# shears¶

Functions for working with shears

Terms used in function names:

• mat : array shape (3, 3) (3D non-homogenous coordinates)
• aff : affine array shape (4, 4) (3D homogenous coordinates)
• striu : shears encoded by vector giving triangular portion above diagonal of NxN array (for ND transformation)
• sadn : shears encoded by angle scalar, direction vector, normal vector (with optional point vector)
 aff2sadn(aff) Return shear angle, direction and plane normal from shear matrix. mat2sadn(mat) Return shear angle, direction and plane normal from shear matrix. sadn2aff(angle, direction, normal[, point]) Affine for shear by angle along vector direction on shear plane. sadn2mat(angle, direction, normal) Matrix for shear by angle along direction vector on shear plane. striu2mat(striu) Construct shear matrix from upper triangular vector

transforms3d.shears.aff2sadn(aff)

Return shear angle, direction and plane normal from shear matrix.

Parameters: mat : array-like, shape (3,3) shear matrix. angle : scalar angle to shear, in radians direction : array, shape (3,) direction along which to shear normal : array, shape (3,) vector normal to shear plane point : array, shape (3,) point that, with normal, defines shear plane.

Examples

>>> A = sadn2aff(0.5, [1, 0, 0], [0, 1, 0])
>>> angle, direction, normal, point = aff2sadn(A)
>>> angle, direction, normal, point
(0.5, array([-1.,  0.,  0.]), array([ 0., -1.,  0.]), array([ 0.,  0.,  0.]))
>>> A_again = sadn2aff(angle, direction, normal, point)
>>> np.allclose(A, A_again)
True


transforms3d.shears.mat2sadn(mat)

Return shear angle, direction and plane normal from shear matrix.

Parameters: mat : array-like, shape (3,3) shear matrix angle : scalar angle to shear, in radians direction : array, shape (3,) direction along which to shear normal : array, shape (3,) vector defining shear plane, where shear plane passes through origin

Examples

>>> M = sadn2mat(0.5, [1, 0, 0], [0, 1, 0])
>>> angle, direction, normal = mat2sadn(M)
>>> angle, direction, normal
(0.5, array([-1.,  0.,  0.]), array([ 0., -1.,  0.]))
>>> M_again = sadn2mat(angle, direction, normal)
>>> np.allclose(M, M_again)
True


transforms3d.shears.sadn2aff(angle, direction, normal, point=None)

Affine for shear by angle along vector direction on shear plane.

The shear plane is defined by a point and normal vector. The direction vector must be orthogonal to the plane’s normal vector.

A point P is transformed by the shear matrix into P” such that the vector P-P” is parallel to the direction vector and its extent is given by the angle of P-P’-P”, where P’ is the orthogonal projection of P onto the shear plane.

Parameters: angle : scalar angle to shear, in radians direction : array-like, shape (3,) direction along which to shear normal : array-like, shape (3,) vector normal to shear-plane point : None or array-like, shape (3,), optional point, that, with normal defines shear plane. Defaults to None, equivalent to shear-plane through origin. aff : array shape (4,4) affine shearing matrix

Examples

>>> angle = (np.random.random() - 0.5) * 4*math.pi
>>> direct = np.random.random(3) - 0.5
>>> normal = np.cross(direct, np.random.random(3))
>>> S = sadn2mat(angle, direct, normal)
>>> np.allclose(1.0, np.linalg.det(S))
True


transforms3d.shears.sadn2mat(angle, direction, normal)

Matrix for shear by angle along direction vector on shear plane.

The shear plane is defined by normal vector normal, and passes through the origin. The direction vector must be orthogonal to the plane’s normal vector.

A point P is transformed by the shear matrix into P” such that the vector P-P” is parallel to the direction vector and its extent is given by the angle of P-P’-P”, where P’ is the orthogonal projection of P onto the shear plane.

Parameters: angle : scalar angle to shear, in radians direction : array-like, shape (3,) direction along which to shear normal : array-like, shape (3,) vector defining shear plane, where shear plane passes through origin mat : array shape (3,3) shear matrix

Examples

>>> angle = (np.random.random() - 0.5) * 4*math.pi
>>> direct = np.random.random(3) - 0.5
>>> normal = np.cross(direct, np.random.random(3))
>>> S = sadn2aff(angle, direct, normal)
>>> np.allclose(1.0, np.linalg.det(S))
True


## striu2mat¶

transforms3d.shears.striu2mat(striu)

Construct shear matrix from upper triangular vector

Parameters: striu : array, shape (N,) vector giving triangle above diagonal of shear matrix. SM : array, shape (N, N) shear matrix

Notes

Shear lengths are triangular numbers.

Examples

>>> S = [0.1, 0.2, 0.3]
>>> striu2mat(S)
array([[ 1. ,  0.1,  0.2],
[ 0. ,  1. ,  0.3],
[ 0. ,  0. ,  1. ]])
>>> striu2mat([1])
array([[ 1.,  1.],
[ 0.,  1.]])
>>> striu2mat([1, 2])
Traceback (most recent call last):
...
ValueError: 2 is a strange number of shear elements