Transformations#

The pygeostat.transformations module provides functions for calculating GSLIB rotations, coordinate transformations, and 3D geometric operations.

GSLIB Rotations#

get_rotation_matrix#

pygeostat.transformations.get_rotation_matrix(azm, dip, tilt)[source]#

Returns the GSLIB rotation matrix given GSLIB angles for azimuth, dip and tilt

principalvectors#

pygeostat.transformations.principalvectors(azm, dip, tilt, lefthandrule=True, majorlen=1.0, minorlen=1.0, vertlen=1.0)[source]#

Returns the GSLIB principal vectors: NOTE: we typically think of things/visualize using the left hand rule in which case the original axes are: major_axis, minor_axis, vert_axis = [1, 0, 0], [0, -1, 0], [0, 0, 1] although the GSLIB rotation matrix actually defines them as: major_axis, minor_axis, vert_axis = [1, 0, 0], [0, 1, 0], [0, 0, 1] This is only an issue in practice for an asymmetric minor axis

azmdip#

pygeostat.transformations.azmdip(point_vector)[source]#

Calculates the azimuth/dip to a point (length 3 vector [x,y,z]) from the origin

principaldirs#

pygeostat.transformations.principaldirs(azm, dip, tilt)[source]#

Calculate the principal directions from a plane of major continuity

Visualization#

Arrow3D Class#

class pygeostat.transformations.Arrow3D(xs, ys, zs, *args, **kwargs)[source]#

Arrow3D from http://stackoverflow.com/questions/11140163/python-matplotlib-plotting-a-3d-cube-a-sphere-and-a-vector

draw(renderer)[source]#

Draw the Artist (and its children) using the given renderer.

This has no effect if the artist is not visible (.Artist.get_visible returns False).

Parameters:

renderer (~matplotlib.backend_bases.RendererBase subclass.)

Notes

This method is overridden in the Artist subclasses.

set(*, agg_filter=<UNSET>, alpha=<UNSET>, animated=<UNSET>, antialiased=<UNSET>, arrowstyle=<UNSET>, capstyle=<UNSET>, clip_box=<UNSET>, clip_on=<UNSET>, clip_path=<UNSET>, color=<UNSET>, connectionstyle=<UNSET>, edgecolor=<UNSET>, facecolor=<UNSET>, fill=<UNSET>, gid=<UNSET>, hatch=<UNSET>, hatch_linewidth=<UNSET>, in_layout=<UNSET>, joinstyle=<UNSET>, label=<UNSET>, linestyle=<UNSET>, linewidth=<UNSET>, mouseover=<UNSET>, mutation_aspect=<UNSET>, mutation_scale=<UNSET>, patchA=<UNSET>, patchB=<UNSET>, path_effects=<UNSET>, picker=<UNSET>, positions=<UNSET>, rasterized=<UNSET>, sketch_params=<UNSET>, snap=<UNSET>, transform=<UNSET>, url=<UNSET>, visible=<UNSET>, zorder=<UNSET>)#

Set multiple properties at once.

Supported properties are

Properties:

agg_filter: a filter function, which takes a (m, n, 3) float array and a dpi value, and returns a (m, n, 3) array and two offsets from the bottom left corner of the image alpha: float or None animated: bool antialiased or aa: bool or None arrowstyle: [ ‘-’ | ‘<-’ | ‘->’ | ‘<->’ | ‘<|-' | '-|>’ | ‘<|-|>’ | ‘]-’ | ‘-[’ | ‘]-[’ | ‘|-|’ | ‘]->’ | ‘<-[’ | ‘simple’ | ‘fancy’ | ‘wedge’ ] capstyle: .CapStyle or {‘butt’, ‘projecting’, ‘round’} clip_box: ~matplotlib.transforms.BboxBase or None clip_on: bool clip_path: Patch or (Path, Transform) or None color: :mpltype:`color` connectionstyle: [ ‘arc3’ | ‘angle3’ | ‘angle’ | ‘arc’ | ‘bar’ ] edgecolor or ec: :mpltype:`color` or None facecolor or fc: :mpltype:`color` or None figure: ~matplotlib.figure.Figure or ~matplotlib.figure.SubFigure fill: bool gid: str hatch: {‘/’, ‘\’, ‘|’, ‘-’, ‘+’, ‘x’, ‘o’, ‘O’, ‘.’, ‘*’} hatch_linewidth: unknown in_layout: bool joinstyle: .JoinStyle or {‘miter’, ‘round’, ‘bevel’} label: object linestyle or ls: {‘-’, ‘–’, ‘-.’, ‘:’, ‘’, (offset, on-off-seq), …} linewidth or lw: float or None mouseover: bool mutation_aspect: float mutation_scale: float patchA: .patches.Patch patchB: .patches.Patch path_effects: list of .AbstractPathEffect picker: None or bool or float or callable positions: unknown rasterized: bool sketch_params: (scale: float, length: float, randomness: float) snap: bool or None transform: ~matplotlib.transforms.Transform url: str visible: bool zorder: float

plotprincipalvectors#

pygeostat.transformations.plotprincipalvectors(axes, azm, dip, tilt, lefthandrule=True, majorlen=1.0, minorlen=1.0, vertlen=1.0)[source]#

Add principal direction unit arrows to plot

drawgsvectorwidget#

pygeostat.transformations.drawgsvectorwidget(legend=True, sidetext=True, majortitle=False)[source]#

Draws an interactive GSLIB vector widget in an IPython notebook.

