1240 lines
51 KiB
Python
1240 lines
51 KiB
Python
# -*- coding: UTF-8 no BOM -*-
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###################################################
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# NOTE: everything here needs to be a np array #
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###################################################
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import math,os
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import numpy as np
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# ******************************************************************************************
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class Rodrigues:
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def __init__(self, vector = np.zeros(3)):
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self.vector = vector
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def asQuaternion(self):
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norm = np.linalg.norm(self.vector)
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halfAngle = np.arctan(norm)
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return Quaternion(np.cos(halfAngle),np.sin(halfAngle)*self.vector/norm)
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def asAngleAxis(self):
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norm = np.linalg.norm(self.vector)
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halfAngle = np.arctan(norm)
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return (2.0*halfAngle,self.vector/norm)
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# ******************************************************************************************
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class Quaternion:
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"""
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Orientation represented as unit quaternion
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All methods and naming conventions based on http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions
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w is the real part, (x, y, z) are the imaginary parts
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Representation of rotation is in ACTIVE form!
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(derived directly or through angleAxis, Euler angles, or active matrix)
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vector "a" (defined in coordinate system "A") is actively rotated to new coordinates "b"
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b = Q * a
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b = np.dot(Q.asMatrix(),a)
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"""
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def __init__(self,
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quatArray = [1.0,0.0,0.0,0.0]):
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"""initializes to identity if not given"""
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self.w, \
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self.x, \
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self.y, \
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self.z = quatArray
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self.homomorph()
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def __iter__(self):
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"""components"""
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return iter([self.w,self.x,self.y,self.z])
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def __copy__(self):
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"""create copy"""
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Q = Quaternion([self.w,self.x,self.y,self.z])
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return Q
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copy = __copy__
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def __repr__(self):
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"""readbable string"""
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return 'Quaternion(real=%+.6f, imag=<%+.6f, %+.6f, %+.6f>)' % \
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(self.w, self.x, self.y, self.z)
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def __pow__(self, exponent):
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"""power"""
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omega = math.acos(self.w)
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vRescale = math.sin(exponent*omega)/math.sin(omega)
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Q = Quaternion()
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Q.w = math.cos(exponent*omega)
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Q.x = self.x * vRescale
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Q.y = self.y * vRescale
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Q.z = self.z * vRescale
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return Q
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def __ipow__(self, exponent):
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"""in place power"""
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omega = math.acos(self.w)
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vRescale = math.sin(exponent*omega)/math.sin(omega)
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self.w = np.cos(exponent*omega)
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self.x *= vRescale
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self.y *= vRescale
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self.z *= vRescale
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return self
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def __mul__(self, other):
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"""multiplication"""
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try: # quaternion
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Aw = self.w
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Ax = self.x
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Ay = self.y
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Az = self.z
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Bw = other.w
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Bx = other.x
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By = other.y
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Bz = other.z
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Q = Quaternion()
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Q.w = - Ax * Bx - Ay * By - Az * Bz + Aw * Bw
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Q.x = + Ax * Bw + Ay * Bz - Az * By + Aw * Bx
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Q.y = - Ax * Bz + Ay * Bw + Az * Bx + Aw * By
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Q.z = + Ax * By - Ay * Bx + Az * Bw + Aw * Bz
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return Q
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except: pass
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try: # vector (perform active rotation, i.e. q*v*q.conjugated)
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w = self.w
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x = self.x
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y = self.y
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z = self.z
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Vx = other[0]
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Vy = other[1]
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Vz = other[2]
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return np.array([\
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w * w * Vx + 2 * y * w * Vz - 2 * z * w * Vy + \
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x * x * Vx + 2 * y * x * Vy + 2 * z * x * Vz - \
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z * z * Vx - y * y * Vx,
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2 * x * y * Vx + y * y * Vy + 2 * z * y * Vz + \
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2 * w * z * Vx - z * z * Vy + w * w * Vy - \
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2 * x * w * Vz - x * x * Vy,
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2 * x * z * Vx + 2 * y * z * Vy + \
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z * z * Vz - 2 * w * y * Vx - y * y * Vz + \
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2 * w * x * Vy - x * x * Vz + w * w * Vz ])
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except: pass
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try: # scalar
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Q = self.copy()
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Q.w *= other
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Q.x *= other
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Q.y *= other
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Q.z *= other
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return Q
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except:
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return self.copy()
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def __imul__(self, other):
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"""in place multiplication"""
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try: # Quaternion
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Ax = self.x
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Ay = self.y
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Az = self.z
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Aw = self.w
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Bx = other.x
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By = other.y
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Bz = other.z
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Bw = other.w
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self.x = Ax * Bw + Ay * Bz - Az * By + Aw * Bx
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self.y = -Ax * Bz + Ay * Bw + Az * Bx + Aw * By
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self.z = Ax * By - Ay * Bx + Az * Bw + Aw * Bz
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self.w = -Ax * Bx - Ay * By - Az * Bz + Aw * Bw
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except: pass
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return self
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def __div__(self, other):
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"""division"""
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if isinstance(other, (int,float)):
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w = self.w / other
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x = self.x / other
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y = self.y / other
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z = self.z / other
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return self.__class__([w,x,y,z])
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else:
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return NotImplemented
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def __idiv__(self, other):
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"""in place division"""
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if isinstance(other, (int,float)):
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self.w /= other
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self.x /= other
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self.y /= other
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self.z /= other
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return self
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def __add__(self, other):
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"""addition"""
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if isinstance(other, Quaternion):
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w = self.w + other.w
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x = self.x + other.x
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y = self.y + other.y
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z = self.z + other.z
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return self.__class__([w,x,y,z])
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else:
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return NotImplemented
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def __iadd__(self, other):
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"""in place division"""
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if isinstance(other, Quaternion):
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self.w += other.w
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self.x += other.x
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self.y += other.y
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self.z += other.z
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return self
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def __sub__(self, other):
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"""subtraction"""
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if isinstance(other, Quaternion):
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Q = self.copy()
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Q.w -= other.w
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Q.x -= other.x
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Q.y -= other.y
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Q.z -= other.z
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return Q
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else:
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return self.copy()
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def __isub__(self, other):
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"""in place subtraction"""
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if isinstance(other, Quaternion):
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self.w -= other.w
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self.x -= other.x
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self.y -= other.y
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self.z -= other.z
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return self
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def __neg__(self):
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"""additive inverse"""
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self.w = -self.w
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self.x = -self.x
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self.y = -self.y
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self.z = -self.z
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return self
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def __abs__(self):
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"""norm"""
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return math.sqrt(self.w ** 2 + \
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self.x ** 2 + \
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self.y ** 2 + \
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self.z ** 2)
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magnitude = __abs__
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def __eq__(self,other):
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"""equal at e-8 precision"""
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return (abs(self.w-other.w) < 1e-8 and \
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abs(self.x-other.x) < 1e-8 and \
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abs(self.y-other.y) < 1e-8 and \
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abs(self.z-other.z) < 1e-8) \
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or \
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(abs(-self.w-other.w) < 1e-8 and \
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abs(-self.x-other.x) < 1e-8 and \
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abs(-self.y-other.y) < 1e-8 and \
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abs(-self.z-other.z) < 1e-8)
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def __ne__(self,other):
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"""not equal at e-8 precision"""
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return not self.__eq__(self,other)
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def __cmp__(self,other):
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"""linear ordering"""
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return cmp(self.Rodrigues(),other.Rodrigues())
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def magnitude_squared(self):
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return self.w ** 2 + \
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self.x ** 2 + \
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self.y ** 2 + \
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self.z ** 2
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def identity(self):
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self.w = 1.
