This class implements 3² matrix. More...

#include <matrix.h>

Public Member Functions
	Matrix ()
	Empty.

	Matrix (const double &)
	Creates a diagonal matrix. More...

	Matrix (const Vector &)
	Create a diagonal matrix with diagonal terms set to the vector entries. More...

	Matrix (const Vector &, const Vector &, const Vector &)
	Creates a matrix given column vectors. More...

	Matrix (const double &, const double &, const double &, const double &, const double &, const double &, const double &, const double &, const double &)
	Creates a matrix with a set of double values. More...

constexpr double &	operator[] (int)
	Direct access to the array of the matrix.

constexpr double	operator[] (int) const
	Overloaded.

constexpr double &	operator() (int, int)
	Get element (i,j) of the matrix. More...

constexpr double	operator() (int, int) const
	Overloaded.

Vector	C (int) const
	Get the i-th column vector of the matrix. More...

Vector	Row (int) const
	Get the i-th row vector of the matrix. More...

Matrix	operator- () const
	Returns the opposite of a matrix -A.

Matrix &	operator+= (const Matrix &)
	Destructive addition operator.

Matrix &	operator-= (const Matrix &)
	Destructive subtraction operator.

Matrix &	*operator=** (const Matrix &)
	Destructive multiplication.

Matrix &	*operator=** (double)
	Destructive multiplication operator.

Matrix &	operator/= (double)
	Destructive division operator.

Vector	operator* (const Vector &) const
	Right multiply by a vector. More...

Vector	GetRotationAngles () const
	Compute the angles of the rotation matrix.

Matrix	Abs () const
	Compute the absolute value of the matrix.

Matrix	T () const
	Transpose the matrix. More...

Matrix	Adjoint () const
	Compute the adjoint, i.e, the comatrix of the matrix. More...

double	Determinant () const
	Computes the determinant of the matrix.

double	Trace () const
	Compute the trace (sum of diagonal terms) of a matrix.

Cubic	Characteristic () const
	Compute the characteristic polynomial.

double	SpectralNorm () const
	Computes the spectral norm of the matrix.

double	InfinityNorm () const
	Compute the infinity norm of the matrix.

double	FrobeniusNorm () const
	Compute the Frobenius norm of the matrix.

void	SingularValueDecomposition (Matrix &, Vector &, Matrix &) const
	Decompose a matrix into singular values. More...

void	QDU (Matrix &, Vector &, Vector &) const
	Perform the QDU decomposition of a matrix. More...

void	ExtractAngleAxis (Vector &, double &) const
	Given a rotation matrix, extract its defining axis and its angle. More...

void	EigenSolveSymmetric (double[3], Vector[3]) const
	Find the eigen values of the matrix. More...

Vector	Eigen () const
	Find the eigen values of the matrix. More...

void	Tridiagonal (double[3], double[2])
	Compute the Householder reduction of a symmetric matrix. More...

int	QLAlgorithm (double[3], double[3])
	iteration with implicit shifting to reduce matrix from tridiagonal to diagonal More...

Static Public Member Functions
static Matrix	Rotation (const Vector &)
	Create a rotation matrix given a vector of angles that specifies the rotation around each world coordinate axis. More...

static Matrix	RotationMaya (const Vector &)
	Create a rotation matrix given a vector of angles that specifies the rotation around each world coordinate axis. More...

static Matrix	Rotation (const Vector &, const double &)
	Create a rotation matrix about an arbitrary axis. More...

static Matrix	Rotation (const Vector &, const Vector &)
	Create a rotation matrix that rotates a normalized vector into another one. More...

static Matrix	RotationCanonical (const Vector &)
	Create a rotation matrix that rotates (0,0,1) to a normalized vector. More...

static Matrix	RotationX (const double &)
	Create a rotation matrix around the x-axis. More...

static Matrix	RotationY (const double &)
	Create a rotation matrix around the y-axis. More...

static Matrix	RotationZ (const double &)
	Create a rotation matrix around the z-axis. More...

static Matrix	Symmetry (const Vector &)
	Create a planar symmetry matrix. More...

static Matrix	Covariance (Vector *, int)
	Compute the covariance matrix of a point cloud. More...

static Matrix	Lerp (const double &, const Matrix &, const Matrix &)
	Spherical interpolation of two rotation matrices. More...

static Matrix	Rotation (RandomFast &=random239)
	Generate uniformly distributed random rotation matrix. More...

static Matrix	Frenet (const Vector &, const Vector &=Vector::Z)
	Compute the Frenet frame. More...

Static Public Attributes
static const Matrix	Null
	Null matrix.

static const Matrix	Identity
	Identity matrix.