Warning

Must be run with either:

%matplotlib inline or %matplotlib notebook activated

Code author: Jared Deutsch 2016-02-21

drawellipsoid#

pygeostat.transformations.drawellipsoid(ax, hmax, hmin, vert, azm=0.0, dip=0.0, tilt=0.0, color='#87CEFA', alpha=0.5)[source]#

Draws an orientatable ellipsoid with GSLIB conventions.

Spherical coordinate calculation from:

Rotation uses GSLIB rotation matrix definition

Code author: Jared Deutsch 2016-03-06

drawgsaniswidget#

pygeostat.transformations.drawgsaniswidget(legend=True, sidetext=True, majortitle=False)[source]#

Draws an interactive anisotropy widget in an IPython notebook.

Warning

Must be run with either:

%matplotlib inline or %matplotlib notebook activated

Code author: Jared Deutsch 2016-03-06

Matrix Operations#

identity_matrix#

pygeostat.transformations.identity_matrix()[source]#

Return 4x4 identity/unit matrix.

>>> I = identity_matrix()
>>> numpy.allclose(I, numpy.dot(I, I))
True
>>> numpy.sum(I), numpy.trace(I)
(4.0, 4.0)
>>> numpy.allclose(I, numpy.identity(4))
True

inverse_matrix#

pygeostat.transformations.inverse_matrix(matrix)[source]#

Return inverse of square transformation matrix.

>>> M0 = random_rotation_matrix()
>>> M1 = inverse_matrix(M0.T)
>>> numpy.allclose(M1, numpy.linalg.inv(M0.T))
True
>>> for size in range(1, 7):
...     M0 = numpy.random.rand(size, size)
...     M1 = inverse_matrix(M0)
...     if not numpy.allclose(M1, numpy.linalg.inv(M0)): print(size)

concatenate_matrices#

pygeostat.transformations.concatenate_matrices(*matrices)[source]#

Return concatenation of series of transformation matrices.

>>> M = numpy.random.rand(16).reshape((4, 4)) - 0.5
>>> numpy.allclose(M, concatenate_matrices(M))
True
>>> numpy.allclose(numpy.dot(M, M.T), concatenate_matrices(M, M.T))
True

decompose_matrix#

pygeostat.transformations.decompose_matrix(matrix)[source]#

Return sequence of transformations from transformation matrix.

matrixarray_like

Non-degenerative homogeneous transformation matrix

Return tuple of:

scale : vector of 3 scaling factors shear : list of shear factors for x-y, x-z, y-z axes angles : list of Euler angles about static x, y, z axes translate : translation vector along x, y, z axes perspective : perspective partition of matrix

Raise ValueError if matrix is of wrong type or degenerative.

>>> T0 = translation_matrix([1, 2, 3])
>>> scale, shear, angles, trans, persp = decompose_matrix(T0)
>>> T1 = translation_matrix(trans)
>>> numpy.allclose(T0, T1)
True
>>> S = scale_matrix(0.123)
>>> scale, shear, angles, trans, persp = decompose_matrix(S)
>>> scale[0]
0.123
>>> R0 = euler_matrix(1, 2, 3)
>>> scale, shear, angles, trans, persp = decompose_matrix(R0)
>>> R1 = euler_matrix(*angles)
>>> numpy.allclose(R0, R1)
True

compose_matrix#

pygeostat.transformations.compose_matrix(scale=None, shear=None, angles=None, translate=None, perspective=None)[source]#

Return transformation matrix from sequence of transformations.

This is the inverse of the decompose_matrix function.

Sequence of transformations:

scale : vector of 3 scaling factors shear : list of shear factors for x-y, x-z, y-z axes angles : list of Euler angles about static x, y, z axes translate : translation vector along x, y, z axes perspective : perspective partition of matrix

>>> scale = numpy.random.random(3) - 0.5
>>> shear = numpy.random.random(3) - 0.5
>>> angles = (numpy.random.random(3) - 0.5) * (2*math.pi)
>>> trans = numpy.random.random(3) - 0.5
>>> persp = numpy.random.random(4) - 0.5
>>> M0 = compose_matrix(scale, shear, angles, trans, persp)
>>> result = decompose_matrix(M0)
>>> M1 = compose_matrix(*result)
>>> is_same_transform(M0, M1)
True

is_same_transform#

pygeostat.transformations.is_same_transform(matrix0, matrix1)[source]#

Return True if two matrices perform same transformation.

>>> is_same_transform(numpy.identity(4), numpy.identity(4))
True
>>> is_same_transform(numpy.identity(4), random_rotation_matrix())
False

Translation#

translation_matrix#

pygeostat.transformations.translation_matrix(direction)[source]#

Return matrix to translate by direction vector.