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self.x = 0.
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self.y = 0.
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self.z = 0.
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return self
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def normalize(self):
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d = self.magnitude()
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if d > 0.0:
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self /= d
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return self
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def conjugate(self):
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self.x = -self.x
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self.y = -self.y
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self.z = -self.z
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return self
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def inverse(self):
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d = self.magnitude()
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if d > 0.0:
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self.conjugate()
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self /= d
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return self
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def homomorph(self):
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if self.w < 0.0:
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self.w = -self.w
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self.x = -self.x
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self.y = -self.y
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self.z = -self.z
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return self
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def normalized(self):
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return self.copy().normalize()
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def conjugated(self):
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return self.copy().conjugate()
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def inversed(self):
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return self.copy().inverse()
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def homomorphed(self):
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return self.copy().homomorph()
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def asList(self):
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return [i for i in self]
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def asM(self): # to find Averaging Quaternions (see F. Landis Markley et al.)
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return np.outer([i for i in self],[i for i in self])
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def asMatrix(self):
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return np.array(
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[[1.0-2.0*(self.y*self.y+self.z*self.z), 2.0*(self.x*self.y-self.z*self.w), 2.0*(self.x*self.z+self.y*self.w)],
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[ 2.0*(self.x*self.y+self.z*self.w), 1.0-2.0*(self.x*self.x+self.z*self.z), 2.0*(self.y*self.z-self.x*self.w)],
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[ 2.0*(self.x*self.z-self.y*self.w), 2.0*(self.x*self.w+self.y*self.z), 1.0-2.0*(self.x*self.x+self.y*self.y)]])
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def asAngleAxis(self,
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degrees = False):
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if self.w > 1:
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self.normalize()
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s = math.sqrt(1. - self.w**2)
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x = 2*self.w**2 - 1.
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y = 2*self.w * s
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angle = math.atan2(y,x)
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if angle < 0.0:
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angle *= -1.
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s *= -1.
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return (np.degrees(angle) if degrees else angle,
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np.array([1.0, 0.0, 0.0] if np.abs(angle) < 1e-6 else [self.x / s, self.y / s, self.z / s]))
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def asRodrigues(self):
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return np.inf*np.ones(3) if self.w == 0.0 else np.array([self.x, self.y, self.z])/self.w
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def asEulers(self,
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type = "bunge",
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degrees = False,
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standardRange = False):
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"""
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Orientation as Bunge-Euler angles
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conversion of ACTIVE rotation to Euler angles taken from:
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Melcher, A.; Unser, A.; Reichhardt, M.; Nestler, B.; Poetschke, M.; Selzer, M.
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Conversion of EBSD data by a quaternion based algorithm to be used for grain structure simulations
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Technische Mechanik 30 (2010) pp 401--413
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"""
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angles = [0.0,0.0,0.0]
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if type.lower() == 'bunge' or type.lower() == 'zxz':
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if abs(self.x) < 1e-4 and abs(self.y) < 1e-4:
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x = self.w**2 - self.z**2
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y = 2.*self.w*self.z
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angles[0] = math.atan2(y,x)
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elif abs(self.w) < 1e-4 and abs(self.z) < 1e-4:
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x = self.x**2 - self.y**2
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y = 2.*self.x*self.y
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angles[0] = math.atan2(y,x)
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angles[1] = math.pi
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else:
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chi = math.sqrt((self.w**2 + self.z**2)*(self.x**2 + self.y**2))
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x = (self.w * self.x - self.y * self.z)/2./chi
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y = (self.w * self.y + self.x * self.z)/2./chi
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angles[0] = math.atan2(y,x)
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x = self.w**2 + self.z**2 - (self.x**2 + self.y**2)
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y = 2.*chi
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angles[1] = math.atan2(y,x)
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x = (self.w * self.x + self.y * self.z)/2./chi
