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)
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Return shear angle, direction and plane normal from shear matrix. |
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Return scalar t, unit vectors a, b so mat = t * np.outer(a, b) |
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Return shear angle, direction and plane normal from shear matrix. |
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Affine for shear by angle along vector direction on shear plane. |
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Matrix for shear by angle along direction vector on shear plane. |
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Construct shear matrix from upper triangular vector |
aff2sadn¶
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transforms3d.shears.aff2sadn(aff)¶ Return shear angle, direction and plane normal from shear matrix.
- Parameters
- matarray-like, shape (3,3)
shear matrix.
- Returns
- anglescalar
angle to shear, in radians
- directionarray, shape (3,)
direction along which to shear
- normalarray, shape (3,)
vector normal to shear plane
- pointarray, shape (3,)
point that, with normal, defines shear plane.
Notes
The translation part of the affine shear matrix is calculated using:
This holds for the ``i``th coordinate:
Then:
This can be compared with the equation of the plane:
where
dis the distance from the plane to the origin.Examples
>>> A = sadn2aff(0.5, [1, 0, 0], [0, 1, 0]) >>> angle, direction, normal, point = aff2sadn(A) >>> angle, direction, normal (0.5, array([1., 0., 0.]), array([0., 1., 0.])) >>> np.all(point == [0, 0, 0]) True >>> A_again = sadn2aff(angle, direction, normal, point) >>> np.allclose(A, A_again) True
inverse_outer¶
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transforms3d.shears.inverse_outer(mat)¶ Return scalar t, unit vectors a, b so mat = t * np.outer(a, b)
- Parameters
- matarray-like, shape (3,3)
shear matrix
- Returns
- tfloat
Scalar such that mat = t * np.outer(a, b)
- aarray, shape (3,)
Unit vector such that mat = t * np.outer(a, b)
- barray, shape (3,)
Unit vector such that mat = t * np.outer(a, b)
mat2sadn¶
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transforms3d.shears.mat2sadn(mat)¶ Return shear angle, direction and plane normal from shear matrix.
- Parameters
- matarray-like, shape (3,3)
shear matrix
- Returns
- anglescalar
angle to shear, in radians
- directionarray, shape (3,)
direction along which to shear
- normalarray, shape (3,)
vector defining shear plane, where shear plane passes through origin
Notes
The shear matrix we are decomposing was calculated using:
So the idea is to use an “inverse outer product” to recover the shears. See
inverse_outer()for the implementation.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
sadn2aff¶
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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
- anglescalar
angle to shear, in radians
- directionarray-like, shape (3,)
direction along which to shear
- normalarray-like, shape (3,)
vector normal to shear-plane
- pointNone or array-like, shape (3,), optional
point, that, with normal defines shear plane. Defaults to None, equivalent to shear-plane through origin.
- Returns
- affarray 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
sadn2mat¶
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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
- anglescalar
angle to shear, in radians
- directionarray-like, shape (3,)
direction along which to shear
- normalarray-like, shape (3,)
vector defining shear plane, where shear plane passes through origin
- Returns
- matarray 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¶
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transforms3d.shears.striu2mat(striu)¶ Construct shear matrix from upper triangular vector
- Parameters
- striuarray, shape (N,)
vector giving triangle above diagonal of shear matrix.
- Returns
- SMarray, shape (N, N)
shear matrix
Notes
Shear lengths are triangular numbers.
See http://en.wikipedia.org/wiki/Triangular_number
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