>>> v = numpy.random.random(3) - 0.5
>>> numpy.allclose(v, translation_matrix(v)[:3, 3])
True

translation_from_matrix#

pygeostat.transformations.translation_from_matrix(matrix)[source]#

Return translation vector from translation matrix.

>>> v0 = numpy.random.random(3) - 0.5
>>> v1 = translation_from_matrix(translation_matrix(v0))
>>> numpy.allclose(v0, v1)
True

Rotation#

rotation_matrix#

pygeostat.transformations.rotation_matrix(angle, direction, point=None)[source]#

Return matrix to rotate about axis defined by point and direction.

>>> R = rotation_matrix(math.pi/2, [0, 0, 1], [1, 0, 0])
>>> numpy.allclose(numpy.dot(R, [0, 0, 0, 1]), [1, -1, 0, 1])
True
>>> angle = (random.random() - 0.5) * (2*math.pi)
>>> direc = numpy.random.random(3) - 0.5
>>> point = numpy.random.random(3) - 0.5
>>> R0 = rotation_matrix(angle, direc, point)
>>> R1 = rotation_matrix(angle-2*math.pi, direc, point)
>>> is_same_transform(R0, R1)
True
>>> R0 = rotation_matrix(angle, direc, point)
>>> R1 = rotation_matrix(-angle, -direc, point)
>>> is_same_transform(R0, R1)
True
>>> I = numpy.identity(4, numpy.float64)
>>> numpy.allclose(I, rotation_matrix(math.pi*2, direc))
True
>>> numpy.allclose(2, numpy.trace(rotation_matrix(math.pi/2,
...                                               direc, point)))
True

rotation_from_matrix#

pygeostat.transformations.rotation_from_matrix(matrix)[source]#

Return rotation angle and axis from rotation matrix.

>>> angle = (random.random() - 0.5) * (2*math.pi)
>>> direc = numpy.random.random(3) - 0.5
>>> point = numpy.random.random(3) - 0.5
>>> R0 = rotation_matrix(angle, direc, point)
>>> angle, direc, point = rotation_from_matrix(R0)
>>> R1 = rotation_matrix(angle, direc, point)
>>> is_same_transform(R0, R1)
True

random_rotation_matrix#

pygeostat.transformations.random_rotation_matrix(rand=None)[source]#

Return uniform random rotation matrix.

rand: array like

Three independent random variables that are uniformly distributed between 0 and 1 for each returned quaternion.

>>> R = random_rotation_matrix()
>>> numpy.allclose(numpy.dot(R.T, R), numpy.identity(4))
True

Reflection#

reflection_matrix#

pygeostat.transformations.reflection_matrix(point, normal)[source]#

Return matrix to mirror at plane defined by point and normal vector.

>>> v0 = numpy.random.random(4) - 0.5
>>> v0[3] = 1.
>>> v1 = numpy.random.random(3) - 0.5
>>> R = reflection_matrix(v0, v1)
>>> numpy.allclose(2, numpy.trace(R))
True
>>> numpy.allclose(v0, numpy.dot(R, v0))
True
>>> v2 = v0.copy()
>>> v2[:3] += v1
>>> v3 = v0.copy()
>>> v2[:3] -= v1
>>> numpy.allclose(v2, numpy.dot(R, v3))
True

reflection_from_matrix#

pygeostat.transformations.reflection_from_matrix(matrix)[source]#

Return mirror plane point and normal vector from reflection matrix.

>>> v0 = numpy.random.random(3) - 0.5
>>> v1 = numpy.random.random(3) - 0.5
>>> M0 = reflection_matrix(v0, v1)
>>> point, normal = reflection_from_matrix(M0)
>>> M1 = reflection_matrix(point, normal)
>>> is_same_transform(M0, M1)
True

Scaling#

scale_matrix#

pygeostat.transformations.scale_matrix(factor, origin=None, direction=None)[source]#

Return matrix to scale by factor around origin in direction.

Use factor -1 for point symmetry.

>>> v = (numpy.random.rand(4, 5) - 0.5) * 20
>>> v[3] = 1
>>> S = scale_matrix(-1.234)
>>> numpy.allclose(numpy.dot(S, v)[:3], -1.234*v[:3])
True
>>> factor = random.random() * 10 - 5
>>> origin = numpy.random.random(3) - 0.5
>>> direct = numpy.random.random(3) - 0.5
>>> S = scale_matrix(factor, origin)
>>> S = scale_matrix(factor, origin, direct)

scale_from_matrix#

pygeostat.transformations.scale_from_matrix(matrix)[source]#

Return scaling factor, origin and direction from scaling matrix.

