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Matrix.py
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Matrix.py
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# Copyright (c) 2018 Ultimaker B.V.
# Uranium is released under the terms of the LGPLv3 or higher.
import math
import numpy
from UM.Math.Vector import Vector
from typing import cast, Dict, List, Optional, Tuple, TYPE_CHECKING, Union
numpy.seterr(divide="ignore")
if TYPE_CHECKING:
from UM.Math.Quaternion import Quaternion
class Matrix:
"""This class is a 4x4 homogeneous matrix wrapper around numpy.
Heavily based (in most cases a straight copy with some refactoring) on the excellent
'library' Transformations.py created by Christoph Gohlke.
"""
# epsilon for testing whether a number is close to zero
_EPS = numpy.finfo(float).eps * 4.0
# map axes strings to/from tuples of inner axis, parity, repetition, frame
# 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.
_AXES2TUPLE = {
"sxyz": (0, 0, 0, 0), "sxyx": (0, 0, 1, 0), "sxzy": (0, 1, 0, 0),
"sxzx": (0, 1, 1, 0), "syzx": (1, 0, 0, 0), "syzy": (1, 0, 1, 0),
"syxz": (1, 1, 0, 0), "syxy": (1, 1, 1, 0), "szxy": (2, 0, 0, 0),
"szxz": (2, 0, 1, 0), "szyx": (2, 1, 0, 0), "szyz": (2, 1, 1, 0),
"rzyx": (0, 0, 0, 1), "rxyx": (0, 0, 1, 1), "ryzx": (0, 1, 0, 1),
"rxzx": (0, 1, 1, 1), "rxzy": (1, 0, 0, 1), "ryzy": (1, 0, 1, 1),
"rzxy": (1, 1, 0, 1), "ryxy": (1, 1, 1, 1), "ryxz": (2, 0, 0, 1),
"rzxz": (2, 0, 1, 1), "rxyz": (2, 1, 0, 1), "rzyz": (2, 1, 1, 1)
} #type: Dict[str, Tuple[int, int, int, int]]
# axis sequences for Euler angles
_NEXT_AXIS = [1, 2, 0, 1]
def __init__(self, data: Optional[Union[List[List[float]], numpy.ndarray]] = None) -> None:
if data is None:
self._data = numpy.identity(4, dtype = numpy.float64)
else:
self._data = numpy.array(data, copy=True, dtype = numpy.float64)
def __deepcopy__(self, memo):
# So, you must be asking yourself, why not let python handle this simple case on it's own? Well, that's because
# we found out that this is about 3x faster.
# Note that actually using Matrix(self._data) (without the deepcopy) is another factor 3 faster.
return Matrix(self._data)
def copy(self) -> "Matrix":
return Matrix(self._data)
def __eq__(self, other: object) -> bool:
if self is other:
return True
if type(other) is not Matrix:
return False
other = cast(Matrix, other)
if self._data is None and other._data is None:
return True
return numpy.array_equal(self._data, other._data)
def at(self, x: int, y: int) -> float:
if x >= 4 or y >= 4 or x < 0 or y < 0:
raise IndexError
return self._data[x,y]
def setRow(self, index: int, value: List[float]) -> None:
if index < 0 or index > 3:
raise IndexError()
self._data[0, index] = value[0]
self._data[1, index] = value[1]
self._data[2, index] = value[2]
if len(value) > 3:
self._data[3, index] = value[3]
else:
self._data[3, index] = 0
def setColumn(self, index: int, value: List[float]) -> None:
if index < 0 or index > 3:
raise IndexError()
self._data[index, 0] = value[0]
self._data[index, 1] = value[1]
self._data[index, 2] = value[2]
if len(value) > 3:
self._data[index, 3] = value[3]
else:
self._data[index, 3] = 0
def multiply(self, other: Union[Vector, "Matrix"], copy: bool = False) -> "Matrix":
if not copy:
self._data = numpy.dot(self._data, other.getData())
return self
else:
return Matrix(data = numpy.dot(self._data, other.getData()))
def preMultiply(self, other: Union[Vector, "Matrix"], copy: bool = False) -> "Matrix":
if not copy:
self._data = numpy.dot(other.getData(), self._data)
return self
else:
return Matrix(data = numpy.dot(other.getData(), self._data))
def getData(self) -> numpy.ndarray:
"""Get raw data.
