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transformation.py
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# -*- coding: utf-8 -*-
# transformations.py
# Copyright (c) 2006-2013, Christoph Gohlke
# Copyright (c) 2006-2013, The Regents of the University of California
# Produced at the Laboratory for Fluorescence Dynamics
# All rights reserved.
#
# Redistribution and use in source and binary forms, with or without
# modification, are permitted provided that the following conditions are met:
#
# * Redistributions of source code must retain the above copyright
# notice, this list of conditions and the following disclaimer.
# * Redistributions in binary form must reproduce the above copyright
# notice, this list of conditions and the following disclaimer in the
# documentation and/or other materials provided with the distribution.
# * Neither the name of the copyright holders nor the names of any
# contributors may be used to endorse or promote products derived
# from this software without specific prior written permission.
#
# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
# AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
# IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
# ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
# LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
# CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
# SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
# CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
# ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
# POSSIBILITY OF SUCH DAMAGE.
"""Homogeneous Transformation Matrices and Quaternions.
A library for calculating 4x4 matrices for translating, rotating, reflecting,
scaling, shearing, projecting, orthogonalizing, and superimposing arrays of
3D homogeneous coordinates as well as for converting between rotation matrices,
Euler angles, and quaternions. Also includes an Arcball control object and
functions to decompose transformation matrices.
:Author:
`Christoph Gohlke <http://www.lfd.uci.edu/~gohlke/>`_
:Organization:
Laboratory for Fluorescence Dynamics, University of California, Irvine
:Version: 2013.06.29
Requirements
------------
* `CPython 2.7 or 3.3 <http://www.python.org>`_
* `Numpy 1.7 <http://www.numpy.org>`_
* `Transformations.c 2013.01.18 <http://www.lfd.uci.edu/~gohlke/>`_
(recommended for speedup of some functions)
Notes
-----
The API is not stable yet and is expected to change between revisions.
This Python code is not optimized for speed. Refer to the transformations.c
module for a faster implementation of some functions.
Documentation in HTML format can be generated with epydoc.
Matrices (M) can be inverted using numpy.linalg.inv(M), be concatenated using
numpy.dot(M0, M1), or transform homogeneous coordinate arrays (v) using
numpy.dot(M, v) for shape (4, \*) column vectors, respectively
numpy.dot(v, M.T) for shape (\*, 4) row vectors ("array of points").
This module follows the "column vectors on the right" and "row major storage"
(C contiguous) conventions. The translation components are in the right column
of the transformation matrix, i.e. M[:3, 3].
The transpose of the transformation matrices may have to be used to interface
with other graphics systems, e.g. with OpenGL's glMultMatrixd(). See also [16].
Calculations are carried out with numpy.float64 precision.
Vector, point, quaternion, and matrix function arguments are expected to be
"array like", i.e. tuple, list, or numpy arrays.
Return types are numpy arrays unless specified otherwise.
Angles are in radians unless specified otherwise.
Quaternions w+ix+jy+kz are represented as [w, x, y, z].
A triple of Euler angles can be applied/interpreted in 24 ways, which can
be specified using a 4 character string or encoded 4-tuple:
*Axes 4-string*: e.g. 'sxyz' or 'ryxy'
- first character : rotations are applied to 's'tatic or 'r'otating frame
- remaining characters : successive rotation axis 'x', 'y', or 'z'
*Axes 4-tuple*: e.g. (0, 0, 0, 0) or (1, 1, 1, 1)
- inner axis: code of axis ('x':0, 'y':1, 'z':2) of rightmost matrix.
- parity : even (0) if inner axis 'x' is followed by 'y', 'y' is followed
by 'z', or 'z' is followed by 'x'. Otherwise odd (1).
- repetition : first and last axis are same (1) or different (0).
- frame : rotations are applied to static (0) or rotating (1) frame.
References
----------
(1) Matrices and transformations. Ronald Goldman.