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y = (self.z * self.x - self.y * self.w)/2./chi
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angles[2] = math.atan2(y,x)
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if standardRange:
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angles[0] %= 2*math.pi
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if angles[1] < 0.0:
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angles[1] += math.pi
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angles[2] *= -1.0
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angles[2] %= 2*math.pi
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return np.degrees(angles) if degrees else angles
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# # Static constructors
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@classmethod
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def fromIdentity(cls):
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return cls()
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@classmethod
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def fromRandom(cls,randomSeed = None):
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if randomSeed is None:
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randomSeed = int(os.urandom(4).encode('hex'), 16)
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np.random.seed(randomSeed)
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r = np.random.random(3)
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w = math.cos(2.0*math.pi*r[0])*math.sqrt(r[2])
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x = math.sin(2.0*math.pi*r[1])*math.sqrt(1.0-r[2])
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y = math.cos(2.0*math.pi*r[1])*math.sqrt(1.0-r[2])
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z = math.sin(2.0*math.pi*r[0])*math.sqrt(r[2])
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return cls([w,x,y,z])
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@classmethod
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def fromRodrigues(cls, rodrigues):
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if not isinstance(rodrigues, np.ndarray): rodrigues = np.array(rodrigues)
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halfangle = math.atan(np.linalg.norm(rodrigues))
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c = math.cos(halfangle)
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w = c
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x,y,z = c*rodrigues
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return cls([w,x,y,z])
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@classmethod
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def fromAngleAxis(cls,
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angle,
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axis,
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degrees = False):
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if not isinstance(axis, np.ndarray): axis = np.array(axis,dtype='d')
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axis = axis.astype(float)/np.linalg.norm(axis)
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angle = np.radians(angle) if degrees else angle
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s = math.sin(0.5 * angle)
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w = math.cos(0.5 * angle)
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x = axis[0] * s
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y = axis[1] * s
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z = axis[2] * s
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return cls([w,x,y,z])
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@classmethod
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def fromEulers(cls,
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eulers,
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type = 'Bunge',
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degrees = False):
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if not isinstance(eulers, np.ndarray): eulers = np.array(eulers,dtype='d')
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eulers = np.radians(eulers) if degrees else eulers
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c = np.cos(0.5 * eulers)
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s = np.sin(0.5 * eulers)
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if type.lower() == 'bunge' or type.lower() == 'zxz':
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w = c[0] * c[1] * c[2] - s[0] * c[1] * s[2]
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x = c[0] * s[1] * c[2] + s[0] * s[1] * s[2]
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y = - c[0] * s[1] * s[2] + s[0] * s[1] * c[2]
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z = c[0] * c[1] * s[2] + s[0] * c[1] * c[2]
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else:
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w = c[0] * c[1] * c[2] - s[0] * s[1] * s[2]
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x = s[0] * s[1] * c[2] + c[0] * c[1] * s[2]
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y = s[0] * c[1] * c[2] + c[0] * s[1] * s[2]
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z = c[0] * s[1] * c[2] - s[0] * c[1] * s[2]
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return cls([w,x,y,z])
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# Modified Method to calculate Quaternion from Orientation Matrix,
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# Source: http://www.euclideanspace.com/maths/geometry/rotations/conversions/matrixToQuaternion/
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@classmethod
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def fromMatrix(cls, m):
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if m.shape != (3,3) and np.prod(m.shape) == 9:
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m = m.reshape(3,3)
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tr = np.trace(m)
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if tr > 1e-8:
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s = math.sqrt(tr + 1.0)*2.0
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return cls(
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[ s*0.25,
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(m[2,1] - m[1,2])/s,
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(m[0,2] - m[2,0])/s,
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(m[1,0] - m[0,1])/s,
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])
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elif m[0,0] > m[1,1] and m[0,0] > m[2,2]:
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t = m[0,0] - m[1,1] - m[2,2] + 1.0
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s = 2.0*math.sqrt(t)
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return cls(
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[ (m[2,1] - m[1,2])/s,
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s*0.25,
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(m[0,1] + m[1,0])/s,
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(m[2,0] + m[0,2])/s,
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])
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elif m[1,1] > m[2,2]:
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t = -m[0,0] + m[1,1] - m[2,2] + 1.0
