This template class is used to describe 3x3 physical matrices. More...

Useful functions
Quantity	m [3][3]
	The components of the physical matrix.

static const SBDTypePhysicalMatrix33< Quantity >	zero
	The zero physical matrix.

static const SBDTypePhysicalMatrix33< Quantity >	identity
	The identity physical matrix.

Quantity	trace () const
	Returns the trace m[0][0] + m[1][1] + m[2][2] of this physical matrix.

void	print () const
	Prints this physical matrix.

void	setIdentity ()
	Sets this physical matrix to the identity matrix.

void	setZero ()
	Sets this physical matrix to zero.

SBDTypePhysicalMatrix33< Quantity >	transpose () const
	Returns the transpose of this physical matrix.

void	toQuaternion (SBQuantity::dimensionless &w, SBQuantity::dimensionless &x, SBQuantity::dimensionless &y, SBQuantity::dimensionless &z) const
	Computes the quaternion [ `w` `x` `y` `z` ] corresponding to this physical matrix.

void	toRotationVector (SBVector3 &rotationVector) const
	Computes the rotation vector corresponding to this dimensionless physical matrix.

void	orthonormalize ()
	Orthonormalizes this dimensionless physical matrix.

void	quasiStaticUpdate (const SBQuantity::dimensionless &wx, const SBQuantity::dimensionless &wy, const SBQuantity::dimensionless &wz, SBQuantity::dimensionless &rotAngle, SBVector3 &rotAxis, SBDTypePhysicalMatrix33< Quantity > &rotA, SBDTypePhysicalMatrix33< Quantity > &rotB, SBDTypePhysicalMatrix33< Quantity > &rotC)
	Performs a quasi-static update of this dimensionless physical matrix using the rotation vector [ `wx` `wy` `wz` ].

void	computeEulerDecompositionZYZ (SBQuantity::dimensionless &phi, SBQuantity::dimensionless &theta, SBQuantity::dimensionless &psi)
	Computes a ZYZ Euler decomposition of this physical matrix.

void	revoluteJointZAxisMatrix (const SBVector3 &axis)
	Sets this physical matrix to an orthonormal matrix whose third column is `axis`.

void	diagonalize (SBDTypePhysicalVector3< Quantity > &ev, SBDTypePhysicalMatrix33< SBQuantity::dimensionless > &p)
	Computes the eigenvalues `ev` and the eigenvectors `p` of this physical matrix.

SBQuantityProduct3< Quantity, Quantity, Quantity >::Type	det () const
	Returns the determinant of this physical matrix.

Quantity	norm () const
	Returns the 2-norm of this physical matrix.

Quantity	cosphi () const
	Returns the cosine of the rotation angle of this rotation matrix.

SBDTypePhysicalMatrix33< typename SBQuantityInverse< Quantity >::Type >	inverse () const
	Returns the inverse of this physical matrix.

void	makeEulerRotationZYZ (const SBQuantity::dimensionless &phi, const SBQuantity::dimensionless &theta, const SBQuantity::dimensionless &psi)
	This function sets this physical matrix from ZYZ Euler angles `phi`, `theta` and `psi`.

void	swapColumns (unsigned int i, unsigned int j)
	Swaps columns `i` and `j`.

void	swapRows (unsigned int i, unsigned int j)
	Swaps rows `i` and `j`.

static SBDTypePhysicalMatrix33< SBQuantity::dimensionless >	fromQuaternion (const SBQuantity::dimensionless &w, const SBQuantity::dimensionless &x, const SBQuantity::dimensionless &y, const SBQuantity::dimensionless &z)
	Returns the dimensionless physical matrix corresponding to quaternion [ `w` `x` `y` `z` ].

static SBDTypePhysicalMatrix33< SBQuantity::dimensionless >	fromAxisAngle (const SBVector3 &axis, const SBQuantity::dimensionless &angle)
	Returns the dimensionless physical matrix corresponding to rotation axis `axis` and rotation angle `angle` (in radians)

static SBDTypePhysicalMatrix33< SBQuantity::dimensionless >	fromAxisAnglePi (const SBVector3 &axis)
	Returns the dimensionless physical matrix corresponding to rotation axis `axis` and a rotation angle equal to Pi.

static SBDTypePhysicalMatrix33< SBQuantity::dimensionless >	fromAlignment (const SBVector3 &from, const SBVector3 &to)
	Returns the dimensionless physical matrix that transforms vector `from` into vector `to`. Precisely, left-multiplying vector `from` by the resulting matrix produces vector `to`.

