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# Copyright (c) 2015 Ultimaker B.V.
# Uranium is released under the terms of the LGPLv3 or higher.
import numpy
import numpy.linalg
import math
import copy
from UM.Math.Vector import Vector
from UM.Math.Float import Float
from UM.Math.Matrix import Matrix
## Unit Quaternion class based on numpy arrays.
#
# This class represents a Unit quaternion that can be used for rotations.
#
# \note The operations that modify this quaternion will ensure the length
# of the quaternion remains 1. This is done to make this class simpler
# to use.
#
class Quaternion(object):
EPS = numpy.finfo(float).eps * 4.0
def __init__(self, x=0.0, y=0.0, z=0.0, w=1.0):
# Components are stored as XYZW
self._data = numpy.array([x, y, z, w], dtype=numpy.float32)
def getData(self):
return self._data
@property
def x(self):
return self._data[0]
@property
def y(self):
return self._data[1]
@property
def z(self):
return self._data[2]
@property
def w(self):
return self._data[3]
## Set quaternion by providing rotation about an axis.
#
# \param angle \type{float} Angle in radians
# \param axis \type{Vector} Axis of rotation
def setByAngleAxis(self, angle, axis):
a = axis.normalized().getData()
halfAngle = angle / 2.0
self._data[3] = math.cos(halfAngle)
self._data[0:3] = a * math.sin(halfAngle)
self.normalize()
def __mul__(self, other):
result = copy.deepcopy(self)
result *= other
return result
def __imul__(self, other):
if type(other) is Quaternion:
v1 = Vector(other.x, other.y, other.z)
v2 = Vector(self.x, self.y, self.z)
w = other.w * self.w - v1.dot(v2)
v = v2 * other.w + v1 * self.w + v2.cross(v1)
self._data[0] = v.x
self._data[1] = v.y
self._data[2] = v.z
self._data[3] = w
elif type(other) is float or type(other) is int:
self._data *= other
else:
raise NotImplementedError()
return self
def __add__(self, other):
result = copy.deepcopy(self)
result += other
return result
def __iadd__(self, other):
if type(other) is Quaternion:
self._data[0] += other._data[0]
self._data[1] += other._data[1]
self._data[2] += other._data[2]
self._data[3] += other._data[3]
else:
raise NotImplementedError()
return self
def __truediv__(self, other):
result = copy.deepcopy(self)
result /= other
return result
def __itruediv__(self, other):
if type(other) is float or type(other) is int:
self._data /= other
else:
raise NotImplementedError()
return self
def __eq__(self, other):
return Float.fuzzyCompare(self.x, other.x, 1e-6) and Float.fuzzyCompare(self.y, other.y, 1e-6) and Float.fuzzyCompare(self.z, other.z, 1e-6) and Float.fuzzyCompare(self.w, other.w, 1e-6)
def __neg__(self):
q = copy.deepcopy(self)
q._data = -q._data
return q
def getInverse(self):
result = copy.deepcopy(self)
result.invert()
return result
def invert(self):
self._data[0:3] = -self._data[0:3]
return self
def rotate(self, vector):
vMult = 2.0 * (self.x * vector.x + self.y * vector.y + self.z * vector.z)
crossMult = 2.0 * self.w
pMult = crossMult * self.w - 1.0
return Vector( pMult * vector.x + vMult * self.x + crossMult * (self.y * vector.z - self.z * vector.y),
pMult * vector.y + vMult * self.y + crossMult * (self.z * vector.x - self.x * vector.z),
pMult * vector.z + vMult * self.z + crossMult * (self.x * vector.y - self.y * vector.x) )
def dot(self, other):
return numpy.dot(self._data, other._data)
def length(self):
return numpy.linalg.norm(self._data)
def normalize(self):
self._data /= numpy.linalg.norm(self._data)