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s = 2.0*math.sqrt(t)
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|
return cls(
|
|
[ (m[0,2] - m[2,0])/s,
|
|
(m[0,1] + m[1,0])/s,
|
|
s*0.25,
|
|
(m[1,2] + m[2,1])/s,
|
|
])
|
|
|
|
else:
|
|
t = -m[0,0] - m[1,1] + m[2,2] + 1.0
|
|
s = 2.0*math.sqrt(t)
|
|
|
|
return cls(
|
|
[ (m[1,0] - m[0,1])/s,
|
|
(m[2,0] + m[0,2])/s,
|
|
(m[1,2] + m[2,1])/s,
|
|
s*0.25,
|
|
])
|
|
|
|
|
|
@classmethod
|
|
def new_interpolate(cls, q1, q2, t):
|
|
"""
|
|
interpolation
|
|
|
|
see http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20070017872_2007014421.pdf
|
|
for (another?) way to interpolate quaternions
|
|
"""
|
|
assert isinstance(q1, Quaternion) and isinstance(q2, Quaternion)
|
|
Q = cls()
|
|
|
|
costheta = q1.w * q2.w + q1.x * q2.x + q1.y * q2.y + q1.z * q2.z
|
|
if costheta < 0.:
|
|
costheta = -costheta
|
|
q1 = q1.conjugated()
|
|
elif costheta > 1:
|
|
costheta = 1
|
|
|
|
theta = math.acos(costheta)
|
|
if abs(theta) < 0.01:
|
|
Q.w = q2.w
|
|
Q.x = q2.x
|
|
Q.y = q2.y
|
|
Q.z = q2.z
|
|
return Q
|
|
|
|
sintheta = math.sqrt(1.0 - costheta * costheta)
|
|
if abs(sintheta) < 0.01:
|
|
Q.w = (q1.w + q2.w) * 0.5
|
|
Q.x = (q1.x + q2.x) * 0.5
|
|
Q.y = (q1.y + q2.y) * 0.5
|
|
Q.z = (q1.z + q2.z) * 0.5
|
|
return Q
|
|
|
|
ratio1 = math.sin((1 - t) * theta) / sintheta
|
|
ratio2 = math.sin(t * theta) / sintheta
|
|
|
|
Q.w = q1.w * ratio1 + q2.w * ratio2
|
|
Q.x = q1.x * ratio1 + q2.x * ratio2
|
|
Q.y = q1.y * ratio1 + q2.y * ratio2
|
|
Q.z = q1.z * ratio1 + q2.z * ratio2
|
|
return Q
|
|
|
|
|
|
# ******************************************************************************************
|
|
class Symmetry:
|
|
|
|
lattices = [None,'orthorhombic','tetragonal','hexagonal','cubic',]
|
|
|
|
def __init__(self, symmetry = None):
|
|
"""lattice with given symmetry, defaults to None"""
|
|
if isinstance(symmetry, str) and symmetry.lower() in Symmetry.lattices:
|
|
self.lattice = symmetry.lower()
|
|
else:
|
|
self.lattice = None
|
|
|
|
|
|
def __copy__(self):
|
|
"""copy"""
|
|
return self.__class__(self.lattice)
|
|
|
|
copy = __copy__
|
|
|
|
|
|
def __repr__(self):
|
|
"""readbable string"""
|
|
return '%s' % (self.lattice)
|
|
|
|
|
|
def __eq__(self, other):
|
|
"""equal"""
|
|
return self.lattice == other.lattice
|
|
|
|
def __neq__(self, other):
|
|
"""not equal"""
|
|
return not self.__eq__(other)
|
|
|
|
def __cmp__(self,other):
|
|
"""linear ordering"""
|
|
return cmp(Symmetry.lattices.index(self.lattice),Symmetry.lattices.index(other.lattice))
|
|
|
|
def symmetryQuats(self,who = []):
|
|
"""List of symmetry operations as quaternions."""
|
|
if self.lattice == 'cubic':
|
|
symQuats = [
|
|
[ 1.0, 0.0, 0.0, 0.0 ],
|
|
[ 0.0, 1.0, 0.0, 0.0 ],
|
|
[ 0.0, 0.0, 1.0, 0.0 ],
|
|
[ 0.0, 0.0, 0.0, 1.0 ],
|
|
[ 0.0, 0.0, 0.5*math.sqrt(2), 0.5*math.sqrt(2) ],
|
|
[ 0.0, 0.0, 0.5*math.sqrt(2),-0.5*math.sqrt(2) ],
|
|
[ 0.0, 0.5*math.sqrt(2), 0.0, 0.5*math.sqrt(2) ],
|
|
[ 0.0, 0.5*math.sqrt(2), 0.0, -0.5*math.sqrt(2) ],
|
|
[ 0.0, 0.5*math.sqrt(2),-0.5*math.sqrt(2), 0.0 ],
|
|
[ 0.0, -0.5*math.sqrt(2),-0.5*math.sqrt(2), 0.0 ],
|
|
[ 0.5, 0.5, 0.5, 0.5 ],
|
|
[-0.5, 0.5, 0.5, 0.5 ],
|
|
[-0.5, 0.5, 0.5, -0.5 ],
|
|
[-0.5, 0.5, -0.5, 0.5 ],
|
|
[-0.5, -0.5, 0.5, 0.5 ],
|
|
[-0.5, -0.5, 0.5, -0.5 ],
|
|
[-0.5, -0.5, -0.5, 0.5 ],
|
|
[-0.5, 0.5, -0.5, -0.5 ],
|
|
[-0.5*math.sqrt(2), 0.0, 0.0, 0.5*math.sqrt(2) ],
|
|
[ 0.5*math.sqrt(2), 0.0, 0.0, 0.5*math.sqrt(2) ],
|
|
[-0.5*math.sqrt(2), 0.0, 0.5*math.sqrt(2), 0.0 ],
|
|
[-0.5*math.sqrt(2), 0.0, -0.5*math.sqrt(2), 0.0 ],
|
|
[-0.5*math.sqrt(2), 0.5*math.sqrt(2), 0.0, 0.0 ],
|
|
[-0.5*math.sqrt(2),-0.5*math.sqrt(2), 0.0, 0.0 ],
|
|
]
|
|
elif self.lattice == 'hexagonal':
|
|
symQuats = [
|
|
[ 1.0,0.0,0.0,0.0 ],
|
|
[-0.5*math.sqrt(3), 0.0, 0.0,-0.5 ],
|
|
[ 0.5, 0.0, 0.0, 0.5*math.sqrt(3) ],
|
|
[ 0.0,0.0,0.0,1.0 ],
|
|
[-0.5, 0.0, 0.0, 0.5*math.sqrt(3) ],
|
|
[-0.5*math.sqrt(3), 0.0, 0.0, 0.5 ],
|
|
[ 0.0,1.0,0.0,0.0 ],
|
|
[ 0.0,-0.5*math.sqrt(3), 0.5, 0.0 ],
|
|
[ 0.0, 0.5,-0.5*math.sqrt(3), 0.0 ],
|
|
[ 0.0,0.0,1.0,0.0 ],
|
|
[ 0.0,-0.5,-0.5*math.sqrt(3), 0.0 ],
|
|
[ 0.0, 0.5*math.sqrt(3), 0.5, 0.0 ],
|
|
]
|
|
elif self.lattice == 'tetragonal':
|
|
symQuats = [
|
|
[ 1.0,0.0,0.0,0.0 ],
|
|
[ 0.0,1.0,0.0,0.0 ],
|
|
[ 0.0,0.0,1.0,0.0 ],
|
|
[ 0.0,0.0,0.0,1.0 ],
|
|
[ 0.0, 0.5*math.sqrt(2), 0.5*math.sqrt(2), 0.0 ],
|
|
[ 0.0,-0.5*math.sqrt(2), 0.5*math.sqrt(2), 0.0 ],
|
|
[ 0.5*math.sqrt(2), 0.0, 0.0, 0.5*math.sqrt(2) ],
|
|
[-0.5*math.sqrt(2), 0.0, 0.0, 0.5*math.sqrt(2) ],
|
|
]
|
|
elif self.lattice == 'orthorhombic':
|
|
symQuats = [
|
|
[ 1.0,0.0,0.0,0.0 ],
|
|
[ 0.0,1.0,0.0,0.0 ],
|
|
[ 0.0,0.0,1.0,0.0 ],
|
|
[ 0.0,0.0,0.0,1.0 ],
|
|
]
|
|
else:
|
|
symQuats = [
|
|
[ 1.0,0.0,0.0,0.0 ],
|
|
]
|
|
|
|
return list(map(Quaternion,
|
|
np.array(symQuats)[np.atleast_1d(np.array(who)) if who != [] else range(len(symQuats))]))
|
|
|
|
|
|
def equivalentQuaternions(self,
|
|
quaternion,
|
|
who = []):
|
|
"""List of symmetrically equivalent quaternions based on own symmetry."""
|
|
return [quaternion*q for q in self.symmetryQuats(who)]
|
|
|
|
|
|
def inFZ(self,R):
|
|
"""Check whether given Rodrigues vector falls into fundamental zone of own symmetry."""
|
|
if isinstance(R, Quaternion): R = R.asRodrigues() # translate accidentially passed quaternion
|
|
# fundamental zone in Rodrigues space is point symmetric around origin
|
|
R = abs(R)
|
|
if self.lattice == 'cubic':
|
|
return math.sqrt(2.0)-1.0 >= R[0] \
|
|
and math.sqrt(2.0)-1.0 >= R[1] \
|
|
and math.sqrt(2.0)-1.0 >= R[2] \
|
|
and 1.0 >= R[0] + R[1] + R[2]
|
|
elif self.lattice == 'hexagonal':
|
|
return 1.0 >= R[0] and 1.0 >= R[1] and 1.0 >= R[2] \
|
|
and 2.0 >= math.sqrt(3)*R[0] + R[1] \
|
|
and 2.0 >= math.sqrt(3)*R[1] + R[0] \
|
|
and 2.0 >= math.sqrt(3) + R[2]
|
|
elif self.lattice == 'tetragonal':
|
|
return 1.0 >= R[0] and 1.0 >= R[1] \
|
|
and math.sqrt(2.0) >= R[0] + R[1] \
|
|
and math.sqrt(2.0) >= R[2] + 1.0
|
|
elif self.lattice == 'orthorhombic':
|
|
return 1.0 >= R[0] and 1.0 >= R[1] and 1.0 >= R[2]
|
|
else:
|
|
return True
|
|
|
|
|
|
def inDisorientationSST(self,R):
|
|
"""
|
|
Check whether given Rodrigues vector (of misorientation) falls into standard stereographic triangle of own symmetry.
|
|
|
|
Determination of disorientations follow the work of A. Heinz and P. Neumann:
|
|
Representation of Orientation and Disorientation Data for Cubic, Hexagonal, Tetragonal and Orthorhombic Crystals
|
|
Acta Cryst. (1991). A47, 780-789
|
|
"""
|
|
if isinstance(R, Quaternion): R = R.asRodrigues() # translate accidentially passed quaternion
|
|
|
|
epsilon = 0.0
|
|
if self.lattice == 'cubic':
|
|
return R[0] >= R[1]+epsilon and R[1] >= R[2]+epsilon and R[2] >= epsilon
|
|
|
|
elif self.lattice == 'hexagonal':
|
|
return R[0] >= math.sqrt(3)*(R[1]-epsilon) and R[1] >= epsilon and R[2] >= epsilon
|
|
|
|
elif self.lattice == 'tetragonal':
|
|
return R[0] >= R[1]-epsilon and R[1] >= epsilon and R[2] >= epsilon
|
|
|
|
elif self.lattice == 'orthorhombic':
|
|
return R[0] >= epsilon and R[1] >= epsilon and R[2] >= epsilon
|
|
|
|
else:
|
|
return True
|
|
|
|
|
|
def inSST(self,
|
|
vector,
|
|
proper = False,
|
|
color = False):
|
|
"""
|
|
Check whether given vector falls into standard stereographic triangle of own symmetry.
|
|
|
|
proper considers only vectors with z >= 0, hence uses two neighboring SSTs.
|
|
Return inverse pole figure color if requested.