In "Graphics Gems I", pp 472-475. Morgan Kaufmann, 1990.
(2) More matrices and transformations: shear and pseudo-perspective.
Ronald Goldman. In "Graphics Gems II", pp 320-323. Morgan Kaufmann, 1991.
(3) Decomposing a matrix into simple transformations. Spencer Thomas.
In "Graphics Gems II", pp 320-323. Morgan Kaufmann, 1991.
(4) Recovering the data from the transformation matrix. Ronald Goldman.
In "Graphics Gems II", pp 324-331. Morgan Kaufmann, 1991.
(5) Euler angle conversion. Ken Shoemake.
In "Graphics Gems IV", pp 222-229. Morgan Kaufmann, 1994.
(6) Arcball rotation control. Ken Shoemake.
In "Graphics Gems IV", pp 175-192. Morgan Kaufmann, 1994.
(7) Representing attitude: Euler angles, unit quaternions, and rotation
vectors. James Diebel. 2006.
(8) A discussion of the solution for the best rotation to relate two sets
of vectors. W Kabsch. Acta Cryst. 1978. A34, 827-828.
(9) Closed-form solution of absolute orientation using unit quaternions.
BKP Horn. J Opt Soc Am A. 1987. 4(4):629-642.
(10) Quaternions. Ken Shoemake.
http://www.sfu.ca/~jwa3/cmpt461/files/quatut.pdf
(11) From quaternion to matrix and back. JMP van Waveren. 2005.
http://www.intel.com/cd/ids/developer/asmo-na/eng/293748.htm
(12) Uniform random rotations. Ken Shoemake.
In "Graphics Gems III", pp 124-132. Morgan Kaufmann, 1992.
(13) Quaternion in molecular modeling. CFF Karney.
J Mol Graph Mod, 25(5):595-604
(14) New method for extracting the quaternion from a rotation matrix.
Itzhack Y Bar-Itzhack, J Guid Contr Dynam. 2000. 23(6): 1085-1087.
(15) Multiple View Geometry in Computer Vision. Hartley and Zissermann.
Cambridge University Press; 2nd Ed. 2004. Chapter 4, Algorithm 4.7, p 130.
(16) Column Vectors vs. Row Vectors.
http://steve.hollasch.net/cgindex/math/matrix/column-vec.html
Examples
--------
>>> alpha, beta, gamma = 0.123, -1.234, 2.345
>>> origin, xaxis, yaxis, zaxis = [0, 0, 0], [1, 0, 0], [0, 1, 0], [0, 0, 1]
>>> I = identity_matrix()
>>> Rx = rotation_matrix(alpha, xaxis)
>>> Ry = rotation_matrix(beta, yaxis)
>>> Rz = rotation_matrix(gamma, zaxis)
>>> R = concatenate_matrices(Rx, Ry, Rz)
>>> euler = euler_from_matrix(R, 'rxyz')
>>> numpy.allclose([alpha, beta, gamma], euler)
True
>>> Re = euler_matrix(alpha, beta, gamma, 'rxyz')
>>> is_same_transform(R, Re)
True
>>> al, be, ga = euler_from_matrix(Re, 'rxyz')
>>> is_same_transform(Re, euler_matrix(al, be, ga, 'rxyz'))
True
>>> qx = quaternion_about_axis(alpha, xaxis)
>>> qy = quaternion_about_axis(beta, yaxis)
>>> qz = quaternion_about_axis(gamma, zaxis)
>>> q = quaternion_multiply(qx, qy)
>>> q = quaternion_multiply(q, qz)
>>> Rq = quaternion_matrix(q)
>>> is_same_transform(R, Rq)
True
>>> S = scale_matrix(1.23, origin)
>>> T = translation_matrix([1, 2, 3])
>>> Z = shear_matrix(beta, xaxis, origin, zaxis)
>>> R = random_rotation_matrix(numpy.random.rand(3))
>>> M = concatenate_matrices(T, R, Z, S)
>>> scale, shear, angles, trans, persp = decompose_matrix(M)
>>> numpy.allclose(scale, 1.23)
True
>>> numpy.allclose(trans, [1, 2, 3])
True
>>> numpy.allclose(shear, [0, math.tan(beta), 0])
True
>>> is_same_transform(R, euler_matrix(axes='sxyz', *angles))
True
>>> M1 = compose_matrix(scale, shear, angles, trans, persp)
>>> is_same_transform(M, M1)
True
>>> v0, v1 = random_vector(3), random_vector(3)
>>> M = rotation_matrix(angle_between_vectors(v0, v1), vector_product(v0, v1))
>>> v2 = numpy.dot(v0, M[:3,:3].T)
>>> numpy.allclose(unit_vector(v1), unit_vector(v2))
True
"""
from __future__ import division, print_function
import math
import numpy
__version__ = '2013.06.29'
__docformat__ = 'restructuredtext en'
__all__ = []
def identity_matrix():
"""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
"""
return numpy.identity(4)
def translation_matrix(direction):
"""Return matrix to translate by direction vector.
