maspack.matrix

Class PolarDecomposition3d

java.lang.Object

maspack.matrix.PolarDecomposition3d

public class PolarDecomposition3d
extends java.lang.Object

Class to produce the polar decompositions of a 3x3 matrix. The right polar decomposition of a matrix M is produced by the factor(maspack.matrix.Matrix3dBase) method and is given by

 M = Q P
 

where Q is orthogonal and P is symmetric positive definite (or indefinite if M is singular). Note that Q is not necessarily right-handed. It is possible to define an alternative decomposition,

 M = R H
 

where R is a right-handed rotation matrix and H is symmetric indefinite. Alternatively, if Q is not right-handed, one can make it so by negating one of its axes, using

 Q' = Q N
 

where N is a diagonal matrix whose elements are either 1 or -1 and whose determinant is -1. At present, this class chooses N with a single -1 at the same location as the diagonal element of Q that is nearest to 1. The idea here is to flip the single axis that is least affected by Q.

The left polar decomposition is produced by the factorLeft(maspack.matrix.Matrix3dBase) method and is given by

 M = P Q
 

with the alternative decomposition,

 M = H R
 

Note that the Q and R matrices associated with the right and left factorizations are identical, whereas the P and H matrices are not.

Constructor Summary

Constructors

Constructor and Description
PolarDecomposition3d()
PolarDecomposition3d(Matrix3dBase M)

Method Summary

Modifier and Type	Method and Description
void	factor(Matrix3dBase M)
void	factorLeft(Matrix3dBase M)
void	getH(Matrix3d H)
int	getMaxQDiagIndex() Returns the index of the maximum diagonal element of Q.
void	getN(Vector3d Ndiag) Return the diagonal elements of the matrix N which is used to flip rows or columns of Q (or some product thereof) in the event that Q is not right-handed.
Matrix3d	getP()
void	getP(Matrix3d P)
Matrix3d	getQ()
void	getQ(Matrix3d Q)
RotationMatrix3d	getR()
void	getR(Matrix3dBase R)
void	getSig(Vector3d sig)
Matrix3d	getV()
boolean	isRightHanded()
void	makeRightHanded(RotationMatrix3d R) Flip an axis of R in order to make it right-handed.

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

PolarDecomposition3d

public PolarDecomposition3d()

PolarDecomposition3d

public PolarDecomposition3d(Matrix3dBase M)

Method Detail

factor

public void factor(Matrix3dBase M)

factorLeft

public void factorLeft(Matrix3dBase M)

getR

public RotationMatrix3d getR()

getR

public void getR(Matrix3dBase R)

getQ

public Matrix3d getQ()

getQ

public void getQ(Matrix3d Q)

getH

public void getH(Matrix3d H)

getP

public Matrix3d getP()

getP

public void getP(Matrix3d P)

getV

public Matrix3d getV()

getSig

public void getSig(Vector3d sig)

isRightHanded

public boolean isRightHanded()

getN

public void getN(Vector3d Ndiag)

Return the diagonal elements of the matrix N which is used to flip rows or columns of Q (or some product thereof) in the event that Q is not right-handed. If det(Q) == 1, then N is the identity matrix. Otherwise, if det(Q) = -1, then N flips the column corresponding to the diagonal element of Q which is nearest to 1.

Parameters:

Ndiag - returns the diagonal elements of N.

getMaxQDiagIndex

public int getMaxQDiagIndex()

Returns the index of the maximum diagonal element of Q. This is used to create the matrix N which is returned by getN(maspack.matrix.Vector3d).

Returns:

index of the maximum diagonal element of Q.

makeRightHanded

public void makeRightHanded(RotationMatrix3d R)

Flip an axis of R in order to make it right-handed. The axis chosen is one that was transformed *least* by Q, which will correspond to a diagonal Q element nearest to 1.

Parameters:

R - unitary matrix