## Set quaternion by providing a homogenous (4x4) rotation matrix.
# \param matrix 4x4 Matrix object
# \param is_precise
def setByMatrix(self, matrix, is_precise = False):
trace = matrix.at(0, 0) + matrix.at(1, 1) + matrix.at(2, 2)
if trace > 0.0:
self._data[0] = matrix.at(2, 1) - matrix.at(1, 2)
self._data[1] = matrix.at(0, 2) - matrix.at(2, 0)
self._data[2] = matrix.at(1, 0) - matrix.at(0, 1)
self._data[3] = trace + 1
else:
i = 0
if matrix.at(1, 1) > matrix.at(0, 0):
i = 1
if matrix.at(2, 2) > matrix.at(i, i):
i = 2
# Yes, this is repeated code. Writing it out however makes the code way
# more readable than any magical index shifting.
if i == 0:
self._data[0] = matrix.at(0, 0) - matrix.at(1, 1) - matrix.at(2, 2) + 1.0
self._data[1] = matrix.at(0, 1) + matrix.at(1, 0)
self._data[2] = matrix.at(0, 2) + matrix.at(2, 0)
self._data[3] = matrix.at(2, 1) - matrix.at(1, 2)
elif i == 1:
self._data[0] = matrix.at(0, 1) + matrix.at(1, 0)
self._data[1] = matrix.at(1, 1) - matrix.at(0, 0) - matrix.at(2, 2) + 1.0
self._data[2] = matrix.at(1, 2) + matrix.at(2, 1)
self._data[3] = matrix.at(0, 2) - matrix.at(2, 0)
else:
self._data[0] = matrix.at(0, 2) + matrix.at(2, 0)
self._data[1] = matrix.at(2, 1) + matrix.at(1, 2)
self._data[1] = matrix.at(2, 2) - matrix.at(0, 0) - matrix.at(1, 1) + 1.0
self._data[3] = matrix.at(1, 0) - matrix.at(0, 1)
self.normalize()
def toMatrix(self):
m = numpy.zeros((4, 4), dtype=numpy.float32)
s = 2.0 / (self.x ** 2 + self.y ** 2 + self.z ** 2 + self.w ** 2)
xs = s * self.x
ys = s * self.y
zs = s * self.z
wx = self.w * xs
wy = self.w * ys
wz = self.w * zs
xx = self.x * xs
xy = self.x * ys
xz = self.x * zs
yy = self.y * ys
yz = self.y * zs
zz = self.z * zs
m[0,0] = 1.0 - (yy + zz)
m[0,1] = xy - wz
m[0,2] = xz + wy
m[1,0] = xy + wz
m[1,1] = 1.0 - (xx + zz)
m[1,2] = yz - wx
m[2,0] = xz - wy
m[2,1] = yz + wx
m[2,2] = 1.0 - (xx + yy)
m[3,3] = 1.0
return Matrix(m)
@staticmethod
def slerp(start, end, amount):
if Float.fuzzyCompare(amount, 0.0):
return start
elif Float.fuzzyCompare(amount, 1.0):
return end
rho = math.acos(start.dot(end))
return (start * math.sin((1 - amount) * rho) + end * math.sin(amount * rho)) / math.sin(rho)
## Returns a quaternion representing the rotation from vector 1 to vector 2.
#
# \param v1 \type{Vector} The vector to rotate from.
# \param v2 \type{Vector} The vector to rotate to.
@staticmethod
def rotationTo(v1, v2):
d = v1.dot(v2)
if d >= 1.0:
return Quaternion() # Vectors are equal, no rotation needed.
q = None
if Float.fuzzyCompare(d, -1.0, 1e-6):
axis = Vector.Unit_X.cross(v1)
if Float.fuzzyCompare(axis.length(), 0.0):
axis = Vector.Unit_Y.cross(v1)
axis.normalize()
q = Quaternion()
q.setByAngleAxis(math.pi, axis)
else:
s = math.sqrt((1.0 + d) * 2.0)
invs = 1.0 / s
c = v1.cross(v2)
q = Quaternion(
c.x * invs,
c.y * invs,
c.z * invs,
s * 0.5
)
q.normalize()
return q
@staticmethod
def fromMatrix(matrix):
q = Quaternion()
q.setByMatrix(matrix)
return q
@staticmethod
def fromAngleAxis(angle, axis):
q = Quaternion()
q.setByAngleAxis(angle, axis)
return q
def __repr__(self):
return "Quaternion(x={0}, y={1}, z={2}, w={3})".format(self.x, self.y, self.z, self.w)
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