>>> v = numpy.random.random(3) - 0.5
>>> numpy.allclose(v, translation_matrix(v)[:3, 3])
True
"""
M = numpy.identity(4)
M[:3, 3] = direction[:3]
return M
def translation_from_matrix(matrix):
"""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
"""
return numpy.array(matrix, copy=False)[:3, 3].copy()
def reflection_matrix(point, normal):
"""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
"""
normal = unit_vector(normal[:3])
M = numpy.identity(4)
M[:3, :3] -= 2.0 * numpy.outer(normal, normal)
M[:3, 3] = (2.0 * numpy.dot(point[:3], normal)) * normal
return M
def reflection_from_matrix(matrix):
"""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
"""
M = numpy.array(matrix, dtype=numpy.float64, copy=False)
# normal: unit eigenvector corresponding to eigenvalue -1
w, V = numpy.linalg.eig(M[:3, :3])
i = numpy.where(abs(numpy.real(w) + 1.0) < 1e-8)[0]
if not len(i):
raise ValueError("no unit eigenvector corresponding to eigenvalue -1")
normal = numpy.real(V[:, i[0]]).squeeze()
# point: any unit eigenvector corresponding to eigenvalue 1
w, V = numpy.linalg.eig(M)
i = numpy.where(abs(numpy.real(w) - 1.0) < 1e-8)[0]
if not len(i):
raise ValueError("no unit eigenvector corresponding to eigenvalue 1")
point = numpy.real(V[:, i[-1]]).squeeze()
point /= point[3]
return point, normal
def rotation_matrix(angle, direction, point=None):
"""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
"""
sina = math.sin(angle)
cosa = math.cos(angle)
direction = unit_vector(direction[:3])
# rotation matrix around unit vector
R = numpy.diag([cosa, cosa, cosa])
R += numpy.outer(direction, direction) * (1.0 - cosa)
direction *= sina
R += numpy.array([[ 0.0, -direction[2], direction[1]],
[ direction[2], 0.0, -direction[0]],
[-direction[1], direction[0], 0.0]])
M = numpy.identity(4)
M[:3, :3] = R
if point is not None:
# rotation not around origin
point = numpy.array(point[:3], dtype=numpy.float64, copy=False)
M[:3, 3] = point - numpy.dot(R, point)
return M
def rotation_from_matrix(matrix):
"""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
"""
R = numpy.array(matrix, dtype=numpy.float64, copy=False)
R33 = R[:3, :3]
# direction: unit eigenvector of R33 corresponding to eigenvalue of 1
w, W = numpy.linalg.eig(R33.T)
i = numpy.where(abs(numpy.real(w) - 1.0) < 1e-8)[0]
if not len(i):
raise ValueError("no unit eigenvector corresponding to eigenvalue 1")
direction = numpy.real(W[:, i[-1]]).squeeze()
# point: unit eigenvector of R33 corresponding to eigenvalue of 1
w, Q = numpy.linalg.eig(R)
i = numpy.where(abs(numpy.real(w) - 1.0) < 1e-8)[0]
if not len(i):
raise ValueError("no unit eigenvector corresponding to eigenvalue 1")
point = numpy.real(Q[:, i[-1]]).squeeze()
point /= point[3]
# rotation angle depending on direction
cosa = (numpy.trace(R33) - 1.0) / 2.0
if abs(direction[2]) > 1e-8:
sina = (R[1, 0] + (cosa-1.0)*direction[0]*direction[1]) / direction[2]
elif abs(direction[1]) > 1e-8:
sina = (R[0, 2] + (cosa-1.0)*direction[0]*direction[2]) / direction[1]
else:
sina = (R[2, 1] + (cosa-1.0)*direction[1]*direction[2]) / direction[0]
angle = math.atan2(sina, cosa)
return angle, direction, point
def scale_matrix(factor, origin=None, direction=None):