|
|
"""
|
|
# basis = {'cubic' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red
|
|
# [1.,0.,1.]/np.sqrt(2.), # direction of green
|
|
# [1.,1.,1.]/np.sqrt(3.)]).transpose()), # direction of blue
|
|
# 'hexagonal' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red
|
|
# [1.,0.,0.], # direction of green
|
|
# [np.sqrt(3.),1.,0.]/np.sqrt(4.)]).transpose()), # direction of blue
|
|
# 'tetragonal' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red
|
|
# [1.,0.,0.], # direction of green
|
|
# [1.,1.,0.]/np.sqrt(2.)]).transpose()), # direction of blue
|
|
# 'orthorhombic' : np.linalg.inv(np.array([[0.,0.,1.], # direction of red
|
|
# [1.,0.,0.], # direction of green
|
|
# [0.,1.,0.]]).transpose()), # direction of blue
|
|
# }
|
|
|
|
if self.lattice == 'cubic':
|
|
basis = {'improper':np.array([ [-1. , 0. , 1. ],
|
|
[ np.sqrt(2.) , -np.sqrt(2.) , 0. ],
|
|
[ 0. , np.sqrt(3.) , 0. ] ]),
|
|
'proper':np.array([ [ 0. , -1. , 1. ],
|
|
[-np.sqrt(2.) , np.sqrt(2.) , 0. ],
|
|
[ np.sqrt(3.) , 0. , 0. ] ]),
|
|
}
|
|
elif self.lattice == 'hexagonal':
|
|
basis = {'improper':np.array([ [ 0. , 0. , 1. ],
|
|
[ 1. , -np.sqrt(3.), 0. ],
|
|
[ 0. , 2. , 0. ] ]),
|
|
'proper':np.array([ [ 0. , 0. , 1. ],
|
|
[-1. , np.sqrt(3.) , 0. ],
|
|
[ np.sqrt(3) , -1. , 0. ] ]),
|
|
}
|
|
elif self.lattice == 'tetragonal':
|
|
basis = {'improper':np.array([ [ 0. , 0. , 1. ],
|
|
[ 1. , -1. , 0. ],
|
|
[ 0. , np.sqrt(2.), 0. ] ]),
|
|
'proper':np.array([ [ 0. , 0. , 1. ],
|
|
[-1. , 1. , 0. ],
|
|
[ np.sqrt(2.) , 0. , 0. ] ]),
|
|
}
|
|
elif self.lattice == 'orthorhombic':
|
|
basis = {'improper':np.array([ [ 0., 0., 1.],
|
|
[ 1., 0., 0.],
|
|
[ 0., 1., 0.] ]),
|
|
'proper':np.array([ [ 0., 0., 1.],
|
|
[-1., 0., 0.],
|
|
[ 0., 1., 0.] ]),
|
|
}
|
|
else:
|
|
basis = {'improper':np.zeros((3,3),dtype=float),
|
|
'proper':np.zeros((3,3),dtype=float),
|
|
}
|
|
|
|
if np.all(basis == 0.0):
|
|
theComponents = -np.ones(3,'d')
|
|
inSST = np.all(theComponents >= 0.0)
|
|
else:
|
|
v = np.array(vector,dtype = float)
|
|
if proper: # check both improper ...
|
|
theComponents = np.dot(basis['improper'],v)
|
|
inSST = np.all(theComponents >= 0.0)
|
|
if not inSST: # ... and proper SST
|
|
theComponents = np.dot(basis['proper'],v)
|
|
inSST = np.all(theComponents >= 0.0)
|
|
else:
|
|
v[2] = abs(v[2]) # z component projects identical
|
|
theComponents = np.dot(basis['improper'],v) # for positive and negative values
|
|
inSST = np.all(theComponents >= 0.0)
|
|
|
|
if color: # have to return color array
|
|
if inSST:
|
|
rgb = np.power(theComponents/np.linalg.norm(theComponents),0.5) # smoothen color ramps
|
|
rgb = np.minimum(np.ones(3,'d'),rgb) # limit to maximum intensity
|
|
rgb /= max(rgb) # normalize to (HS)V = 1
|
|
else:
|
|
rgb = np.zeros(3,'d')
|
|
return (inSST,rgb)
|
|
else:
|
|
return inSST
|
|
|
|
# code derived from http://pyeuclid.googlecode.com/svn/trunk/euclid.py
|
|
# suggested reading: http://web.mit.edu/2.998/www/QuaternionReport1.pdf
|
|
|
|
|
|
|
|
# ******************************************************************************************
|
|
class Orientation:
|
|
|
|
__slots__ = ['quaternion','symmetry']
|
|
|
|
def __init__(self,
|
|
quaternion = Quaternion.fromIdentity(),
|
|
Rodrigues = None,
|
|
angleAxis = None,
|
|
matrix = None,
|
|
Eulers = None,
|
|
random = False, # integer to have a fixed seed or True for real random
|
|
symmetry = None,
|
|
degrees = False,
|
|
):
|
|
if random: # produce random orientation
|
|
if isinstance(random, bool ):
|
|
self.quaternion = Quaternion.fromRandom()
|
|
else:
|
|
self.quaternion = Quaternion.fromRandom(randomSeed=random)
|
|
elif isinstance(Eulers, np.ndarray) and Eulers.shape == (3,): # based on given Euler angles
|
|
self.quaternion = Quaternion.fromEulers(Eulers,type='bunge',degrees=degrees)
|
|
elif isinstance(matrix, np.ndarray) : # based on given rotation matrix
|
|
self.quaternion = Quaternion.fromMatrix(matrix)
|
|
elif isinstance(angleAxis, np.ndarray) and angleAxis.shape == (4,): # based on given angle and rotation axis
|
|
self.quaternion = Quaternion.fromAngleAxis(angleAxis[0],angleAxis[1:4],degrees=degrees)
|
|
elif isinstance(Rodrigues, np.ndarray) and Rodrigues.shape == (3,): # based on given Rodrigues vector
|
|
self.quaternion = Quaternion.fromRodrigues(Rodrigues)
|
|
elif isinstance(quaternion, Quaternion): # based on given quaternion
|
|
self.quaternion = quaternion.homomorphed()
|
|
elif isinstance(quaternion, np.ndarray) and quaternion.shape == (4,): # based on given quaternion-like array
|
|
self.quaternion = Quaternion(quaternion).homomorphed()
|
|
|
|
self.symmetry = Symmetry(symmetry)
|
|
|
|
def __copy__(self):
|
|
"""copy"""
|
|
return self.__class__(quaternion=self.quaternion,symmetry=self.symmetry.lattice)
|
|
|
|
copy = __copy__
|
|
|
|
|
|
def __repr__(self):
|
|
"""value as all implemented representations"""
|
|
return 'Symmetry: %s\n' % (self.symmetry) + \
|
|
'Quaternion: %s\n' % (self.quaternion) + \
|
|
'Matrix:\n%s\n' % ( '\n'.join(['\t'.join(map(str,self.asMatrix()[i,:])) for i in range(3)]) ) + \
|
|
'Bunge Eulers / deg: %s' % ('\t'.join(map(str,self.asEulers('bunge',degrees=True))) )
|
|
|
|
def asQuaternion(self):
|
|
return self.quaternion.asList()
|
|
|
|
def asEulers(self,
|
|
type = 'bunge',
|
|
degrees = False,
|
|
standardRange = False):
|
|
return self.quaternion.asEulers(type, degrees, standardRange)
|
|
eulers = property(asEulers)
|
|
|
|
def asRodrigues(self):
|
|
return self.quaternion.asRodrigues()
|
|
rodrigues = property(asRodrigues)
|
|
|
|
def asAngleAxis(self,
|
|
degrees = False):
|
|
return self.quaternion.asAngleAxis(degrees)
|
|
angleAxis = property(asAngleAxis)
|
|
|
|
def asMatrix(self):
|
|
return self.quaternion.asMatrix()
|
|
matrix = property(asMatrix)
|
|
|
|
def inFZ(self):
|
|
return self.symmetry.inFZ(self.quaternion.asRodrigues())
|
|
infz = property(inFZ)
|
|
|
|
def equivalentQuaternions(self,
|
|
who = []):
|
|
return self.symmetry.equivalentQuaternions(self.quaternion,who)
|
|
|
|
def equivalentOrientations(self,
|
|
who = []):
|
|
return [Orientation(quaternion = q, symmetry = self.symmetry.lattice) for q in self.equivalentQuaternions(who)]
|
|
|
|
def reduced(self):
|
|
"""Transform orientation to fall into fundamental zone according to symmetry"""
|
|
for me in self.symmetry.equivalentQuaternions(self.quaternion):
|
|
if self.symmetry.inFZ(me.asRodrigues()): break
|
|
|
|
return Orientation(quaternion=me,symmetry=self.symmetry.lattice)
|
|
|
|
|
|
def disorientation(self,
|
|
other,
|
|
SST = True):
|
|
"""
|
|
Disorientation between myself and given other orientation.