Constructors
	SBDTypePhysicalMatrix33 ()
	Constructs a physical matrix with all components set to zero.

	SBDTypePhysicalMatrix33 (const SBDTypePhysicalVector3< Quantity > &d)
	Constructs the diagonal physical matrix with diagonal [ `d.v`[0] `d.v`[1] `d.v`[2] ].

	SBDTypePhysicalMatrix33 (const SBDTypePhysicalVector3< Quantity > &c0, const SBDTypePhysicalVector3< Quantity > &c1, const SBDTypePhysicalVector3< Quantity > &c2)
	Constructs the physical matrix with columns `c0`, `c1` and `c2`.

	SBDTypePhysicalMatrix33 (const Quantity &v)
	Constructs the physical matrix with all components set to `v`.

	SBDTypePhysicalMatrix33 (Quantity mat[3][3])
	Constructs the physical matrix from the quantity array `mat`.

	SBDTypePhysicalMatrix33 (const Quantity &m00, const Quantity &m01, const Quantity &m02, const Quantity &m10, const Quantity &m11, const Quantity &m12, const Quantity &m20, const Quantity &m21, const Quantity &m22)
	Constructs the physical matrix from components `m00`, `m01`, `m02`, `m10`, `m11`, `m12`, `m20`, `m21` and `m22`.

	SBDTypePhysicalMatrix33 (double d)
	Constructs the dimensionless physical matrix with all components set to `d`.

	SBDTypePhysicalMatrix33 (double mat[3][3])
	Constructs the dimensionless physical matrix from double array `mat`.

	SBDTypePhysicalMatrix33 (double m00, double m01, double m02, double m10, double m11, double m12, double m20, double m21, double m22)
	Constructs the dimensionless physical matrix from components `m00`, `m01`, `m02`, `m10`, `m11`, `m12`, `m20`, `m21` and `m22`.

	SBDTypePhysicalMatrix33 (const SBDTypePhysicalMatrix33< Quantity > &mat)
	Copy constructor.

Serialization
void	serialize (SBCSerializer *serializer, const SBVersionNumber &sdkVersionNumber=SB_SDK_VERSION_NUMBER) const
	Serializes the physical matrix using the provided `serializer`.

void	unserialize (SBCSerializer *serializer, const SBVersionNumber &sdkVersionNumber=SB_SDK_VERSION_NUMBER)
	Unserializes the physical matrix using the provided `serializer`.

Accessors
SBDTypePhysicalVector3< Quantity >	getE1 () const
	Returns the first column of this physical matrix.

void	setE1 (const SBDTypePhysicalVector3< Quantity > &v)
	Sets the first column of this physical matrix.

SBDTypePhysicalVector3< Quantity >	getE2 () const
	Returns the second column of this physical matrix.

void	setE2 (const SBDTypePhysicalVector3< Quantity > &v)
	Sets the second column of this physical matrix.

SBDTypePhysicalVector3< Quantity >	getE3 () const
	Returns the third column of this physical matrix.

void	setE3 (const SBDTypePhysicalVector3< Quantity > &v)
	Sets the third column of this physical matrix.

SBDTypePhysicalMatrix33< SBQuantity::dimensionless >	getValue () const
	Returns a dimensionless physical matrix whose components are equal to those of this physical matrix.

void	setValue (const SBDTypePhysicalMatrix33< SBQuantity::dimensionless > &u)
	Sets the components of this physical matrix equal to those of the dimensionless physical matrix `u`.

Operators
SBDTypePhysicalMatrix33< Quantity > &	operator= (const SBDTypePhysicalMatrix33< Quantity > &mat)
	Copy assignment.

SBDTypePhysicalMatrix33< Quantity >	operator+ (const SBDTypePhysicalMatrix33< Quantity > &mat) const
	Returns the sum of this physical matrix with physical matrix `mat`.

SBDTypePhysicalMatrix33< Quantity >	operator- (const SBDTypePhysicalMatrix33< Quantity > &mat) const
	Returns the difference of this physical matrix with physical matrix `mat`.

template<typename QuantityB >
SBDTypePhysicalMatrix33< typename SBQuantityProduct2< Quantity, QuantityB >::Type >	operator* (const SBDTypePhysicalMatrix33< QuantityB > &mat) const
	Returns the product of this physical matrix with physical matrix `mat`.

SBDTypePhysicalMatrix33< Quantity >	operator* (double d) const
	Returns the product of this physical matrix with double `d`.