"""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)
"""
if direction is None:
# uniform scaling
M = numpy.diag([factor, factor, factor, 1.0])
if origin is not None:
M[:3, 3] = origin[:3]
M[:3, 3] *= 1.0 - factor
else:
# nonuniform scaling
direction = unit_vector(direction[:3])
factor = 1.0 - factor
M = numpy.identity(4)
M[:3, :3] -= factor * numpy.outer(direction, direction)
if origin is not None:
M[:3, 3] = (factor * numpy.dot(origin[:3], direction)) * direction
return M
def scale_from_matrix(matrix):
"""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
"""
M = numpy.array(matrix, dtype=numpy.float64, copy=False)
M33 = M[:3, :3]
factor = numpy.trace(M33) - 2.0
try:
# direction: unit eigenvector corresponding to eigenvalue factor
w, V = numpy.linalg.eig(M33)
i = numpy.where(abs(numpy.real(w) - factor) < 1e-8)[0][0]
direction = numpy.real(V[:, i]).squeeze()
direction /= vector_norm(direction)
except IndexError:
# uniform scaling
factor = (factor + 2.0) / 3.0
direction = None
# origin: any eigenvector corresponding to eigenvalue 1
w, V = numpy.linalg.eig(M)
i = numpy.where(abs(numpy.real(w) - 1.0) < 1e-8)[0]
if not len(i):
raise ValueError("no eigenvector corresponding to eigenvalue 1")
origin = numpy.real(V[:, i[-1]]).squeeze()
origin /= origin[3]
return factor, origin, direction
def projection_matrix(point, normal, direction=None,
perspective=None, pseudo=False):
"""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
"""
M = numpy.identity(4)
point = numpy.array(point[:3], dtype=numpy.float64, copy=False)
normal = unit_vector(normal[:3])
if perspective is not None:
# perspective projection
perspective = numpy.array(perspective[:3], dtype=numpy.float64,
copy=False)
M[0, 0] = M[1, 1] = M[2, 2] = numpy.dot(perspective-point, normal)
M[:3, :3] -= numpy.outer(perspective, normal)
if pseudo:
# preserve relative depth
M[:3, :3] -= numpy.outer(normal, normal)
M[:3, 3] = numpy.dot(point, normal) * (perspective+normal)
else:
M[:3, 3] = numpy.dot(point, normal) * perspective
M[3, :3] = -normal
M[3, 3] = numpy.dot(perspective, normal)
elif direction is not None:
# parallel projection
direction = numpy.array(direction[:3], dtype=numpy.float64, copy=False)
scale = numpy.dot(direction, normal)
M[:3, :3] -= numpy.outer(direction, normal) / scale
M[:3, 3] = direction * (numpy.dot(point, normal) / scale)
else:
# orthogonal projection
M[:3, :3] -= numpy.outer(normal, normal)
M[:3, 3] = numpy.dot(point, normal) * normal
return M
def projection_from_matrix(matrix, pseudo=False):
"""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
"""
M = numpy.array(matrix, dtype=numpy.float64, copy=False)
M33 = M[:3, :3]
w, V = numpy.linalg.eig(M)
i = numpy.where(abs(numpy.real(w) - 1.0) < 1e-8)[0]
if not pseudo and len(i):
# point: any eigenvector corresponding to eigenvalue 1
point = numpy.real(V[:, i[-1]]).squeeze()
point /= point[3]
# direction: unit eigenvector corresponding to eigenvalue 0
w, V = numpy.linalg.eig(M33)
i = numpy.where(abs(numpy.real(w)) < 1e-8)[0]
if not len(i):
raise ValueError("no eigenvector corresponding to eigenvalue 0")
direction = numpy.real(V[:, i[0]]).squeeze()