|
|
|
|
Rotation axis falls into SST if SST == True.
|
|
(Currently requires same symmetry for both orientations.
|
|
Look into A. Heinz and P. Neumann 1991 for cases with differing sym.)
|
|
"""
|
|
if self.symmetry != other.symmetry: raise TypeError('disorientation between different symmetry classes not supported yet.')
|
|
|
|
misQ = self.quaternion.conjugated()*other.quaternion
|
|
mySymQs = self.symmetry.symmetryQuats() if SST else self.symmetry.symmetryQuats()[:1] # take all or only first sym operation
|
|
otherSymQs = other.symmetry.symmetryQuats()
|
|
|
|
for i,sA in enumerate(mySymQs):
|
|
for j,sB in enumerate(otherSymQs):
|
|
theQ = sA.conjugated()*misQ*sB
|
|
for k in range(2):
|
|
theQ.conjugate()
|
|
breaker = self.symmetry.inFZ(theQ) \
|
|
and (not SST or other.symmetry.inDisorientationSST(theQ))
|
|
if breaker: break
|
|
if breaker: break
|
|
if breaker: break
|
|
|
|
# disorientation, own sym, other sym, self-->other: True, self<--other: False
|
|
return (Orientation(quaternion = theQ,symmetry = self.symmetry.lattice),
|
|
i,j,k == 1)
|
|
|
|
|
|
def inversePole(self,
|
|
axis,
|
|
proper = False,
|
|
SST = True):
|
|
"""axis rotated according to orientation (using crystal symmetry to ensure location falls into SST)"""
|
|
if SST: # pole requested to be within SST
|
|
for i,q in enumerate(self.symmetry.equivalentQuaternions(self.quaternion)): # test all symmetric equivalent quaternions
|
|
pole = q.conjugated()*axis # align crystal direction to axis
|
|
if self.symmetry.inSST(pole,proper): break # found SST version
|
|
else:
|
|
pole = self.quaternion.conjugated()*axis # align crystal direction to axis
|
|
|
|
return (pole,i if SST else 0)
|
|
|
|
def IPFcolor(self,axis):
|
|
"""TSL color of inverse pole figure for given axis"""
|
|
color = np.zeros(3,'d')
|
|
|
|
for q in self.symmetry.equivalentQuaternions(self.quaternion):
|
|
pole = q.conjugated()*axis # align crystal direction to axis
|
|
inSST,color = self.symmetry.inSST(pole,color=True)
|
|
if inSST: break
|
|
|
|
return color
|
|
|
|
@classmethod
|
|
def average(cls,
|
|
orientations,
|
|
multiplicity = []):
|
|
"""
|
|
average orientation
|
|
|
|
ref: F. Landis Markley, Yang Cheng, John Lucas Crassidis, and Yaakov Oshman.
|
|
Averaging Quaternions,
|
|
Journal of Guidance, Control, and Dynamics, Vol. 30, No. 4 (2007), pp. 1193-1197.
|
|
doi: 10.2514/1.28949
|
|
usage:
|
|
a = Orientation(Eulers=np.radians([10, 10, 0]), symmetry='hexagonal')
|
|
b = Orientation(Eulers=np.radians([20, 0, 0]), symmetry='hexagonal')
|
|
avg = Orientation.average([a,b])
|
|
"""
|
|
if not all(isinstance(item, Orientation) for item in orientations):
|
|
raise TypeError("Only instances of Orientation can be averaged.")
|
|
|
|
N = len(orientations)
|
|
if multiplicity == [] or not multiplicity:
|
|
multiplicity = np.ones(N,dtype='i')
|
|
|
|
reference = orientations[0] # take first as reference
|
|
for i,(o,n) in enumerate(zip(orientations,multiplicity)):
|
|
closest = o.equivalentOrientations(reference.disorientation(o,SST = False)[2])[0] # select sym orientation with lowest misorientation
|
|
M = closest.quaternion.asM() * n if i == 0 else M + closest.quaternion.asM() * n # noqa add (multiples) of this orientation to average noqa
|
|
eig, vec = np.linalg.eig(M/N)
|
|
|
|
return Orientation(quaternion = Quaternion(quatArray = np.real(vec.T[eig.argmax()])),
|
|
symmetry = reference.symmetry.lattice)
|
|
|
|
|
|
def related(self,
|
|
relationModel,
|
|
direction,
|
|
targetSymmetry = None):