:returns: 4x4 numpy array
"""
return self._data.astype(numpy.float32)
def getFlatData(self):
return self._data.flatten()
def setToIdentity(self) -> None:
"""Create a 4x4 identity matrix. This overwrites any existing data."""
self._data = numpy.identity(4, dtype = numpy.float64)
def invert(self) -> None:
"""Invert the matrix"""
self._data = numpy.linalg.inv(self._data)
def pseudoinvert(self) -> None:
"""
Invert the matrix in-place with a pseudoinverse.
The pseudoinverse is guaranteed to succeed, but if the matrix was singular is not a true inverse. Just something
that approaches the inverse.
"""
self._data = numpy.linalg.pinv(self._data)
def getInverse(self) -> "Matrix":
"""Return a inverted copy of the matrix.
:returns: The invertex matrix.
"""
try:
return Matrix(numpy.linalg.inv(self._data))
except:
return Matrix(self._data)
def getTransposed(self) -> "Matrix":
"""Return the transpose of the matrix."""
try:
return Matrix(self._data.transpose())
except:
return Matrix(self._data)
def transpose(self) -> None:
self._data = self._data.transpose()
def translate(self, direction: Vector) -> None:
"""Translate the matrix based on Vector.
:param direction: The vector by which the matrix needs to be translated.
"""
translation_matrix = Matrix()
translation_matrix.setByTranslation(direction)
self.multiply(translation_matrix)
def setByTranslation(self, direction: Vector) -> None:
"""Set the matrix by translation vector. This overwrites any existing data.
:param direction: The vector by which the (unit) matrix needs to be translated.
"""
M = numpy.identity(4, dtype = numpy.float64)
M[:3, 3] = direction.getData()[:3]
self._data = M
def setTranslation(self, translation: Union[Vector, "Matrix"]) -> None:
self._data[:3, 3] = translation.getData()
def getTranslation(self) -> Vector:
return Vector(data = self._data[:3, 3])
def rotateByAxis(self, angle: float, direction: Vector, point: Optional[List[float]] = None) -> None:
"""Rotate the matrix based on rotation axis
:param angle: The angle by which matrix needs to be rotated.
:param direction: Axis by which the matrix needs to be rotated about.
:param point: Point where from where the rotation happens. If None, origin is used.
"""
rotation_matrix = Matrix()
rotation_matrix.setByRotationAxis(angle, direction, point)
self.multiply(rotation_matrix)
def setByRotationAxis(self, angle: float, direction: Vector, point: Optional[List[float]] = None) -> None:
"""Set the matrix based on rotation axis. This overwrites any existing data.
:param angle: The angle by which matrix needs to be rotated in radians.
:param direction: Axis by which the matrix needs to be rotated about.
:param point: Point where from where the rotation happens. If None, origin is used.
"""
sina = math.sin(angle)
cosa = math.cos(angle)
direction_data = cast(numpy.ndarray, self._unitVector(direction.getData()))
# rotation matrix around unit vector
R = numpy.diag([cosa, cosa, cosa])
R += numpy.outer(direction_data, direction_data) * (1.0 - cosa)
direction_data *= sina
R += numpy.array([[ 0.0, -direction_data[2], direction_data[1]],
[ direction_data[2], 0.0, -direction_data[0]],
[-direction_data[1], direction_data[0], 0.0]], dtype = numpy.float64)
M = numpy.identity(4)
M[:3, :3] = R
if point is not None:
# rotation not around origin
point2 = numpy.array(point[:3], dtype = numpy.float64, copy=False)
M[:3, 3] = point2 - numpy.dot(R, point2)
self._data = M
def compose(self, scale: Vector = None, shear: Vector = None, angles: Vector = None, translate: Vector = None, perspective: Vector = None, mirror: Vector = None) -> None:
"""Return transformation matrix from sequence of transformations.
This is the inverse of the decompose_matrix function.
