py3dframe.matrix.SO3_project

SO3_project(matrix)

Project a matrix to the special orthogonal group \(SO(3)\) using SVD and minimisation of the frobenius norm.

The orthogonal group SO(3) is the set of 3x3 orthonormal matrices with determinant equal to 1.

To project a matrix to SO(3), the SVD is computed and the orthogonal matrix is obtained by:

\[\mathbf{O} = \mathbf{U} \mathbf{V}^T\]

If the determinant of the orthogonal matrix is -1, the last column of the component \(\mathbf{U}\) is multiplied by -1.

where \(\mathbf{U}\) and \(\mathbf{V}\) are the left and right singular vectors of the matrix such as:

\[\mathbf{M} = \mathbf{U} \mathbf{\Sigma} \mathbf{V}^T\]

See also

• function py3dframe.matrix.is_SO3() for the check of special orthogonality.

• function py3dframe.matrix.O3_project() for the projection of a matrix to the orthogonal group \(O(3)\).

Parameters:

matrix (array_like) – A 3x3 matrix to be projected.

Returns:

The SO(3) projection of the matrix.

Return type:

numpy.ndarray

Raises:

ValueError – If the matrix is not 3x3.

Examples

>>> import numpy
>>> from py3dframe import SO3_project
>>> e1 = numpy.array([1, 1, 0])
>>> e2 = numpy.array([-1, 1, 0])
>>> e3 = numpy.array([0, 0, 1])
>>> matrix = numpy.column_stack((e1, e2, e3))
>>> print(SO3_project(matrix))
[[ 0.70710678  -0.70710678  0.        ]
 [ 0.70710678  0.70710678  0.        ]
 [ 0.          0.          1.        ]]