Protected Attributes
double	r [9] = { 1.0,0.0,0.0,0.0,1.0,0.0,0.0,0.0,1.0 }
	The array storing the coefficients of the matrix.

Friends
Matrix	operator+ (const Matrix &, const Matrix &)
	Overloaded.

Matrix	operator- (const Matrix &, const Matrix &)
	Overloaded.

Matrix	operator* (const Matrix &, const Matrix &)
	Multiplication. More...

Matrix	operator* (const Matrix &, const double &)
	Right multiply by a double.

Matrix	operator* (const double &, const Matrix &)
	Left multiply by a double.

Matrix	operator/ (const Matrix &, const double &)
	Right divide by a double.

Matrix	operator/ (const double &, const Matrix &)
	Computes the inverse matrix and scales it by a double. More...

std::ostream &	operator<< (std::ostream &, const Matrix &)
	Overloaded. More...

Matrix	Inverse (const Matrix &)
	Compute the inverse of a matrix A^-1. More...

Detailed Description

This class implements 3² matrix.

Operators have been overloaded so as to behave as expected with the Vector class.

Many functions have been coded inline out of efficiency.

The constructors of this class are used to create matrices in the general case. See the static member functions Matrix::Rotation() to create different kinds of rotation matrices. Canonical symmetry matrices should be created as follows:

Matrix planar=Matrix(Vector(1.0,-1.0,1.0)); // Symmetry, plane y=0

Matrix origin=Matrix(Vector(-1.0,-1.0,-1.0)); // Symmetry around origin

Matrix

This class implements 32 matrix.

Definition: matrix.h:14

Matrix::Matrix

Matrix()

Empty.

Definition: matrix.h:19

Vector

Vectors in three dimensions.

Definition: evector.h:21

More general symmetries can be created using static member functions:

Matrix planar=Matrix::Symmetry(Vector(2.0,-3.0,1.0)); // Symmetry, plane with normal vector as argument

Matrix::Symmetry

static Matrix Symmetry(const Vector &)

Create a planar symmetry matrix.

Definition: matrix.cpp:185

Components are stored in a single dimension array, sorting components by column.

Constructor & Destructor Documentation

◆ Matrix() [1/4]

Matrix::Matrix ( const double & a )

inlineexplicit

Creates a diagonal matrix.

Parameters

a	The diagonal value.

◆ Matrix() [2/4]

Matrix::Matrix ( const Vector & a )

inlineexplicit

Create a diagonal matrix with diagonal terms set to the vector entries.

Parameters

a	Vector of diagonal values.

◆ Matrix() [3/4]

Matrix::Matrix	(	const Vector &	a,
		const Vector &	b,
		const Vector &	c
	)

inlineexplicit

Creates a matrix given column vectors.

Parameters

a,b,c Column vectors.

◆ Matrix() [4/4]

Matrix::Matrix	(	const double &	a00,
		const double &	a01,
		const double &	a02,
		const double &	a10,
		const double &	a11,
		const double &	a12,
		const double &	a20,
		const double &	a21,
		const double &	a22
	)

inlineexplicit

Creates a matrix with a set of double values.

Coefficients are given in column order.

Member Function Documentation

◆ Adjoint()

Matrix Matrix::Adjoint ( ) const

Compute the adjoint, i.e, the comatrix of the matrix.

See also: Inverse()

◆ C()

Vector Matrix::C ( int i ) const

inline

Get the i-th column vector of the matrix.

Parameters

i Index.

◆ Covariance()

Matrix Matrix::Covariance	(	Vector *	p,
		int	n
	)

static

Compute the covariance matrix of a point cloud.

This matrix is useful for computing the principal directions of a point of clouds. The principal directions are defined as the eigen vectors of the matrix.

See also: EigenSolveSymmetric

Parameters

p	Array of vertices.
n	Number of vertices.

◆ Eigen()

Vector Matrix::Eigen ( ) const

Find the eigen values of the matrix.

Eigen values are sorted and returned as a vector.

◆ EigenSolveSymmetric()

void Matrix::EigenSolveSymmetric	(	double	a[3],
		Vector	e[3]
	)		const

Find the eigen values of the matrix.

Return eigen vectors as well.

Parameters

a	An array of three eigen values.
e	The array containing the three eigen vectors.

◆ ExtractAngleAxis()

void Matrix::ExtractAngleAxis	(	Vector &	v,
		double &	angle
	)		const

Given a rotation matrix, extract its defining axis and its angle.

This is the same as computing its quaternion representation.

Parameters

angle	Angle.
v	Vector.