SBDTypePhysicalMatrix33< Quantity >	operator/ (double d) const
	Returns the division of this physical matrix by double `d`.

void	operator+= (const SBDTypePhysicalMatrix33< Quantity > &mat)
	Adds physical matrix `mat` to this physical matrix.

void	operator-= (const SBDTypePhysicalMatrix33< Quantity > &mat)
	Subtracts physical matrix `mat` from this physical matrix.

SBDTypePhysicalMatrix33< Quantity >	operator- () const
	Returns the opposite of this physical matrix.

template<typename QuantityB >
SBDTypePhysicalMatrix33< Quantity > &	operator*= (QuantityB d)
	Multiplies this physical matrix with physical quantity `d`.

template<typename QuantityB >
SBDTypePhysicalMatrix33< Quantity > &	operator/= (QuantityB d)
	Divides this physical matrix by physical quantity `d`.

template<typename QuantityB >
SBDTypePhysicalVector3< typename SBQuantityProduct2< Quantity, QuantityB >::Type >	operator* (const SBDTypePhysicalVector3< QuantityB > &v) const
	Returns the product of this physical matrix with physical vector `v`.

bool	operator== (const SBDTypePhysicalMatrix33< Quantity > &mat) const
	Returns true if this physical matrix is equal to physical matrix `mat` (component-wise)

bool	operator!= (const SBDTypePhysicalMatrix33< Quantity > &mat) const
	Returns true if this physical matrix is different from physical matrix `mat` (component-wise)

Detailed Description

template<typename Quantity>
class SBDTypePhysicalMatrix33< Quantity >

Template Parameters

Quantity The quantity type of the physical matrix

This template class describes 3x3 physical matrices.

Physical matrices are physical quantities, and thus use SAMSON's unit system. In a physical matrix (e.g. an inertia tensor), all components have the same unit, so that physical matrices are defined by only one unit, and the SBDTypePhysicalMatrix33 template class is parameterized by only one type: Quantity.

Most of the time, developers of SAMSON Extensions do not have to use this template, but may directly use some predefined types that correspond to specific instantiations of this template class, e.g. SBMatrix33 (dimensionless matrix), SBMass33 (mass matrix), SBInertiaTensor33 (inertia tensor), etc.

Dimensionless 3x3 matrices (SBMatrix33) are especially useful since they are used to represent 3D reference frames, rotation matrices, rigid-body transforms, etc.

Short name: SBPhysicalMatrix33

See also: SAMSON's unit system; The SBDType library; SBDTypePhysicalVector3; SBDTypeSpatialTransform

Member Function Documentation

◆ computeEulerDecompositionZYZ()

template<typename Quantity >

void SBDTypePhysicalMatrix33< Quantity >::computeEulerDecompositionZYZ	(	SBQuantity::dimensionless &	phi,
		SBQuantity::dimensionless &	theta,
		SBQuantity::dimensionless &	psi
	)

inline

Parameters

phi	The first Z rotation angle
theta	The Y rotation angle
psi	The second Z rotation angle

This function computes a ZYZ Euler decomposition of this physical matrix:

\(\mathbf{M}=\mathbf{R}(z,psi)\mathbf{R}(y,theta)\mathbf{R}(z,phi)\)

where \(\mathbf{M}\) denotes this physical matrix, \(\mathbf{R}(z,phi)\) denotes the first rotation around the Z axis, \(\mathbf{R}(y,theta)\) denotes the rotation around the Y axis, and \(\mathbf{R}(z,psi)\) denotes the second rotation around the Z axis.

Note that this function may only be used with dimensionless physical matrices.

◆ cosphi()

template<typename Quantity >

Quantity SBDTypePhysicalMatrix33< Quantity >::cosphi ( ) const

inline

This function assumes this physical matrix is a rotation matrix, and returns the cosine of the corresponding rotation angle

◆ diagonalize()

template<typename Quantity >

void SBDTypePhysicalMatrix33< Quantity >::diagonalize	(	SBDTypePhysicalVector3< Quantity > &	ev,
		SBDTypePhysicalMatrix33< SBQuantity::dimensionless > &	p
	)

inline

Parameters

ev	Stores the eigenvalues when this function returns
p	Stores the eigenvectors when this function returns

This function computes the eigenvalues ev and the eigenvectors p of this physical matrix. Note that the eigenvalues have the same units as the physical matrix, while the eigenvectors are stored in a dimensionless matrix.