direction /= vector_norm(direction)
# normal: unit eigenvector of M33.T corresponding to eigenvalue 0
w, V = numpy.linalg.eig(M33.T)
i = numpy.where(abs(numpy.real(w)) < 1e-8)[0]
if len(i):
# parallel projection
normal = numpy.real(V[:, i[0]]).squeeze()
normal /= vector_norm(normal)
return point, normal, direction, None, False
else:
# orthogonal projection, where normal equals direction vector
return point, direction, None, None, False
else:
# perspective projection
i = numpy.where(abs(numpy.real(w)) > 1e-8)[0]
if not len(i):
raise ValueError(
"no eigenvector not corresponding to eigenvalue 0")
point = numpy.real(V[:, i[-1]]).squeeze()
point /= point[3]
normal = - M[3, :3]
perspective = M[:3, 3] / numpy.dot(point[:3], normal)
if pseudo:
perspective -= normal
return point, normal, None, perspective, pseudo
def clip_matrix(left, right, bottom, top, near, far, perspective=False):
"""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.])
"""
if left >= right or bottom >= top or near >= far:
raise ValueError("invalid frustum")
if perspective:
if near <= _EPS:
raise ValueError("invalid frustum: near <= 0")
t = 2.0 * near
M = [[t/(left-right), 0.0, (right+left)/(right-left), 0.0],
[0.0, t/(bottom-top), (top+bottom)/(top-bottom), 0.0],
[0.0, 0.0, (far+near)/(near-far), t*far/(far-near)],
[0.0, 0.0, -1.0, 0.0]]
else:
M = [[2.0/(right-left), 0.0, 0.0, (right+left)/(left-right)],
[0.0, 2.0/(top-bottom), 0.0, (top+bottom)/(bottom-top)],
[0.0, 0.0, 2.0/(far-near), (far+near)/(near-far)],
[0.0, 0.0, 0.0, 1.0]]
return numpy.array(M)
def shear_matrix(angle, direction, point, normal):
"""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
"""
normal = unit_vector(normal[:3])
direction = unit_vector(direction[:3])
if abs(numpy.dot(normal, direction)) > 1e-6:
raise ValueError("direction and normal vectors are not orthogonal")
angle = math.tan(angle)
M = numpy.identity(4)
M[:3, :3] += angle * numpy.outer(direction, normal)
M[:3, 3] = -angle * numpy.dot(point[:3], normal) * direction
return M
def shear_from_matrix(matrix):
"""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
"""
M = numpy.array(matrix, dtype=numpy.float64, copy=False)
M33 = M[:3, :3]
# normal: cross independent eigenvectors corresponding to the eigenvalue 1
w, V = numpy.linalg.eig(M33)
i = numpy.where(abs(numpy.real(w) - 1.0) < 1e-4)[0]
if len(i) < 2:
raise ValueError("no two linear independent eigenvectors found %s" % w)
V = numpy.real(V[:, i]).squeeze().T
lenorm = -1.0
for i0, i1 in ((0, 1), (0, 2), (1, 2)):
n = numpy.cross(V[i0], V[i1])
w = vector_norm(n)
if w > lenorm:
lenorm = w
normal = n
normal /= lenorm
# direction and angle
direction = numpy.dot(M33 - numpy.identity(3), normal)
angle = vector_norm(direction)
direction /= angle
angle = math.atan(angle)
# point: eigenvector corresponding to eigenvalue 1
w, V = numpy.linalg.eig(M)
i = numpy.where(abs(numpy.real(w) - 1.0) < 1e-8)[0]
if not len(i):
raise ValueError("no eigenvector corresponding to eigenvalue 1")
point = numpy.real(V[:, i[-1]]).squeeze()
point /= point[3]
return angle, direction, point, normal
def decompose_matrix(matrix):
"""Return sequence of transformations from transformation matrix.