>>> factor = random.random() * 10 - 5
>>> origin = numpy.random.random(3) - 0.5
>>> direct = numpy.random.random(3) - 0.5
>>> S0 = scale_matrix(factor, origin)
>>> factor, origin, direction = scale_from_matrix(S0)
>>> S1 = scale_matrix(factor, origin, direction)
>>> is_same_transform(S0, S1)
True
>>> S0 = scale_matrix(factor, origin, direct)
>>> factor, origin, direction = scale_from_matrix(S0)
>>> S1 = scale_matrix(factor, origin, direction)
>>> is_same_transform(S0, S1)
True

Projection#

projection_matrix#

pygeostat.transformations.projection_matrix(point, normal, direction=None, perspective=None, pseudo=False)[source]#

Return matrix to project onto plane defined by point and normal.

Using either perspective point, projection direction, or none of both.

If pseudo is True, perspective projections will preserve relative depth such that Perspective = dot(Orthogonal, PseudoPerspective).

>>> P = projection_matrix([0, 0, 0], [1, 0, 0])
>>> numpy.allclose(P[1:, 1:], numpy.identity(4)[1:, 1:])
True
>>> point = numpy.random.random(3) - 0.5
>>> normal = numpy.random.random(3) - 0.5
>>> direct = numpy.random.random(3) - 0.5
>>> persp = numpy.random.random(3) - 0.5
>>> P0 = projection_matrix(point, normal)
>>> P1 = projection_matrix(point, normal, direction=direct)
>>> P2 = projection_matrix(point, normal, perspective=persp)
>>> P3 = projection_matrix(point, normal, perspective=persp, pseudo=True)
>>> is_same_transform(P2, numpy.dot(P0, P3))
True
>>> P = projection_matrix([3, 0, 0], [1, 1, 0], [1, 0, 0])
>>> v0 = (numpy.random.rand(4, 5) - 0.5) * 20
>>> v0[3] = 1
>>> v1 = numpy.dot(P, v0)
>>> numpy.allclose(v1[1], v0[1])
True
>>> numpy.allclose(v1[0], 3-v1[1])
True

projection_from_matrix#

pygeostat.transformations.projection_from_matrix(matrix, pseudo=False)[source]#

Return projection plane and perspective point from projection matrix.

Return values are same as arguments for projection_matrix function: point, normal, direction, perspective, and pseudo.

>>> point = numpy.random.random(3) - 0.5
>>> normal = numpy.random.random(3) - 0.5
>>> direct = numpy.random.random(3) - 0.5
>>> persp = numpy.random.random(3) - 0.5
>>> P0 = projection_matrix(point, normal)
>>> result = projection_from_matrix(P0)
>>> P1 = projection_matrix(*result)
>>> is_same_transform(P0, P1)
True
>>> P0 = projection_matrix(point, normal, direct)
>>> result = projection_from_matrix(P0)
>>> P1 = projection_matrix(*result)
>>> is_same_transform(P0, P1)
True
>>> P0 = projection_matrix(point, normal, perspective=persp, pseudo=False)
>>> result = projection_from_matrix(P0, pseudo=False)
>>> P1 = projection_matrix(*result)
>>> is_same_transform(P0, P1)
True
>>> P0 = projection_matrix(point, normal, perspective=persp, pseudo=True)
>>> result = projection_from_matrix(P0, pseudo=True)
>>> P1 = projection_matrix(*result)
>>> is_same_transform(P0, P1)
True

clip_matrix#

pygeostat.transformations.clip_matrix(left, right, bottom, top, near, far, perspective=False)[source]#

Return matrix to obtain normalized device coordinates from frustum.

The frustum bounds are axis-aligned along x (left, right), y (bottom, top) and z (near, far).

Normalized device coordinates are in range [-1, 1] if coordinates are inside the frustum.

If perspective is True the frustum is a truncated pyramid with the perspective point at origin and direction along z axis, otherwise an orthographic canonical view volume (a box).

Homogeneous coordinates transformed by the perspective clip matrix need to be dehomogenized (divided by w coordinate).

>>> frustum = numpy.random.rand(6)
>>> frustum[1] += frustum[0]
>>> frustum[3] += frustum[2]
>>> frustum[5] += frustum[4]
>>> M = clip_matrix(perspective=False, *frustum)
>>> numpy.dot(M, [frustum[0], frustum[2], frustum[4], 1])
array([-1., -1., -1.,  1.])
>>> numpy.dot(M, [frustum[1], frustum[3], frustum[5], 1])
array([ 1.,  1.,  1.,  1.])
>>> M = clip_matrix(perspective=True, *frustum)
>>> v = numpy.dot(M, [frustum[0], frustum[2], frustum[4], 1])
>>> v / v[3]
array([-1., -1., -1.,  1.])
>>> v = numpy.dot(M, [frustum[1], frustum[3], frustum[4], 1])
>>> v / v[3]
array([ 1.,  1., -1.,  1.])

Shear#

shear_matrix#

pygeostat.transformations.shear_matrix(angle, direction, point, normal)[source]#

Return matrix to shear by angle along direction vector 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.

>>> angle = (random.random() - 0.5) * 4*math.pi
>>> direct = numpy.random.random(3) - 0.5
>>> point = numpy.random.random(3) - 0.5
>>> normal = numpy.cross(direct, numpy.random.random(3))
>>> S = shear_matrix(angle, direct, point, normal)
>>> numpy.allclose(1, numpy.linalg.det(S))
True

shear_from_matrix#

pygeostat.transformations.shear_from_matrix(matrix)[source]#

Return shear angle, direction and plane from shear matrix.

