw an exception if the scaling in `mat`

Definition at line 1102 of file ImathMatrixAlgo.h.

template<class T >

Matrix44< T > sansScalingAndShear	(	const Matrix44< T > &	mat,
		bool	exc = `true`
	)

Return the given 4x4 matrix with scaling and shear removed.

Parameters

[in]	mat	The input matrix
[in]	exc	If true, throw an exception if the scaling in `mat` is very close to zero.

Definition at line 485 of file ImathMatrixAlgo.h.

template<class T >

void sansScalingAndShear	(	Matrix44< T > &	result,
		const Matrix44< T > &	mat,
		bool	exc = `true`
	)

Extract scaling and shear from the given 4x4 matrix in-place.

Parameters

[in,out]	result	The output matrix
[in]	mat	The return value if `result` is degenerate
[in]	exc	If true, throw an exception if the scaling in `mat` is very close to zero.