Let (x,y,z) the unit-length axis and let a an angle of rotation. The rotation matrix is R=I+sin(a).P+(1-cos(A)).P² where I is the identity and P a 0 diagonal matrix with (x,y,z) located at theit right index locations.

Some algebra will show that:
cos(a)=(trace(R)-1)/2 and R-R^t=2.sin(a).P
In the event that a=π, R-R^t=0 which prevents us from extracting the axis through P. Instead note that R=I+2.P² when a=π, so P²=(R-I)/2. The diagonal entries of P² are x²-1, y²-1, and z²-1. We can solve these for axis (x,y,z). Because the angle is π, it does not matter which sign you choose on the square roots.

◆ Frenet()

Matrix Matrix::Frenet	(	const Vector &	t,
		const Vector &	u = `Vector::Z`
	)

static

Compute the Frenet frame.

Given tangent t and up vector u, computes the matrix (t,t^u,u)). This function stores the tangent, the binormal and the normal vectors in the matrix. The normalization of the vectors is performed internally.

Parameters

t	Tangent, which need not be normalized.
u	Up vector.

◆ Lerp()

Matrix Matrix::Lerp	(	const double &	t,
		const Matrix &	a,
		const Matrix &	b
	)

static

Spherical interpolation of two rotation matrices.

This is an expensive function that first converts rotations to quaternions, then interpolates the quaternions, before finaly reconstructing the corresponding matrix.

See also: Quaternion Lerp(const double&, const Quaternion&, const Quaternion&);

Parameters

t	Interpolation parameter.
a,b	Rotation matrix.

◆ operator()()

constexpr double & Matrix::operator()	(	int	i,
		int	j
	)

inlineconstexpr

Get element (i,j) of the matrix.

Parameters

i	Row.
j	Column.

◆ operator*()

Vector Matrix::operator* ( const Vector & v ) const

inline

Right multiply by a vector.

Right multiplication by a vector is a member function and not a friend to avoid left/right multiplication ambiguities since matrix multiplication is not commutative.

Parameters

v

Vector.

Vector v=Matrix::Rotation(Vector(0.0,Math::Pi/3.0,Math::Pi/4))*Vector(1.0,1.0,1.0);

Math::Pi

static const double Pi

π.

Definition: mathematics.h:174

Matrix::Rotation

static Matrix Rotation(const Vector &)

Create a rotation matrix given a vector of angles that specifies the rotation around each world coord...

Definition: matrix.cpp:117

◆ QDU()

void Matrix::QDU	(	Matrix &	a,
		Vector &	D,
		Vector &	U
	)		const

Perform the QDU decomposition of a matrix.

Q is orthogonal, D is diagonal, and U is upper triangular with ones on its diagonal.

The algorithm uses Gram-Schmidt orthogonalization (the QR algorithm).

◆ QLAlgorithm()

int Matrix::QLAlgorithm	(	double	diag[3],
		double	subd[3]
	)

iteration with implicit shifting to reduce matrix from tridiagonal to diagonal

Parameters

diag	Diagonal elements.
subd	Triangular sub-matrix.

◆ Rotation() [1/4]

Matrix Matrix::Rotation ( const Vector & v )

static

Create a rotation matrix given a vector of angles that specifies the rotation around each world coordinate axis.

Rotations are concatenated, starting by rotating around the z axis, then y and eventually x.

Parameters

v	Vector of angles in radian.

◆ Rotation() [2/4]

Matrix Matrix::Rotation	(	const Vector &	u,
		const double &	a
	)

static

Create a rotation matrix about an arbitrary axis.

Matrix R=Matrix::Rotation(Normalized(Vector(1,2,-1)),Math::DegreeToRadian(30.0));

Math::DegreeToRadian

static constexpr double DegreeToRadian(const double &)

Convert degrees to randians.

Definition: mathematics.h:478

Parameters

u	Rotation axis, which should be unit.
a	Angle, in radian.

◆ Rotation() [3/4]

Matrix Matrix::Rotation	(	const Vector &	a,
		const Vector &	b
	)

static

Create a rotation matrix that rotates a normalized vector into another one.

Parameters

a,b	Initial and final vector (should be normalized).

◆ Rotation() [4/4]

Matrix Matrix::Rotation ( RandomFast & ra = random239 )

static

Generate uniformly distributed random rotation matrix.

See Fast Random Rotation Matrices, J. Arvo. Graphics Gems III, 1992.

Parameters

ra	Fast random number generator.

◆ RotationCanonical()

Matrix Matrix::RotationCanonical ( const Vector & v )

static

Create a rotation matrix that rotates (0,0,1) to a normalized vector.

