thirdparty

Class PolarDecomposition

java.lang.Object

thirdparty.PolarDecomposition

public final class PolarDecomposition
extends java.lang.Object

References:

• Paper: Ken Shoemake and Tom Duff. Matrix Animation and Polar Decomposition.
  Proceedings of Graphics Interface 1992.
• Book: Graphic Gems IV III.4 Polar Matrix Decomposition.
• c implementation by Ken Shoemake, 1993 (Decompose.c, Decompose.h and QuatTypes.h).

Author:

Ken Shoemake, 1993 (original c implementation found in Decompose.c and Decompose.h)

Method Summary

Modifier and Type	Method and Description
static void	decomp_affine(Mat4x4 M, Transform dst) Decompose 4x4 affine matrix A as TFRUK(U transpose), where t contains the translation components, q contains the rotation R, u contains U, k contains scale factors, and f contains the sign of the determinant.
protected static float	polar_decomp(Mat4x4 M, Mat4x4 Q, Mat4x4 S) Polar Decomposition of 3x3 matrix in 4x4, M = QS.
protected static Quat	snuggle(Quat q_, Vec3 k, Quat dst) Given a unit quaternion, q, and a scale vector, k, find a unit quaternion, p, which permutes the axes and turns freely in the plane of duplicate scale factors, such that q p has the largest possible w component, i.e. the smallest possible angle.
protected static void	spect_decomp(Vec3 dst, Mat4x4 S, Mat4x4 U) Compute the spectral decomposition of symmetric positive semi-definite S.

Methods inherited from class java.lang.Object

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

Method Detail

decomp_affine

public static void decomp_affine(Mat4x4 M,
                                 Transform dst)

Decompose 4x4 affine matrix A as TFRUK(U transpose), where t contains the translation components, q contains the rotation R, u contains U, k contains scale factors, and f contains the sign of the determinant.
Assumes A transforms column vectors in right-handed coordinates.

See Ken Shoemake and Tom Duff. Matrix Animation and Polar Decomposition.
Proceedings of Graphics Interface 1992.

snuggle

protected static Quat snuggle(Quat q_,
                              Vec3 k,
                              Quat dst)

Given a unit quaternion, q, and a scale vector, k, find a unit quaternion, p, which permutes the axes and turns freely in the plane of duplicate scale factors, such that q p has the largest possible w component, i.e. the smallest possible angle. Permutes k's components to go with q p instead of q.

See Ken Shoemake and Tom Duff. Matrix Animation and Polar Decomposition.
Proceedings of Graphics Interface 1992. Details on p. 262-263.

spect_decomp

protected static void spect_decomp(Vec3 dst,
                                   Mat4x4 S,
                                   Mat4x4 U)

Compute the spectral decomposition of symmetric positive semi-definite S. Returns rotation in U and scale factors in result, so that if K is a diagonal matrix of the scale factors, then S = U K (U transpose). Uses Jacobi method.

See Gene H. Golub and Charles F. Van Loan. Matrix Computations. Hopkins 1983.

polar_decomp

protected static float polar_decomp(Mat4x4 M,
                                    Mat4x4 Q,
                                    Mat4x4 S)

Polar Decomposition of 3x3 matrix in 4x4, M = QS.

See Nicholas Higham and Robert S. Schreiber, Fast Polar Decomposition of An Arbitrary Matrix, Technical Report 88-942, October 1988, Department of Computer Science, Cornell University.