:param scale : vector of 3 scaling factors
:param shear : list of shear factors for x-y, x-z, y-z axes
:param angles : list of Euler angles about static x, y, z axes
:param translate : translation vector along x, y, z axes
:param perspective : perspective partition of matrix
:param mirror: vector with mirror factors (1 if that axis is not mirrored, -1 if it is)
"""
M = numpy.identity(4)
if perspective is not None:
P = numpy.identity(4)
P[3, :] = perspective.getData()[:4]
M = numpy.dot(M, P)
if translate is not None:
T = numpy.identity(4)
T[:3, 3] = translate.getData()[:3]
M = numpy.dot(M, T)
if angles is not None:
R = Matrix()
R.setByEuler(angles.x, angles.y, angles.z, "sxyz")
M = numpy.dot(M, R.getData())
if shear is not None:
Z = numpy.identity(4)
Z[1, 2] = shear.x
Z[0, 2] = shear.y
Z[0, 1] = shear.z
M = numpy.dot(M, Z)
if scale is not None:
S = numpy.identity(4)
S[0, 0] = scale.x
S[1, 1] = scale.y
S[2, 2] = scale.z
M = numpy.dot(M, S)
if mirror is not None:
mir = numpy.identity(4)
mir[0, 0] *= mirror.x
mir[1, 1] *= mirror.y
mir[2, 2] *= mirror.z
M = numpy.dot(M, mir)
M /= M[3, 3]
self._data = M
def getEuler(self, axes: str = "sxyz") -> Vector:
"""Return Euler angles from rotation matrix for specified axis sequence.
:param axes: One of 24 axis sequences as string or encoded tuple
Note that many Euler angle triplets can describe one matrix.
"""
firstaxis, parity, repetition, frame = self._AXES2TUPLE[axes.lower()]
i = firstaxis
j = self._NEXT_AXIS[i + parity]
k = self._NEXT_AXIS[i - parity + 1]
M = numpy.array(self._data, dtype = numpy.float64, copy = False)[:3, :3]
if repetition:
sy = math.sqrt(M[i, j] * M[i, j] + M[i, k] * M[i, k])
if sy > self._EPS:
ax = math.atan2( M[i, j], M[i, k])
ay = math.atan2( sy, M[i, i])
az = math.atan2( M[j, i], -M[k, i])
else:
ax = math.atan2(-M[j, k], M[j, j])
ay = math.atan2( sy, M[i, i])
az = 0.0
else:
cy = math.sqrt(M[i, i] * M[i, i] + M[j, i] * M[j, i])
if cy > self._EPS:
ax = math.atan2( M[k, j], M[k, k])
ay = math.atan2(-M[k, i], cy)
az = math.atan2( M[j, i], M[i, i])
else:
ax = math.atan2(-M[j, k], M[j, j])
ay = math.atan2(-M[k, i], cy)
az = 0.0
if parity:
ax, ay, az = -ax, -ay, -az
if frame:
ax, az = az, ax
return Vector(ax, ay, az)
def setByEuler(self, ai: float, aj: float, ak: float, axes: str = "sxyz") -> None:
"""Return homogeneous rotation matrix from Euler angles and axis sequence.
:param ai: Eulers roll
:param aj: Eulers pitch
:param ak: Eulers yaw
:param axes: One of 24 axis sequences as string or encoded tuple
"""
firstaxis, parity, repetition, frame = self._AXES2TUPLE[axes.lower()]
i = firstaxis
j = self._NEXT_AXIS[i + parity]
k = self._NEXT_AXIS[i - parity + 1]
if frame:
ai, ak = ak, ai
if parity:
ai, aj, ak = -ai, -aj, -ak
si, sj, sk = math.sin(ai), math.sin(aj), math.sin(ak)
ci, cj, ck = math.cos(ai), math.cos(aj), math.cos(ak)
cc, cs = ci * ck, ci * sk
sc, ss = si * ck, si * sk
M = numpy.identity(4)
if repetition:
M[i, i] = cj
M[i, j] = sj * si
M[i, k] = sj * ci
M[j, i] = sj * sk
M[j, j] = -cj * ss + cc
M[j, k] = -cj * cs - sc
M[k, i] = -sj * ck
M[k, j] = cj * sc + cs
M[k, k] = cj * cc - ss
else:
M[i, i] = cj * ck
M[i, j] = sj * sc - cs
M[i, k] = sj * cc + ss
M[j, i] = cj * sk
M[j, j] = sj * ss + cc
M[j, k] = sj * cs - sc
M[k, i] = -sj
M[k, j] = cj * si
M[k, k] = cj * ci
self._data = M
def scaleByFactor(self, factor: float, origin: Optional[List[float]] = None, direction: Optional[Vector] = None) -> None:
"""Scale the matrix by factor wrt origin & direction.
