polar_decomposition

Specification

Location	Author	Maintained by
isl/math/matrix3.py	Kenny Erleben	DIKU

Purpose & Params

Polar Decomposition of matrix A (as described by Etzmuss et. al in ``A Fast Finite Solution for Cloth Modelling’’)

Properties

$R \cdot R^T = I$

$det(R) = 1$

Example

    M = np.random.rand(3,3)
    R, S = mat3.polar_decomposition(M)

    print("------ M -----")
    print(M)
    print("----- S^T -----")
    print(np.transpose(S))
    print(np.dot(R, np.transpose(R)))
    print(np.dot(R,S))

output:

   ------ M ------
   [[0.67751452 0.29537368 0.81107846]
    [0.7250961  0.16420613 0.87102864]
    [0.71635474 0.0468772  0.53182239]]
   ----- S^T -----
   [[0.92180171 0.14036562 0.79280104]
    [0.14036562 0.18668312 0.24870392]
    [0.79280104 0.24870392 1.00448349]]
   --- R * R^T ---
   [[ 1.00000000e+00  5.27355937e-15 -7.71605002e-15]
    [ 5.27355937e-15  1.00000000e+00 -5.82867088e-16]
    [-7.71605002e-15 -5.82867088e-16  1.00000000e+00]]
   ---- R * S ----
   [[0.67751452 0.29537368 0.81107846]
    [0.7250961  0.16420613 0.87102864]
    [0.71635474 0.0468772  0.53182239]]