matrix : array_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
"""
M = numpy.array(matrix, dtype=numpy.float64, copy=True).T
if abs(M[3, 3]) < _EPS:
raise ValueError("M[3, 3] is zero")
M /= M[3, 3]
P = M.copy()
P[:, 3] = 0.0, 0.0, 0.0, 1.0
if not numpy.linalg.det(P):
raise ValueError("matrix is singular")
scale = numpy.zeros((3, ))
shear = [0.0, 0.0, 0.0]
angles = [0.0, 0.0, 0.0]
if any(abs(M[:3, 3]) > _EPS):
perspective = numpy.dot(M[:, 3], numpy.linalg.inv(P.T))
M[:, 3] = 0.0, 0.0, 0.0, 1.0
else:
perspective = numpy.array([0.0, 0.0, 0.0, 1.0])
translate = M[3, :3].copy()
M[3, :3] = 0.0
row = M[:3, :3].copy()
scale[0] = vector_norm(row[0])
row[0] /= scale[0]
shear[0] = numpy.dot(row[0], row[1])
row[1] -= row[0] * shear[0]
scale[1] = vector_norm(row[1])
row[1] /= scale[1]
shear[0] /= scale[1]
shear[1] = numpy.dot(row[0], row[2])
row[2] -= row[0] * shear[1]
shear[2] = numpy.dot(row[1], row[2])
row[2] -= row[1] * shear[2]
scale[2] = vector_norm(row[2])
row[2] /= scale[2]
shear[1:] /= scale[2]
if numpy.dot(row[0], numpy.cross(row[1], row[2])) < 0:
numpy.negative(scale, scale)
numpy.negative(row, row)
angles[1] = math.asin(-row[0, 2])
if math.cos(angles[1]):
angles[0] = math.atan2(row[1, 2], row[2, 2])
angles[2] = math.atan2(row[0, 1], row[0, 0])
else:
#angles[0] = math.atan2(row[1, 0], row[1, 1])
angles[0] = math.atan2(-row[2, 1], row[1, 1])
angles[2] = 0.0
return scale, shear, angles, translate, perspective
def compose_matrix(scale=None, shear=None, angles=None, translate=None,
perspective=None):
"""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
"""
M = numpy.identity(4)
if perspective is not None:
P = numpy.identity(4)
P[3, :] = perspective[:4]
M = numpy.dot(M, P)
if translate is not None:
T = numpy.identity(4)
T[:3, 3] = translate[:3]
M = numpy.dot(M, T)
if angles is not None:
R = euler_matrix(angles[0], angles[1], angles[2], 'sxyz')
M = numpy.dot(M, R)
if shear is not None:
Z = numpy.identity(4)
Z[1, 2] = shear[2]
Z[0, 2] = shear[1]
Z[0, 1] = shear[0]
M = numpy.dot(M, Z)
if scale is not None:
S = numpy.identity(4)
S[0, 0] = scale[0]
S[1, 1] = scale[1]
S[2, 2] = scale[2]
M = numpy.dot(M, S)
M /= M[3, 3]
return M
def orthogonalization_matrix(lengths, angles):
"""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
"""
a, b, c = lengths
angles = numpy.radians(angles)
sina, sinb, _ = numpy.sin(angles)
cosa, cosb, cosg = numpy.cos(angles)
co = (cosa * cosb - cosg) / (sina * sinb)
return numpy.array([