>>> angle = (random.random() - 0.5) * 4*math.pi
>>> direct = numpy.random.random(3) - 0.5
>>> point = numpy.random.random(3) - 0.5
>>> normal = numpy.cross(direct, numpy.random.random(3))
>>> S0 = shear_matrix(angle, direct, point, normal)
>>> angle, direct, point, normal = shear_from_matrix(S0)
>>> S1 = shear_matrix(angle, direct, point, normal)
>>> is_same_transform(S0, S1)
True

Affine Transformations#

affine_matrix_from_points#

pygeostat.transformations.affine_matrix_from_points(v0, v1, shear=True, scale=True, usesvd=True)[source]#

Return affine transform matrix to register two point sets.

v0 and v1 are shape (ndims, *) arrays of at least ndims non-homogeneous coordinates, where ndims is the dimensionality of the coordinate space.

If shear is False, a similarity transformation matrix is returned. If also scale is False, a rigid/Euclidean transformation matrix is returned.

By default the algorithm by Hartley and Zissermann [15] is used. If usesvd is True, similarity and Euclidean transformation matrices are calculated by minimizing the weighted sum of squared deviations (RMSD) according to the algorithm by Kabsch [8]. Otherwise, and if ndims is 3, the quaternion based algorithm by Horn [9] is used, which is slower when using this Python implementation.

The returned matrix performs rotation, translation and uniform scaling (if specified).

>>> v0 = [[0, 1031, 1031, 0], [0, 0, 1600, 1600]]
>>> v1 = [[675, 826, 826, 677], [55, 52, 281, 277]]
>>> affine_matrix_from_points(v0, v1)
array([[   0.14549,    0.00062,  675.50008],
       [   0.00048,    0.14094,   53.24971],
       [   0.     ,    0.     ,    1.     ]])
>>> T = translation_matrix(numpy.random.random(3)-0.5)
>>> R = random_rotation_matrix(numpy.random.random(3))
>>> S = scale_matrix(random.random())
>>> M = concatenate_matrices(T, R, S)
>>> v0 = (numpy.random.rand(4, 100) - 0.5) * 20
>>> v0[3] = 1
>>> v1 = numpy.dot(M, v0)
>>> v0[:3] += numpy.random.normal(0, 1e-8, 300).reshape(3, -1)
>>> M = affine_matrix_from_points(v0[:3], v1[:3])
>>> numpy.allclose(v1, numpy.dot(M, v0))
True

More examples in superimposition_matrix()

superimposition_matrix#

pygeostat.transformations.superimposition_matrix(v0, v1, scale=False, usesvd=True)[source]#

Return matrix to transform given 3D point set into second point set.

v0 and v1 are shape (3, *) or (4, *) arrays of at least 3 points.

The parameters scale and usesvd are explained in the more general affine_matrix_from_points function.

The returned matrix is a similarity or Euclidean transformation matrix. This function has a fast C implementation in transformations.c.

>>> v0 = numpy.random.rand(3, 10)
>>> M = superimposition_matrix(v0, v0)
>>> numpy.allclose(M, numpy.identity(4))
True
>>> R = random_rotation_matrix(numpy.random.random(3))
>>> v0 = [[1,0,0], [0,1,0], [0,0,1], [1,1,1]]
>>> v1 = numpy.dot(R, v0)
>>> M = superimposition_matrix(v0, v1)
>>> numpy.allclose(v1, numpy.dot(M, v0))
True
>>> v0 = (numpy.random.rand(4, 100) - 0.5) * 20
>>> v0[3] = 1
>>> v1 = numpy.dot(R, v0)
>>> M = superimposition_matrix(v0, v1)
>>> numpy.allclose(v1, numpy.dot(M, v0))
True
>>> S = scale_matrix(random.random())
>>> T = translation_matrix(numpy.random.random(3)-0.5)
>>> M = concatenate_matrices(T, R, S)
>>> v1 = numpy.dot(M, v0)
>>> v0[:3] += numpy.random.normal(0, 1e-9, 300).reshape(3, -1)
>>> M = superimposition_matrix(v0, v1, scale=True)
>>> numpy.allclose(v1, numpy.dot(M, v0))
True
>>> M = superimposition_matrix(v0, v1, scale=True, usesvd=False)
>>> numpy.allclose(v1, numpy.dot(M, v0))
True
>>> v = numpy.empty((4, 100, 3))
>>> v[:, :, 0] = v0
>>> M = superimposition_matrix(v0, v1, scale=True, usesvd=False)
>>> numpy.allclose(v1, numpy.dot(M, v[:, :, 0]))
True

orthogonalization_matrix#

pygeostat.transformations.orthogonalization_matrix(lengths, angles)[source]#

Return orthogonalization matrix for crystallographic cell coordinates.

Angles are expected in degrees.

The de-orthogonalization matrix is the inverse.

