838 lines
29 KiB
Python
838 lines
29 KiB
Python
# -*- coding: UTF-8 no BOM -*-
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import numpy,math,random
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# ******************************************************************************************
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class Rodrigues:
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# ******************************************************************************************
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def __init__(self, vector = numpy.zeros(3)):
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self.vector = vector
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def asQuaternion(self):
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norm = numpy.linalg.norm(self.vector)
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halfAngle = numpy.arctan(norm)
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return Quaternion(numpy.cos(halfAngle),numpy.sin(halfAngle)*self.vector/norm)
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def asAngleAxis(self):
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norm = numpy.linalg.norm(self.vector)
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halfAngle = numpy.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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# All methods and naming conventions based off
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# 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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def __init__(self, quatArray=[1.0,0.0,0.0,0.0]):
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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 = self.homomorph()
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def __iter__(self):
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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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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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return 'Quaternion(real=%+.4f, imag=<%+.4f, %+.4f, %+.4f>)' % \
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(self.w, self.x, self.y, self.z)
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def __pow__(self, exponent):
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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.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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Q.w = math.cos(exponent*omega)
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return Q
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def __ipow__(self, exponent):
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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.x *= vRescale
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self.y *= vRescale
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self.z *= vRescale
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self.w = numpy.cos(exponent*omega)
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return self
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def __mul__(self, other):
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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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Q = Quaternion()
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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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Q.w = - Ax * Bx - Ay * By - Az * Bz + Aw * Bw
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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 numpy.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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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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if isinstance(other, (int,float,long)):
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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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if isinstance(other, (int,float,long)):
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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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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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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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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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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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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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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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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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return not __eq__(self,other)
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def __cmp__(self,other):
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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 rotateBy_angleaxis(self, angle, axis):
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self *= Quaternion.fromAngleAxis(angle, axis)
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return self
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def rotateBy_Eulers(self, heading, attitude, bank):
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self *= Quaternion.fromEulers(eulers, type)
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return self
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def rotateBy_matrix(self, m):
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self *= Quaternion.fromMatrix(m)
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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 numpy.outer([i for i in self],[i for i in self])
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def asMatrix(self):
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return numpy.array([[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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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 < 1e-3:
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return angle, numpy.array([1.0, 0.0, 0.0])
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else:
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return angle, numpy.array([self.x / s, self.y / s, self.z / s])
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def asRodrigues(self):
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if self.w != 0.0:
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return numpy.array([self.x, self.y, self.z])/self.w
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else:
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return numpy.array([float('inf')]*3)
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def asEulers(self,type='bunge'):
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'''
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conversion taken from:
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Melcher, A.; Unser, A.; Reichhardt, M.; Nestler, B.; Pötschke, 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 - self.y) < 1e-8:
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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 - self.z) < 1e-8:
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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 angles[0] < 0.0:
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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
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# if angles[2] < 0.0:
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# angles[2] += 2*math.pi
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return 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):
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r1 = random.random()
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r2 = random.random()