[ a*sinb*math.sqrt(1.0-co*co), 0.0, 0.0, 0.0],
[-a*sinb*co, b*sina, 0.0, 0.0],
[ a*cosb, b*cosa, c, 0.0],
[ 0.0, 0.0, 0.0, 1.0]])
def affine_matrix_from_points(v0, v1, shear=True, scale=True, usesvd=True):
"""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()
"""
v0 = numpy.array(v0, dtype=numpy.float64, copy=True)
v1 = numpy.array(v1, dtype=numpy.float64, copy=True)
ndims = v0.shape[0]
if ndims < 2 or v0.shape[1] < ndims or v0.shape != v1.shape:
raise ValueError("input arrays are of wrong shape or type")
# move centroids to origin
t0 = -numpy.mean(v0, axis=1)
M0 = numpy.identity(ndims+1)
M0[:ndims, ndims] = t0
v0 += t0.reshape(ndims, 1)
t1 = -numpy.mean(v1, axis=1)
M1 = numpy.identity(ndims+1)
M1[:ndims, ndims] = t1
v1 += t1.reshape(ndims, 1)
if shear:
# Affine transformation
A = numpy.concatenate((v0, v1), axis=0)
u, s, vh = numpy.linalg.svd(A.T)
vh = vh[:ndims].T
B = vh[:ndims]
C = vh[ndims:2*ndims]
t = numpy.dot(C, numpy.linalg.pinv(B))
t = numpy.concatenate((t, numpy.zeros((ndims, 1))), axis=1)
M = numpy.vstack((t, ((0.0,)*ndims) + (1.0,)))
elif usesvd or ndims != 3:
# Rigid transformation via SVD of covariance matrix
u, s, vh = numpy.linalg.svd(numpy.dot(v1, v0.T))
# rotation matrix from SVD orthonormal bases
R = numpy.dot(u, vh)
if numpy.linalg.det(R) < 0.0:
# R does not constitute right handed system
R -= numpy.outer(u[:, ndims-1], vh[ndims-1, :]*2.0)
s[-1] *= -1.0
# homogeneous transformation matrix
M = numpy.identity(ndims+1)
M[:ndims, :ndims] = R
else:
# Rigid transformation matrix via quaternion
# compute symmetric matrix N
xx, yy, zz = numpy.sum(v0 * v1, axis=1)
xy, yz, zx = numpy.sum(v0 * numpy.roll(v1, -1, axis=0), axis=1)
xz, yx, zy = numpy.sum(v0 * numpy.roll(v1, -2, axis=0), axis=1)
N = [[xx+yy+zz, 0.0, 0.0, 0.0],
[yz-zy, xx-yy-zz, 0.0, 0.0],
[zx-xz, xy+yx, yy-xx-zz, 0.0],
[xy-yx, zx+xz, yz+zy, zz-xx-yy]]
# quaternion: eigenvector corresponding to most positive eigenvalue
w, V = numpy.linalg.eigh(N)
q = V[:, numpy.argmax(w)]
q /= vector_norm(q) # unit quaternion
# homogeneous transformation matrix
M = quaternion_matrix(q)
if scale and not shear:
# Affine transformation; scale is ratio of RMS deviations from centroid
v0 *= v0
v1 *= v1
M[:ndims, :ndims] *= math.sqrt(numpy.sum(v1) / numpy.sum(v0))
# move centroids back
M = numpy.dot(numpy.linalg.inv(M1), numpy.dot(M, M0))
M /= M[ndims, ndims]
return M
def superimposition_matrix(v0, v1, scale=False, usesvd=True):
"""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.