>>> O = orthogonalization_matrix([10, 10, 10], [90, 90, 90])
>>> numpy.allclose(O[:3, :3], numpy.identity(3, float) * 10)
True
>>> O = orthogonalization_matrix([9.8, 12.0, 15.5], [87.2, 80.7, 69.7])
>>> numpy.allclose(numpy.sum(O), 43.063229)
True

Euler Angles#

euler_matrix#

pygeostat.transformations.euler_matrix(ai, aj, ak, axes='sxyz')[source]#

Return homogeneous rotation matrix from Euler angles and axis sequence.

ai, aj, ak : Euler’s roll, pitch and yaw angles axes : One of 24 axis sequences as string or encoded tuple

>>> R = euler_matrix(1, 2, 3, 'syxz')
>>> numpy.allclose(numpy.sum(R[0]), -1.34786452)
True
>>> R = euler_matrix(1, 2, 3, (0, 1, 0, 1))
>>> numpy.allclose(numpy.sum(R[0]), -0.383436184)
True
>>> ai, aj, ak = (4*math.pi) * (numpy.random.random(3) - 0.5)
>>> for axes in _AXES2TUPLE.keys():
...    R = euler_matrix(ai, aj, ak, axes)
>>> for axes in _TUPLE2AXES.keys():
...    R = euler_matrix(ai, aj, ak, axes)

euler_from_matrix#

pygeostat.transformations.euler_from_matrix(matrix, axes='sxyz')[source]#

Return Euler angles from rotation matrix for specified axis sequence.

axes : One of 24 axis sequences as string or encoded tuple

Note that many Euler angle triplets can describe one matrix.

>>> R0 = euler_matrix(1, 2, 3, 'syxz')
>>> al, be, ga = euler_from_matrix(R0, 'syxz')
>>> R1 = euler_matrix(al, be, ga, 'syxz')
>>> numpy.allclose(R0, R1)
True
>>> angles = (4*math.pi) * (numpy.random.random(3) - 0.5)
>>> for axes in _AXES2TUPLE.keys():
...    R0 = euler_matrix(axes=axes, *angles)
...    R1 = euler_matrix(axes=axes, *euler_from_matrix(R0, axes))
...    if not numpy.allclose(R0, R1): print(axes, "failed")

euler_from_quaternion#

pygeostat.transformations.euler_from_quaternion(quaternion, axes='sxyz')[source]#

Return Euler angles from quaternion for specified axis sequence.

>>> angles = euler_from_quaternion([0.99810947, 0.06146124, 0, 0])
>>> numpy.allclose(angles, [0.123, 0, 0])
True

Quaternions#

quaternion_from_euler#

pygeostat.transformations.quaternion_from_euler(ai, aj, ak, axes='sxyz')[source]#

Return quaternion from Euler angles and axis sequence.

ai, aj, ak : Euler’s roll, pitch and yaw angles axes : One of 24 axis sequences as string or encoded tuple

>>> q = quaternion_from_euler(1, 2, 3, 'ryxz')
>>> numpy.allclose(q, [0.435953, 0.310622, -0.718287, 0.444435])
True

quaternion_about_axis#

pygeostat.transformations.quaternion_about_axis(angle, axis)[source]#

Return quaternion for rotation about axis.

>>> q = quaternion_about_axis(0.123, [1, 0, 0])
>>> numpy.allclose(q, [0.99810947, 0.06146124, 0, 0])
True

quaternion_matrix#

pygeostat.transformations.quaternion_matrix(quaternion)[source]#

Return homogeneous rotation matrix from quaternion.

>>> M = quaternion_matrix([0.99810947, 0.06146124, 0, 0])
>>> numpy.allclose(M, rotation_matrix(0.123, [1, 0, 0]))
True
>>> M = quaternion_matrix([1, 0, 0, 0])
>>> numpy.allclose(M, numpy.identity(4))
True
>>> M = quaternion_matrix([0, 1, 0, 0])
>>> numpy.allclose(M, numpy.diag([1, -1, -1, 1]))
True

quaternion_from_matrix#

pygeostat.transformations.quaternion_from_matrix(matrix, isprecise=False)[source]#

Return quaternion from rotation matrix.

If isprecise is True, the input matrix is assumed to be a precise rotation matrix and a faster algorithm is used.

>>> q = quaternion_from_matrix(numpy.identity(4), True)
>>> numpy.allclose(q, [1, 0, 0, 0])
True
>>> q = quaternion_from_matrix(numpy.diag([1, -1, -1, 1]))
>>> numpy.allclose(q, [0, 1, 0, 0]) or numpy.allclose(q, [0, -1, 0, 0])
True
>>> R = rotation_matrix(0.123, (1, 2, 3))
>>> q = quaternion_from_matrix(R, True)
>>> numpy.allclose(q, [0.9981095, 0.0164262, 0.0328524, 0.0492786])
True
>>> R = [[-0.545, 0.797, 0.260, 0], [0.733, 0.603, -0.313, 0],
...      [-0.407, 0.021, -0.913, 0], [0, 0, 0, 1]]
>>> q = quaternion_from_matrix(R)
>>> numpy.allclose(q, [0.19069, 0.43736, 0.87485, -0.083611])
True
>>> R = [[0.395, 0.362, 0.843, 0], [-0.626, 0.796, -0.056, 0],
...      [-0.677, -0.498, 0.529, 0], [0, 0, 0, 1]]
>>> q = quaternion_from_matrix(R)
>>> numpy.allclose(q, [0.82336615, -0.13610694, 0.46344705, -0.29792603])
True
>>> R = random_rotation_matrix()
>>> q = quaternion_from_matrix(R)
>>> is_same_transform(R, quaternion_matrix(q))
True
>>> R = euler_matrix(0.0, 0.0, numpy.pi/2.0)
>>> numpy.allclose(quaternion_from_matrix(R, isprecise=False),
...                quaternion_from_matrix(R, isprecise=True))
True