See also: SAMSON's unit system

◆ inverse()

template<typename Quantity >

SBDTypePhysicalMatrix33< typename SBQuantityInverse< Quantity >::Type > SBDTypePhysicalMatrix33< Quantity >::inverse ( ) const

inline

This function assumes the physical matrix is invertible and returns its inverse.

◆ makeEulerRotationZYZ()

template<typename Quantity >

void SBDTypePhysicalMatrix33< Quantity >::makeEulerRotationZYZ	(	const SBQuantity::dimensionless &	phi,
		const SBQuantity::dimensionless &	theta,
		const SBQuantity::dimensionless &	psi
	)

inline

Parameters

phi	The first Z rotation angle
theta	The Y rotation angle
psi	The second Z rotation angle

This function sets this physical matrix from ZYZ Euler angles:

\(\mathbf{M}=\mathbf{R}(z,psi)\mathbf{R}(y,theta)\mathbf{R}(z,phi)\)

where \(\mathbf{M}\) denotes this physical matrix, \(\mathbf{R}(z,phi)\) denotes the first rotation around the Z axis, \(\mathbf{R}(y,theta)\) denotes the rotation around the Y axis, and \(\mathbf{R}(z,psi)\) denotes the second rotation around the Z axis.

Note that this function may only be used with dimensionless physical matrices.

◆ operator*=()

template<typename Quantity >

template<typename QuantityB >

SBDTypePhysicalMatrix33< Quantity > & SBDTypePhysicalMatrix33< Quantity >::operator*= ( QuantityB d )

inline

Parameters

d	A dimensionless physical quantity or a numeric type (i.e. int, float, double, etc)

◆ operator/=()

template<typename Quantity >

template<typename QuantityB >

SBDTypePhysicalMatrix33< Quantity > & SBDTypePhysicalMatrix33< Quantity >::operator/= ( QuantityB d )

inline

Parameters

d	A dimensionless physical quantity or a numeric type (i.e. int, float, double, etc)

◆ quasiStaticUpdate()

template<typename Quantity >

void SBDTypePhysicalMatrix33< Quantity >::quasiStaticUpdate	(	const SBQuantity::dimensionless &	wx,
		const SBQuantity::dimensionless &	wy,
		const SBQuantity::dimensionless &	wz,
		SBQuantity::dimensionless &	rotAngle,
		SBVector3 &	rotAxis,
		SBDTypePhysicalMatrix33< Quantity > &	rotA,
		SBDTypePhysicalMatrix33< Quantity > &	rotB,
		SBDTypePhysicalMatrix33< Quantity > &	rotC
	)

inline

Parameters

wx	The first component of the rotation vector
wy	The second component of the rotation vector
wz	The third component of the rotation vector
rotAngle	The rotation angle
rotAxis	The rotation axis
rotA	The rotation axis
rotB	The rotation axis
rotC	The rotation axis

This function performs a quasi-static update of this dimensionless physical matrix using the rotation vector [ wx wy wz ]. During the update, The rotation vector [ wx wy wz ] is converted to a rotation angle and rotation axis, respectively stored into rotAngle and rotAxis. If \(\mathbf{M}\) denotes the original matrix, the final matrix is \(\cos(w)\mathbf{A}+\sin(w)\mathbf{B}+\mathbf{C}\), where \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\) are physical matrices computed from \(\mathbf{M}\) and the rotation axis \(\mathbf{v}\), and \(w\) is the rotation angle:

\(\mathbf{A}=(\mathbf{I}-\mathbf{v}\mathbf{v}^T)\mathbf{M}\)
\(\mathbf{B}=\tilde{\mathbf{v}}\mathbf{M}\)
\(\mathbf{C}=\mathbf{v}\mathbf{v}^T\mathbf{M}\)

where \(\tilde{\mathbf{v}}\) is the matrix such that \(\tilde{\mathbf{v}}\mathbf{x}=\mathbf{v}\times\mathbf{x}\) for any \(\mathbf{x}\).

When the function returns, \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\) are stored into rotA, rotB and rotC, respectively.

Note that this function may only be used with dimensionless physical matrices.

Useful functions

Constructors

Serialization

Accessors

Operators

Detailed Description

Member Function Documentation

◆ computeEulerDecompositionZYZ()

◆ cosphi()

◆ diagonalize()

◆ inverse()

◆ makeEulerRotationZYZ()

◆ operator*=()

◆ operator/=()

◆ quasiStaticUpdate()