:param factor: The factor by which to scale
:param origin: From where does the scaling need to be done
:param direction: In what direction is the scaling (if None, it's uniform)
"""
scale_matrix = Matrix()
scale_matrix.setByScaleFactor(factor, origin, direction)
self.multiply(scale_matrix)
def setByScaleFactor(self, factor: float, origin: Optional[List[float]] = None, direction: Optional[Vector] = None) -> None:
"""Set the matrix by scale by factor wrt origin & direction. This overwrites any existing data
:param factor: The factor by which to scale
:param origin: From where does the scaling need to be done
:param direction: In what direction is the scaling (if None, it's uniform)
"""
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_data = direction.getData()
factor = 1.0 - factor
M = numpy.identity(4, dtype = numpy.float64)
M[:3, :3] -= factor * numpy.outer(direction_data, direction_data)
if origin is not None:
M[:3, 3] = (factor * numpy.dot(origin[:3], direction_data)) * direction_data
self._data = M
def setByScaleVector(self, scale: Vector) -> None:
self._data = numpy.diag([scale.x, scale.y, scale.z, 1.0])
def getScale(self) -> Vector:
x = numpy.linalg.norm(self._data[0,0:3])
y = numpy.linalg.norm(self._data[1,0:3])
z = numpy.linalg.norm(self._data[2,0:3])
return Vector(x, y, z)
def setOrtho(self, left: float, right: float, bottom: float, top: float, near: float, far: float) -> None:
"""Set the matrix to an orthographic projection. This overwrites any existing data.
:param left: The left edge of the projection
:param right: The right edge of the projection
:param top: The top edge of the projection
:param bottom: The bottom edge of the projection
:param near: The near plane of the projection
:param far: The far plane of the projection
"""
self.setToIdentity()
self._data[0, 0] = 2 / (right - left)
self._data[1, 1] = 2 / (top - bottom)
self._data[2, 2] = -2 / (far - near)
self._data[3, 0] = -((right + left) / (right - left))
self._data[3, 1] = -((top + bottom) / (top - bottom))
self._data[3, 2] = -((far + near) / (far - near))
def setPerspective(self, fovy: float, aspect: float, near: float, far: float) -> None:
"""Set the matrix to a perspective projection. This overwrites any existing data.
:param fovy: Field of view in the Y direction
:param aspect: The aspect ratio
:param near: Distance to the near plane
:param far: Distance to the far plane
"""
self.setToIdentity()
f = 2. / math.tan(math.radians(fovy) / 2.)
self._data[0, 0] = f / aspect
self._data[1, 1] = f
self._data[2, 2] = (far + near) / (near - far)
self._data[2, 3] = -1.
self._data[3, 2] = (2. * far * near) / (near - far)
def decompose(self) -> Tuple[Vector, "Matrix", Vector, Vector]:
"""
SOURCE: https://github.com/matthew-brett/transforms3d/blob/e402e56686648d9a88aa048068333b41daa69d1a/transforms3d/affines.py
Decompose 4x4 homogenous affine matrix into parts.
The parts are translations, rotations, zooms, shears.
This is the same as :func:`decompose` but specialized for 4x4 affines.
Decomposes `A44` into ``T, R, Z, S``, such that::
Smat = np.array([[1, S[0], S[1]],
[0, 1, S[2]],
[0, 0, 1]])
RZS = np.dot(R, np.dot(np.diag(Z), Smat))
A44 = np.eye(4)
A44[:3,:3] = RZS
A44[:-1,-1] = T
The order of transformations is therefore shears, followed by
zooms, followed by rotations, followed by translations.