quaternion_multiply#

pygeostat.transformations.quaternion_multiply(quaternion1, quaternion0)[source]#

Return multiplication of two quaternions.

>>> q = quaternion_multiply([4, 1, -2, 3], [8, -5, 6, 7])
>>> numpy.allclose(q, [28, -44, -14, 48])
True

quaternion_conjugate#

pygeostat.transformations.quaternion_conjugate(quaternion)[source]#

Return conjugate of quaternion.

>>> q0 = random_quaternion()
>>> q1 = quaternion_conjugate(q0)
>>> q1[0] == q0[0] and all(q1[1:] == -q0[1:])
True

quaternion_inverse#

pygeostat.transformations.quaternion_inverse(quaternion)[source]#

Return inverse of quaternion.

>>> q0 = random_quaternion()
>>> q1 = quaternion_inverse(q0)
>>> numpy.allclose(quaternion_multiply(q0, q1), [1, 0, 0, 0])
True

quaternion_real#

pygeostat.transformations.quaternion_real(quaternion)[source]#

Return real part of quaternion.

>>> quaternion_real([3, 0, 1, 2])
3.0

quaternion_imag#

pygeostat.transformations.quaternion_imag(quaternion)[source]#

Return imaginary part of quaternion.

>>> quaternion_imag([3, 0, 1, 2])
array([ 0.,  1.,  2.])

quaternion_slerp#

pygeostat.transformations.quaternion_slerp(quat0, quat1, fraction, spin=0, shortestpath=True)[source]#

Return spherical linear interpolation between two quaternions.

>>> q0 = random_quaternion()
>>> q1 = random_quaternion()
>>> q = quaternion_slerp(q0, q1, 0)
>>> numpy.allclose(q, q0)
True
>>> q = quaternion_slerp(q0, q1, 1, 1)
>>> numpy.allclose(q, q1)
True
>>> q = quaternion_slerp(q0, q1, 0.5)
>>> angle = math.acos(numpy.dot(q0, q))
>>> numpy.allclose(2, math.acos(numpy.dot(q0, q1)) / angle) or         numpy.allclose(2, math.acos(-numpy.dot(q0, q1)) / angle)
True

random_quaternion#

pygeostat.transformations.random_quaternion(rand=None)[source]#

Return uniform random unit quaternion.

rand: array like or None

Three independent random variables that are uniformly distributed between 0 and 1.

>>> q = random_quaternion()
>>> numpy.allclose(1, vector_norm(q))
True
>>> q = random_quaternion(numpy.random.random(3))
>>> len(q.shape), q.shape[0]==4
(1, True)

Arcball#

Arcball Class#

class pygeostat.transformations.Arcball(initial=None)[source]#

Virtual Trackball Control.

>>> ball = Arcball()
>>> ball = Arcball(initial=numpy.identity(4))
>>> ball.place([320, 320], 320)
>>> ball.down([500, 250])
>>> ball.drag([475, 275])
>>> R = ball.matrix()
>>> numpy.allclose(numpy.sum(R), 3.90583455)
True
>>> ball = Arcball(initial=[1, 0, 0, 0])
>>> ball.place([320, 320], 320)
>>> ball.setaxes([1, 1, 0], [-1, 1, 0])
>>> ball.constrain = True
>>> ball.down([400, 200])
>>> ball.drag([200, 400])
>>> R = ball.matrix()
>>> numpy.allclose(numpy.sum(R), 0.2055924)
True
>>> ball.next()
property constrain#

Return state of constrain to axis mode.

down(point)[source]#

Set initial cursor window coordinates and pick constrain-axis.

drag(point)[source]#

Update current cursor window coordinates.

matrix()[source]#

Return homogeneous rotation matrix.

next(acceleration=0.0)[source]#

Continue rotation in direction of last drag.

place(center, radius)[source]#

Place Arcball, e.g. when window size changes.

centersequence[2]

Window coordinates of trackball center.

radiusfloat

Radius of trackball in window coordinates.

setaxes(*axes)[source]#

Set axes to constrain rotations.

arcball_map_to_sphere#

pygeostat.transformations.arcball_map_to_sphere(point, center, radius)[source]#

Return unit sphere coordinates from window coordinates.

arcball_constrain_to_axis#

pygeostat.transformations.arcball_constrain_to_axis(point, axis)[source]#

Return sphere point perpendicular to axis.

arcball_nearest_axis#

pygeostat.transformations.arcball_nearest_axis(point, axes)[source]#

Return axis, which arc is nearest to point.