|
|
|
|
if relationModel not in ['KS','GT','GTdash','NW','Pitsch','Bain']: return None
|
|
if int(direction) == 0: return None
|
|
|
|
# KS from S. Morito et al./Journal of Alloys and Compounds 5775 (2013) S587-S592 DOES THIS PAPER EXISTS?
|
|
# GT from Y. He et al./Journal of Applied Crystallography (2006). 39, 72-81
|
|
# GT' from Y. He et al./Journal of Applied Crystallography (2006). 39, 72-81
|
|
# NW from H. Kitahara et al./Materials Characterization 54 (2005) 378-386
|
|
# Pitsch from Y. He et al./Acta Materialia 53 (2005) 1179-1190
|
|
# Bain from Y. He et al./Journal of Applied Crystallography (2006). 39, 72-81
|
|
|
|
variant = int(abs(direction))-1
|
|
(me,other) = (0,1) if direction > 0 else (1,0)
|
|
|
|
planes = {'KS': \
|
|
np.array([[[ 1, 1, 1],[ 0, 1, 1]],
|
|
[[ 1, 1, 1],[ 0, 1, 1]],
|
|
[[ 1, 1, 1],[ 0, 1, 1]],
|
|
[[ 1, 1, 1],[ 0, 1, 1]],
|
|
[[ 1, 1, 1],[ 0, 1, 1]],
|
|
[[ 1, 1, 1],[ 0, 1, 1]],
|
|
[[ 1, -1, 1],[ 0, 1, 1]],
|
|
[[ 1, -1, 1],[ 0, 1, 1]],
|
|
[[ 1, -1, 1],[ 0, 1, 1]],
|
|
[[ 1, -1, 1],[ 0, 1, 1]],
|
|
[[ 1, -1, 1],[ 0, 1, 1]],
|
|
[[ 1, -1, 1],[ 0, 1, 1]],
|
|
[[ -1, 1, 1],[ 0, 1, 1]],
|
|
[[ -1, 1, 1],[ 0, 1, 1]],
|
|
[[ -1, 1, 1],[ 0, 1, 1]],
|
|
[[ -1, 1, 1],[ 0, 1, 1]],
|
|
[[ -1, 1, 1],[ 0, 1, 1]],
|
|
[[ -1, 1, 1],[ 0, 1, 1]],
|
|
[[ 1, 1, -1],[ 0, 1, 1]],
|
|
[[ 1, 1, -1],[ 0, 1, 1]],
|
|
[[ 1, 1, -1],[ 0, 1, 1]],
|
|
[[ 1, 1, -1],[ 0, 1, 1]],
|
|
[[ 1, 1, -1],[ 0, 1, 1]],
|
|
[[ 1, 1, -1],[ 0, 1, 1]]]),
|
|
'GT': \
|
|
np.array([[[ 1, 1, 1],[ 1, 0, 1]],
|
|
[[ 1, 1, 1],[ 1, 1, 0]],
|
|
[[ 1, 1, 1],[ 0, 1, 1]],
|
|
[[ -1, -1, 1],[ -1, 0, 1]],
|
|
[[ -1, -1, 1],[ -1, -1, 0]],
|
|
[[ -1, -1, 1],[ 0, -1, 1]],
|
|
[[ -1, 1, 1],[ -1, 0, 1]],
|
|
[[ -1, 1, 1],[ -1, 1, 0]],
|
|
[[ -1, 1, 1],[ 0, 1, 1]],
|
|
[[ 1, -1, 1],[ 1, 0, 1]],
|
|
[[ 1, -1, 1],[ 1, -1, 0]],
|
|
[[ 1, -1, 1],[ 0, -1, 1]],
|
|
[[ 1, 1, 1],[ 1, 1, 0]],
|
|
[[ 1, 1, 1],[ 0, 1, 1]],
|
|
[[ 1, 1, 1],[ 1, 0, 1]],
|
|
[[ -1, -1, 1],[ -1, -1, 0]],
|
|
[[ -1, -1, 1],[ 0, -1, 1]],
|
|
[[ -1, -1, 1],[ -1, 0, 1]],
|
|
[[ -1, 1, 1],[ -1, 1, 0]],
|
|
[[ -1, 1, 1],[ 0, 1, 1]],
|
|
[[ -1, 1, 1],[ -1, 0, 1]],
|
|
[[ 1, -1, 1],[ 1, -1, 0]],
|
|
[[ 1, -1, 1],[ 0, -1, 1]],
|
|
[[ 1, -1, 1],[ 1, 0, 1]]]),
|
|
'GTdash': \
|
|
np.array([[[ 7, 17, 17],[ 12, 5, 17]],
|
|
[[ 17, 7, 17],[ 17, 12, 5]],
|
|
[[ 17, 17, 7],[ 5, 17, 12]],
|
|
[[ -7,-17, 17],[-12, -5, 17]],
|
|
[[-17, -7, 17],[-17,-12, 5]],
|
|
[[-17,-17, 7],[ -5,-17, 12]],
|
|
[[ 7,-17,-17],[ 12, -5,-17]],
|
|
[[ 17, -7,-17],[ 17,-12, -5]],
|
|
[[ 17,-17, -7],[ 5,-17,-12]],
|
|
[[ -7, 17,-17],[-12, 5,-17]],
|
|
[[-17, 7,-17],[-17, 12, -5]],
|
|
[[-17, 17, -7],[ -5, 17,-12]],
|
|
[[ 7, 17, 17],[ 12, 17, 5]],
|
|
[[ 17, 7, 17],[ 5, 12, 17]],
|
|
[[ 17, 17, 7],[ 17, 5, 12]],
|
|
[[ -7,-17, 17],[-12,-17, 5]],
|
|
[[-17, -7, 17],[ -5,-12, 17]],
|
|
[[-17,-17, 7],[-17, -5, 12]],
|
|
[[ 7,-17,-17],[ 12,-17, -5]],
|
|
[[ 17, -7,-17],[ 5, -12,-17]],
|
|
[[ 17,-17, 7],[ 17, -5,-12]],
|
|
[[ -7, 17,-17],[-12, 17, -5]],
|
|
[[-17, 7,-17],[ -5, 12,-17]],
|
|
[[-17, 17, -7],[-17, 5,-12]]]),
|
|
'NW': \
|
|
np.array([[[ 1, 1, 1],[ 0, 1, 1]],
|
|
[[ 1, 1, 1],[ 0, 1, 1]],
|
|
[[ 1, 1, 1],[ 0, 1, 1]],
|
|
[[ -1, 1, 1],[ 0, 1, 1]],
|
|
[[ -1, 1, 1],[ 0, 1, 1]],
|
|
[[ -1, 1, 1],[ 0, 1, 1]],
|
|
[[ 1, -1, 1],[ 0, 1, 1]],
|
|
[[ 1, -1, 1],[ 0, 1, 1]],
|
|
[[ 1, -1, 1],[ 0, 1, 1]],
|
|
[[ -1, -1, 1],[ 0, 1, 1]],
|
|
[[ -1, -1, 1],[ 0, 1, 1]],
|
|
[[ -1, -1, 1],[ 0, 1, 1]]]),
|
|
'Pitsch': \
|
|
np.array([[[ 0, 1, 0],[ -1, 0, 1]],
|
|
[[ 0, 0, 1],[ 1, -1, 0]],
|
|
[[ 1, 0, 0],[ 0, 1, -1]],
|
|
[[ 1, 0, 0],[ 0, -1, -1]],
|
|
[[ 0, 1, 0],[ -1, 0, -1]],
|
|
[[ 0, 0, 1],[ -1, -1, 0]],
|
|
[[ 0, 1, 0],[ -1, 0, -1]],
|
|
[[ 0, 0, 1],[ -1, -1, 0]],
|
|
[[ 1, 0, 0],[ 0, -1, -1]],
|
|
[[ 1, 0, 0],[ 0, -1, 1]],
|
|
[[ 0, 1, 0],[ 1, 0, -1]],
|
|
[[ 0, 0, 1],[ -1, 1, 0]]]),
|
|
'Bain': \
|
|
np.array([[[ 1, 0, 0],[ 1, 0, 0]],
|
|
[[ 0, 1, 0],[ 0, 1, 0]],
|
|
[[ 0, 0, 1],[ 0, 0, 1]]]),
|
|
}
|
|
|
|
normals = {'KS': \
|
|
np.array([[[ -1, 0, 1],[ -1, -1, 1]],
|
|
[[ -1, 0, 1],[ -1, 1, -1]],
|
|
[[ 0, 1, -1],[ -1, -1, 1]],
|
|
[[ 0, 1, -1],[ -1, 1, -1]],
|
|