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r3 = random.random()
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w = math.cos(2.0*math.pi*r1)*math.sqrt(r3)
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x = math.sin(2.0*math.pi*r2)*math.sqrt(1.0-r3)
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y = math.cos(2.0*math.pi*r2)*math.sqrt(1.0-r3)
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z = math.sin(2.0*math.pi*r1)*math.sqrt(r3)
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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, numpy.ndarray): rodrigues = numpy.array(rodrigues)
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halfangle = math.atan(numpy.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, angle, axis):
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if not isinstance(axis, numpy.ndarray): axis = numpy.array(axis)
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axis /= numpy.linalg.norm(axis)
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s = math.sin(angle / 2.0)
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w = math.cos(angle / 2.0)
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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, eulers, type = 'Bunge'):
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c1 = math.cos(eulers[0] / 2.0)
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s1 = math.sin(eulers[0] / 2.0)
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c2 = math.cos(eulers[1] / 2.0)
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s2 = math.sin(eulers[1] / 2.0)
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c3 = math.cos(eulers[2] / 2.0)
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s3 = math.sin(eulers[2] / 2.0)
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if type.lower() == 'bunge' or type.lower() == 'zxz':
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w = c1 * c2 * c3 - s1 * c2 * s3
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x = c1 * s2 * c3 + s1 * s2 * s3
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y = - c1 * s2 * s3 + s1 * s2 * c3
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z = c1 * c2 * s3 + s1 * c2 * c3
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else:
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# print 'unknown Euler convention'
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w = c1 * c2 * c3 - s1 * s2 * s3
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x = s1 * s2 * c3 + c1 * c2 * s3
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y = s1 * c2 * c3 + c1 * s2 * s3
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z = c1 * s2 * c3 - s1 * c2 * s3
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return cls([w,x,y,z])
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@classmethod
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def fromMatrix(cls, m):
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if m[0,0] + m[1,1] + m[2,2] > 0.00000001:
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t = m[0,0] + m[1,1] + m[2,2] + 1.0
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s = 0.5/math.sqrt(t)
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return cls(
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[ s*t,
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(m[1,2] - m[2,1])*s,
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(m[2,0] - m[0,2])*s,
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(m[0,1] - m[1,0])*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 = 0.5/math.sqrt(t)
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return cls(
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[ (m[1,2] - m[2,1])*s,
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s*t,
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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 = 0.5/math.sqrt(t)
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return cls(
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[ (m[2,0] - m[0,2])*s,
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(m[0,1] + m[1,0])*s,
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s*t,
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(m[1,2] + m[2,1])*s,
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])
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else:
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t = -m[0,0] - m[1,1] + m[2,2] + 1.0
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s = 0.5/math.sqrt(t)
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return cls(
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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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(m[1,2] + m[2,1])*s,
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s*t,
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])
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@classmethod
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def new_interpolate(cls, q1, q2, t):
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# see http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20070017872_2007014421.pdf for (another?) way to interpolate quaternions
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assert isinstance(q1, Quaternion) and isinstance(q2, Quaternion)
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Q = cls()
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costheta = q1.w * q2.w + q1.x * q2.x + q1.y * q2.y + q1.z * q2.z
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if costheta < 0.:
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costheta = -costheta
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q1 = q1.conjugated()
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elif costheta > 1:
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costheta = 1
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theta = math.acos(costheta)
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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):
|
|
if isinstance(symmetry, basestring) and symmetry.lower() in Symmetry.lattices:
|
|
self.lattice = symmetry.lower()
|
|
else:
|
|
self.lattice = None
|
|
|
|
|
|
def __copy__(self):
|
|
return self.__class__(self.lattice)
|
|
|
|
copy = __copy__
|
|
|
|
|
|
def __repr__(self):
|
|
return '%s' % (self.lattice)
|
|
|
|
|
|
def __eq__(self, other):
|
|
return self.lattice == other.lattice
|
|
|
|
|
|
def __neq__(self, other):
|
|
return not self.__eq__(other)
|
|
|
|
def __cmp__(self,other):
|
|
return cmp(Symmetry.lattices.index(self.lattice),Symmetry.lattices.index(other.lattice))
|
|
|
|
def equivalentQuaternions(self,quaternion):
|
|
'''
|
|
List of symmetrically equivalent quaternions based on own symmetry.
|
|
'''
|
|
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 ],
|
|
[ 0.0,1.0,0.0,0.0 ],
|
|
[ 0.0,0.0,1.0,0.0 ],
|
|
[ 0.0,0.0,0.0,1.0 ],
|
|
[-0.5*math.sqrt(3), 0.0, 0.0, 0.5 ],
|
|
[-0.5*math.sqrt(3), 0.0, 0.0,-0.5 ],
|
|
[ 0.0, 0.5*math.sqrt(3), 0.5, 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,-0.5,-0.5*math.sqrt(3), 0.0 ],
|
|
[ 0.5, 0.0, 0.0, 0.5*math.sqrt(3) ],
|
|
[-0.5, 0.0, 0.0, 0.5*math.sqrt(3) ],
|
|
]
|
|
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 [quaternion*Quaternion(q) for q in symQuats]
|
|
|
|
|
|
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
|
|
R = abs(R) # fundamental zone in Rodrigues space is point symmetric around origin
|
|
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 and self.inFZ(R)
|
|
|
|
elif self.lattice == 'hexagonal':
|
|
return R[0] >= math.sqrt(3)*(R[1]+epsilon) and R[1] >= epsilon and R[2] >= epsilon and self.inFZ(R)
|
|
|
|
elif self.lattice == 'tetragonal':
|
|
return R[0] >= R[1]+epsilon and R[1] >= epsilon and R[2] >= epsilon and self.inFZ(R)
|
|
|
|
elif self.lattice == 'orthorhombic':
|
|
return R[0] >= epsilon and R[1] >= epsilon and R[2] >= epsilon and self.inFZ(R)
|
|
|
|
else:
|
|
return True
|
|
|
|
|
|
def inSST(self,vector,color = False):
|
|
'''
|
|
Check whether given vector falls into standard stereographic triangle of own symmetry.
|
|
Return inverse pole figure color if requested.