Vector Operations#

vector_norm#

pygeostat.transformations.vector_norm(data, axis=None, out=None)[source]#

Return length, i.e. Euclidean norm, of ndarray along axis.

>>> v = numpy.random.random(3)
>>> n = vector_norm(v)
>>> numpy.allclose(n, numpy.linalg.norm(v))
True
>>> v = numpy.random.rand(6, 5, 3)
>>> n = vector_norm(v, axis=-1)
>>> numpy.allclose(n, numpy.sqrt(numpy.sum(v*v, axis=2)))
True
>>> n = vector_norm(v, axis=1)
>>> numpy.allclose(n, numpy.sqrt(numpy.sum(v*v, axis=1)))
True
>>> v = numpy.random.rand(5, 4, 3)
>>> n = numpy.empty((5, 3))
>>> vector_norm(v, axis=1, out=n)
>>> numpy.allclose(n, numpy.sqrt(numpy.sum(v*v, axis=1)))
True
>>> vector_norm([])
0.0
>>> vector_norm([1])
1.0

unit_vector#

pygeostat.transformations.unit_vector(data, axis=None, out=None)[source]#

Return ndarray normalized by length, i.e. Euclidean norm, along axis.

>>> v0 = numpy.random.random(3)
>>> v1 = unit_vector(v0)
>>> numpy.allclose(v1, v0 / numpy.linalg.norm(v0))
True
>>> v0 = numpy.random.rand(5, 4, 3)
>>> v1 = unit_vector(v0, axis=-1)
>>> v2 = v0 / numpy.expand_dims(numpy.sqrt(numpy.sum(v0*v0, axis=2)), 2)
>>> numpy.allclose(v1, v2)
True
>>> v1 = unit_vector(v0, axis=1)
>>> v2 = v0 / numpy.expand_dims(numpy.sqrt(numpy.sum(v0*v0, axis=1)), 1)
>>> numpy.allclose(v1, v2)
True
>>> v1 = numpy.empty((5, 4, 3))
>>> unit_vector(v0, axis=1, out=v1)
>>> numpy.allclose(v1, v2)
True
>>> list(unit_vector([]))
[]
>>> list(unit_vector([1]))
[1.0]

random_vector#

pygeostat.transformations.random_vector(size)[source]#

Return array of random doubles in the half-open interval [0.0, 1.0).

>>> v = random_vector(10000)
>>> numpy.all(v >= 0) and numpy.all(v < 1)
True
>>> v0 = random_vector(10)
>>> v1 = random_vector(10)
>>> numpy.any(v0 == v1)
False

vector_product#

pygeostat.transformations.vector_product(v0, v1, axis=0)[source]#

Return vector perpendicular to vectors.

>>> v = vector_product([2, 0, 0], [0, 3, 0])
>>> numpy.allclose(v, [0, 0, 6])
True
>>> v0 = [[2, 0, 0, 2], [0, 2, 0, 2], [0, 0, 2, 2]]
>>> v1 = [[3], [0], [0]]
>>> v = vector_product(v0, v1)
>>> numpy.allclose(v, [[0, 0, 0, 0], [0, 0, 6, 6], [0, -6, 0, -6]])
True
>>> v0 = [[2, 0, 0], [2, 0, 0], [0, 2, 0], [2, 0, 0]]
>>> v1 = [[0, 3, 0], [0, 0, 3], [0, 0, 3], [3, 3, 3]]
>>> v = vector_product(v0, v1, axis=1)
>>> numpy.allclose(v, [[0, 0, 6], [0, -6, 0], [6, 0, 0], [0, -6, 6]])
True

angle_between_vectors#

pygeostat.transformations.angle_between_vectors(v0, v1, directed=True, axis=0)[source]#

Return angle between vectors.

If directed is False, the input vectors are interpreted as undirected axes, i.e. the maximum angle is pi/2.

>>> a = angle_between_vectors([1, -2, 3], [-1, 2, -3])
>>> numpy.allclose(a, math.pi)
True
>>> a = angle_between_vectors([1, -2, 3], [-1, 2, -3], directed=False)
>>> numpy.allclose(a, 0)
True
>>> v0 = [[2, 0, 0, 2], [0, 2, 0, 2], [0, 0, 2, 2]]
>>> v1 = [[3], [0], [0]]
>>> a = angle_between_vectors(v0, v1)
>>> numpy.allclose(a, [0, 1.5708, 1.5708, 0.95532])
True
>>> v0 = [[2, 0, 0], [2, 0, 0], [0, 2, 0], [2, 0, 0]]
>>> v1 = [[0, 3, 0], [0, 0, 3], [0, 0, 3], [3, 3, 3]]
>>> a = angle_between_vectors(v0, v1, axis=1)
>>> numpy.allclose(a, [1.5708, 1.5708, 1.5708, 0.95532])
True
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