This is the same as:

Vector v;

Matrix r=Matrix::Rotation(Vector::Z,v);

Matrix::r

double r[9]

The array storing the coefficients of the matrix.

Definition: matrix.h:16

Vector::Z

static const Vector Z

Vector(0,0,1).

Definition: evector.h:142

Parameters

v	Final vector (should be normalized).

◆ RotationMaya()

Matrix Matrix::RotationMaya ( const Vector & v )

static

Create a rotation matrix given a vector of angles that specifies the rotation around each world coordinate axis.

Rotations are concatenated, starting by rotating around the z axis, then y and eventually x.

Parameters

v	Vector of angles in radian.

◆ RotationX()

Matrix Matrix::RotationX ( const double & a )

static

Create a rotation matrix around the x-axis.

This static member function is provided out of efficiency as it is much faster than any other general Matrix::Rotation() member.

Parameters

a	Angle (in radian).

◆ RotationY()

Matrix Matrix::RotationY ( const double & a )

static

Create a rotation matrix around the y-axis.

See also: Matrix::RotationX(const double&)

Parameters

a	Angle (in radian).

◆ RotationZ()

Matrix Matrix::RotationZ ( const double & a )

static

Create a rotation matrix around the z-axis.

See also: Matrix::RotationX(const double&)

Parameters

a	Angle (in radian).

◆ Row()

Vector Matrix::Row ( int i ) const

inline

Get the i-th row vector of the matrix.

Parameters

i Index.

◆ SingularValueDecomposition()

void Matrix::SingularValueDecomposition	(	Matrix &	L,
		Vector &	S,
		Matrix &	R
	)		const

Decompose a matrix into singular values.

The input matrix M will be equal to M = L S R.

Parameters

L	Lower matrix.
S	Singular values.
R	Matrix.

◆ Symmetry()

Matrix Matrix::Symmetry ( const Vector & n )

static

Create a planar symmetry matrix.

Parameters

n	Normal to the plane.

◆ T()

Matrix Matrix::T ( ) const

inline

Transpose the matrix.

Note that if the matrix is a rotation matrix, then the transpose is equal to its inverse.

See also: T(const Matrix&)

◆ Tridiagonal()

void Matrix::Tridiagonal	(	double	diag[3],
		double	subd[2]
	)

Compute the Householder reduction of a symmetric matrix.

Decompose a matrix A into Q^t M Q.

Parameters

diag	Diagonal entries of A.
subd	Sub-diagonal entries of A.

Friends And Related Function Documentation

◆ Inverse

Matrix Inverse ( const Matrix & A )

friend

Compute the inverse of a matrix A^-1.

Recall that A^-1 can be defined as the transposed adjoint matrix divided by the determinant.

This function returns the null matrix if A cannot be inversed.

The threshold value involved in the singular matrix detection is set to 10^-18.

Parameters

A Matrix.

◆ operator*

Matrix operator*	(	const Matrix &	u,
		const Matrix &	v
	)

friend

Multiplication.

Parameters

u,v	Input matrices.

◆ operator/

Matrix operator/	(	const double &	x,
		const Matrix &	a
	)

friend

Computes the inverse matrix and scales it by a double.

This function calls Matrix::Inverse().

Parameters

x	Real.
a	Matrix.

◆ operator<<

std::ostream & operator<<	(	std::ostream &	s,
		const Matrix &	matrix
	)

friend

Overloaded.

Parameters

s	Stream.
matrix	The matrix.

Public Member Functions

Static Public Member Functions

Static Public Attributes

Protected Attributes

Friends

Detailed Description

Constructor & Destructor Documentation

◆ Matrix() [1/4]

◆ Matrix() [2/4]

◆ Matrix() [3/4]

◆ Matrix() [4/4]

Member Function Documentation

◆ Adjoint()

◆ C()

◆ Covariance()

◆ Eigen()

◆ EigenSolveSymmetric()

◆ ExtractAngleAxis()

◆ Frenet()

◆ Lerp()

◆ operator()()

◆ operator*()

◆ QDU()

◆ QLAlgorithm()

◆ Rotation() [1/4]

◆ Rotation() [2/4]

◆ Rotation() [3/4]

◆ Rotation() [4/4]

◆ RotationCanonical()

◆ RotationMaya()

◆ RotationX()

◆ RotationY()

◆ RotationZ()

◆ Row()

◆ SingularValueDecomposition()

◆ Symmetry()

◆ T()

◆ Tridiagonal()

Friends And Related Function Documentation

◆ Inverse

◆ operator*

◆ operator/

◆ operator<<