|
|
'''
|
|
# basis = {'cubic' : numpy.linalg.inv(numpy.array([[0.,0.,1.], # direction of red
|
|
# [1.,0.,1.]/numpy.sqrt(2.), # direction of green
|
|
# [1.,1.,1.]/numpy.sqrt(3.)]).transpose()), # direction of blue
|
|
# 'hexagonal' : numpy.linalg.inv(numpy.array([[0.,0.,1.], # direction of red
|
|
# [1.,0.,0.], # direction of green
|
|
# [numpy.sqrt(3.),1.,0.]/numpy.sqrt(4.)]).transpose()), # direction of blue
|
|
# 'tetragonal' : numpy.linalg.inv(numpy.array([[0.,0.,1.], # direction of red
|
|
# [1.,0.,0.], # direction of green
|
|
# [1.,1.,0.]/numpy.sqrt(2.)]).transpose()), # direction of blue
|
|
# 'orthorhombic' : numpy.linalg.inv(numpy.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 = numpy.array([ [-1. , 0. , 1. ],
|
|
[ numpy.sqrt(2.), -numpy.sqrt(2.), 0. ],
|
|
[ 0. , numpy.sqrt(3.), 0. ] ])
|
|
elif self.lattice == 'hexagonal':
|
|
basis = numpy.array([ [ 0. , 0. , 1. ],
|
|
[ 1. , -numpy.sqrt(3.), 0. ],
|
|
[ 0. , 2. , 0. ] ])
|
|
elif self.lattice == 'tetragonal':
|
|
basis = numpy.array([ [ 0. , 0. , 1. ],
|
|
[ 1. , -1. , 0. ],
|
|
[ 0. , numpy.sqrt(2.), 0. ] ])
|
|
elif self.lattice == 'orthorhombic':
|
|
basis = numpy.array([ [ 0., 0., 1.],
|
|
[ 1., 0., 0.],
|
|
[ 0., 1., 0.] ])
|
|
else:
|
|
basis = None
|
|
|
|
if basis == None:
|
|
theComponents = -numpy.ones(3,'d')
|
|
else:
|
|
theComponents = numpy.dot(basis,numpy.array([vector[0],vector[1],abs(vector[2])]))
|
|
|
|
inSST = numpy.all(theComponents >= 0.0)
|
|
|
|
if color: # have to return color array
|
|
if inSST:
|
|
rgb = numpy.power(theComponents/numpy.linalg.norm(theComponents),0.5) # smoothen color ramps
|
|
rgb = numpy.minimum(numpy.ones(3,'d'),rgb) # limit to maximum intensity
|
|
rgb /= max(rgb) # normalize to (HS)V = 1
|
|
else:
|
|
rgb = numpy.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,
|
|
symmetry = None,
|
|
):
|
|
if random: # produce random orientation
|
|
self.quaternion = Quaternion.fromRandom()
|
|
elif isinstance(Eulers, numpy.ndarray) and Eulers.shape == (3,): # based on given Euler angles
|
|
self.quaternion = Quaternion.fromEulers(Eulers,'bunge')
|
|
elif isinstance(matrix, numpy.ndarray) and matrix.shape == (3,3): # based on given rotation matrix
|
|
self.quaternion = Quaternion.fromMatrix(matrix)
|
|
elif isinstance(angleAxis, numpy.ndarray) and angleAxis.shape == (4,): # based on given angle and rotation axis
|
|
self.quaternion = Quaternion.fromAngleAxis(angleAxis[0],angleAxis[1:4])
|
|
elif isinstance(Rodrigues, numpy.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, numpy.ndarray) and quaternion.shape == (4,): # based on given quaternion
|
|
self.quaternion = Quaternion(quaternion).homomorphed()
|
|
|
|
self.symmetry = Symmetry(symmetry)
|
|
|
|
def __copy__(self):
|
|
return self.__class__(quaternion=self.quaternion,symmetry=self.symmetry.lattice)
|
|
|
|
copy = __copy__
|
|
|
|
|
|
def __repr__(self):
|
|
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(lambda x:str(numpy.degrees(x)),self.asEulers('bunge'))) )
|
|
|
|
def asQuaternion(self):
|
|
return self.quaternion.asList()
|
|
|
|
|
|
def asEulers(self,type='bunge'):
|
|
return self.quaternion.asEulers(type)
|
|
|
|
|
|
def asRodrigues(self):
|
|
return self.quaternion.asRodrigues()
|
|
|
|
|
|
def asMatrix(self):
|
|
return self.quaternion.asMatrix()
|
|
|
|
|
|
def reduced(self):
|
|
'''
|
|
Transform orientation to fall into fundamental zone according to own (or given) 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):
|
|
'''
|
|
Disorientation between myself and given other orientation
|
|
(either reduced according to my own symmetry or given one)
|
|
'''
|
|
|
|
lowerSymmetry = min(self.symmetry,other.symmetry)
|
|
breaker = False
|
|
|
|
for me in self.symmetry.equivalentQuaternions(self.quaternion):
|
|
me.conjugate()
|
|
for they in other.symmetry.equivalentQuaternions(other.quaternion):
|
|
# theQ = me * they
|
|
theQ = they * me
|
|
# if theQ.x < 0.0 or theQ.y < 0.0 or theQ.z < 0.0: theQ.conjugate() # speed up scanning since minimum angle is usually found for positive x,y,z
|
|
breaker = lowerSymmetry.inDisorientationSST(theQ.asRodrigues()) #\
|
|
# or lowerSymmetry.inDisorientationSST(theQ.conjugated().asRodrigues())
|
|
if breaker: break
|
|
if breaker: break
|
|
|
|
return Orientation(quaternion=theQ,symmetry=self.symmetry.lattice) #, me.conjugated(), they
|
|
|
|
|
|
def IPFcolor(self,axis):
|
|
'''
|
|
TSL color of inverse pole figure for given axis
|
|
'''
|
|
|
|
color = numpy.zeros(3,'d')
|
|
|
|
for i,q in enumerate(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
|