[[ 1, -1, 0],[ -1, -1, 1]],
|
|
[[ 1, -1, 0],[ -1, 1, -1]],
|
|
[[ 1, 0, -1],[ -1, -1, 1]],
|
|
[[ 1, 0, -1],[ -1, 1, -1]],
|
|
[[ -1, -1, 0],[ -1, -1, 1]],
|
|
[[ -1, -1, 0],[ -1, 1, -1]],
|
|
[[ 0, 1, 1],[ -1, -1, 1]],
|
|
[[ 0, 1, 1],[ -1, 1, -1]],
|
|
[[ 0, -1, 1],[ -1, -1, 1]],
|
|
[[ 0, -1, 1],[ -1, 1, -1]],
|
|
[[ -1, 0, -1],[ -1, -1, 1]],
|
|
[[ -1, 0, -1],[ -1, 1, -1]],
|
|
[[ 1, 1, 0],[ -1, -1, 1]],
|
|
[[ 1, 1, 0],[ -1, 1, -1]],
|
|
[[ -1, 1, 0],[ -1, -1, 1]],
|
|
[[ -1, 1, 0],[ -1, 1, -1]],
|
|
[[ 0, -1, -1],[ -1, -1, 1]],
|
|
[[ 0, -1, -1],[ -1, 1, -1]],
|
|
[[ 1, 0, 1],[ -1, -1, 1]],
|
|
[[ 1, 0, 1],[ -1, 1, -1]]]),
|
|
'GT': \
|
|
np.array([[[ -5,-12, 17],[-17, -7, 17]],
|
|
[[ 17, -5,-12],[ 17,-17, -7]],
|
|
[[-12, 17, -5],[ -7, 17,-17]],
|
|
[[ 5, 12, 17],[ 17, 7, 17]],
|
|
[[-17, 5,-12],[-17, 17, -7]],
|
|
[[ 12,-17, -5],[ 7,-17,-17]],
|
|
[[ -5, 12,-17],[-17, 7,-17]],
|
|
[[ 17, 5, 12],[ 17, 17, 7]],
|
|
[[-12,-17, 5],[ -7,-17, 17]],
|
|
[[ 5,-12,-17],[ 17, -7,-17]],
|
|
[[-17, -5, 12],[-17,-17, 7]],
|
|
[[ 12, 17, 5],[ 7, 17, 17]],
|
|
[[ -5, 17,-12],[-17, 17, -7]],
|
|
[[-12, -5, 17],[ -7,-17, 17]],
|
|
[[ 17,-12, -5],[ 17, -7,-17]],
|
|
[[ 5,-17,-12],[ 17,-17, -7]],
|
|
[[ 12, 5, 17],[ 7, 17, 17]],
|
|
[[-17, 12, -5],[-17, 7,-17]],
|
|
[[ -5,-17, 12],[-17,-17, 7]],
|
|
[[-12, 5,-17],[ -7, 17,-17]],
|
|
[[ 17, 12, 5],[ 17, 7, 17]],
|
|
[[ 5, 17, 12],[ 17, 17, 7]],
|
|
[[ 12, -5,-17],[ 7,-17,-17]],
|
|
[[-17,-12, 5],[-17, 7, 17]]]),
|
|
'GTdash': \
|
|
np.array([[[ 0, 1, -1],[ 1, 1, -1]],
|
|
[[ -1, 0, 1],[ -1, 1, 1]],
|
|
[[ 1, -1, 0],[ 1, -1, 1]],
|
|
[[ 0, -1, -1],[ -1, -1, -1]],
|
|
[[ 1, 0, 1],[ 1, -1, 1]],
|
|
[[ 1, -1, 0],[ 1, -1, -1]],
|
|
[[ 0, 1, -1],[ -1, 1, -1]],
|
|
[[ 1, 0, 1],[ 1, 1, 1]],
|
|
[[ -1, -1, 0],[ -1, -1, 1]],
|
|
[[ 0, -1, -1],[ 1, -1, -1]],
|
|
[[ -1, 0, 1],[ -1, -1, 1]],
|
|
[[ -1, -1, 0],[ -1, -1, -1]],
|
|
[[ 0, -1, 1],[ 1, -1, 1]],
|
|
[[ 1, 0, -1],[ 1, 1, -1]],
|
|
[[ -1, 1, 0],[ -1, 1, 1]],
|
|
[[ 0, 1, 1],[ -1, 1, 1]],
|
|
[[ -1, 0, -1],[ -1, -1, -1]],
|
|
[[ -1, 1, 0],[ -1, 1, -1]],
|
|
[[ 0, -1, 1],[ -1, -1, 1]],
|
|
[[ -1, 0, -1],[ -1, 1, -1]],
|
|
[[ 1, 1, 0],[ 1, 1, 1]],
|
|
[[ 0, 1, 1],[ 1, 1, 1]],
|
|
[[ 1, 0, -1],[ 1, -1, -1]],
|
|
[[ 1, 1, 0],[ 1, 1, -1]]]),
|
|
'NW': \
|
|
np.array([[[ 2, -1, -1],[ 0, -1, 1]],
|
|
[[ -1, 2, -1],[ 0, -1, 1]],
|
|
[[ -1, -1, 2],[ 0, -1, 1]],
|
|
[[ -2, -1, -1],[ 0, -1, 1]],
|
|
[[ 1, 2, -1],[ 0, -1, 1]],
|
|
[[ 1, -1, 2],[ 0, -1, 1]],
|
|
[[ 2, 1, -1],[ 0, -1, 1]],
|
|
[[ -1, -2, -1],[ 0, -1, 1]],
|
|
[[ -1, 1, 2],[ 0, -1, 1]],
|
|
[[ -1, 2, 1],[ 0, -1, 1]],
|
|
[[ -1, 2, 1],[ 0, -1, 1]],
|
|
[[ -1, -1, -2],[ 0, -1, 1]]]),
|
|
'Pitsch': \
|
|
np.array([[[ 1, 0, 1],[ 1, -1, 1]],
|
|
[[ 1, 1, 0],[ 1, 1, -1]],
|
|
[[ 0, 1, 1],[ -1, 1, 1]],
|
|
[[ 0, 1, -1],[ -1, 1, -1]],
|
|
[[ -1, 0, 1],[ -1, -1, 1]],
|
|
[[ 1, -1, 0],[ 1, -1, -1]],
|
|
[[ 1, 0, -1],[ 1, -1, -1]],
|
|
[[ -1, 1, 0],[ -1, 1, -1]],
|
|
[[ 0, -1, 1],[ -1, -1, 1]],
|
|
[[ 0, 1, 1],[ -1, 1, 1]],
|
|
[[ 1, 0, 1],[ 1, -1, 1]],
|
|
[[ 1, 1, 0],[ 1, 1, -1]]]),
|
|
'Bain': \
|
|
np.array([[[ 0, 1, 0],[ 0, 1, 1]],
|
|
[[ 0, 0, 1],[ 1, 0, 1]],
|
|
[[ 1, 0, 0],[ 1, 1, 0]]]),
|
|
}
|
|
myPlane = [float(i) for i in planes[relationModel][variant,me]] # map(float, planes[...]) does not work in python 3
|
|
myPlane /= np.linalg.norm(myPlane)
|
|
myNormal = [float(i) for i in normals[relationModel][variant,me]] # map(float, planes[...]) does not work in python 3
|
|
myNormal /= np.linalg.norm(myNormal)
|
|
myMatrix = np.array([myPlane,myNormal,np.cross(myPlane,myNormal)])
|
|
|
|
otherPlane = [float(i) for i in planes[relationModel][variant,other]] # map(float, planes[...]) does not work in python 3
|
|
otherPlane /= np.linalg.norm(otherPlane)
|
|
otherNormal = [float(i) for i in normals[relationModel][variant,other]] # map(float, planes[...]) does not work in python 3
|
|
otherNormal /= np.linalg.norm(otherNormal)
|
|
otherMatrix = np.array([otherPlane,otherNormal,np.cross(otherPlane,otherNormal)])
|
|
|
|
rot=np.dot(otherMatrix.T,myMatrix)
|
|
|
|
return Orientation(matrix=np.dot(rot,self.asMatrix())) # no symmetry information ??
|