This routine only works for shape (4,4) matrices
Parameters
----------
A44 : array shape (4,4)
Returns
-------
T : array, shape (3,)
Translation vector
R : array shape (3,3)
rotation matrix
Z : array, shape (3,)
Zoom vector. May have one negative zoom to prevent need for negative
determinant R matrix above
S : array, shape (3,)
Shear vector, such that shears fill upper triangle above
diagonal to form shear matrix (type ``striu``).
"""
A44 = numpy.asarray(self._data, dtype = numpy.float64)
T = A44[:-1, -1]
RZS = A44[:-1, :-1]
# compute scales and shears
M0, M1, M2 = numpy.array(RZS).T
# extract x scale and normalize
sx = math.sqrt(numpy.sum(M0 ** 2))
M0 /= sx
# orthogonalize M1 with respect to M0
sx_sxy = numpy.dot(M0, M1)
M1 -= sx_sxy * M0
# extract y scale and normalize
sy = math.sqrt(numpy.sum(M1 ** 2))
M1 /= sy
sxy = sx_sxy / sx
# orthogonalize M2 with respect to M0 and M1
sx_sxz = numpy.dot(M0, M2)
sy_syz = numpy.dot(M1, M2)
M2 -= (sx_sxz * M0 + sy_syz * M1)
# extract z scale and normalize
sz = math.sqrt(numpy.sum(M2 ** 2))
M2 /= sz
sxz = sx_sxz / sx
syz = sy_syz / sy
# Reconstruct rotation matrix, ensure positive determinant
Rmat = numpy.array([M0, M1, M2]).T
# The original code ensures that the determinant is positive, but I can't find a single situation where this
# is actualy used / needed by us. It is, however, one of the more expensive parts of this function.
#if numpy.linalg.det(Rmat) < 0:
# sx *= -1
# Rmat[:, 0] *= -1
return Vector(data = T), Matrix(data=Rmat), Vector(data = numpy.array([sx, sy, sz])), Vector(data=numpy.array([sxy, sxz, syz]))
def _unitVector(self, data: numpy.ndarray, axis: Optional[int] = None, out: Optional[numpy.ndarray] = None) -> Optional[numpy.ndarray]:
"""Return ndarray normalized by length, i.e. Euclidean norm, along axis.
>>> matrix = Matrix()
>>> v0 = numpy.random.random(3)
>>> v1 = matrix._unitVector(v0)
>>> numpy.allclose(v1, v0 / numpy.linalg.norm(v0))
True
>>> v0 = numpy.random.rand(5, 4, 3)
>>> v1 = matrix._unitVector(v0, axis=-1)
>>> v2 = v0 / numpy.expand_dims(numpy.sqrt(numpy.sum(v0 * v0, axis=2)), 2)
>>> numpy.allclose(v1, v2)
True
>>> v1 = matrix._unitVector(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))
>>> matrix._unitVector(v0, axis=1, out=v1)
>>> numpy.allclose(v1, v2)
True
>>> list(matrix._unitVector([]))
[]
>>> list(matrix._unitVector([1]))
[1.0]
"""
if out is None:
data = numpy.array(data, dtype = numpy.float64, copy = True)
if data.ndim == 1:
data /= math.sqrt(numpy.dot(data, data))
return data
else:
if out is not data:
out[:] = numpy.array(data, copy = False)
data = out
length = numpy.atleast_1d(numpy.sum(data * data, axis)) # type: ignore
numpy.sqrt(length, length)
if axis is not None:
length = numpy.expand_dims(length, axis)
data /= length
if out is None:
return data
return None
def __repr__(self) -> str:
return "Matrix( {0} )".format(self._data)
@staticmethod
def fromPositionOrientationScale(position: Vector, orientation: "Quaternion", scale: Vector) -> "Matrix":
s = numpy.identity(4, dtype = numpy.float64)
s[0, 0] = scale.x
s[1, 1] = scale.y
s[2, 2] = scale.z
r = orientation.toMatrix().getData()
t = numpy.identity(4, dtype = numpy.float64)
t[0, 3] = position.x
t[1, 3] = position.y
t[2, 3] = position.z
return Matrix(data = numpy.dot(numpy.dot(t, r), s))