From 6c3b5f17eb6d05a8637f18e454fa03117b5b7fd0 Mon Sep 17 00:00:00 2001 From: Philip Eisenlohr Date: Thu, 6 Sep 2018 14:19:09 -0400 Subject: [PATCH] updated orientation/quaternion class to follow Rowenhorst_etal2015 --- lib/damask/orientation.py | 160 +++++++++++++------------------------- 1 file changed, 53 insertions(+), 107 deletions(-) diff --git a/lib/damask/orientation.py b/lib/damask/orientation.py index 7d97d2c81..17e5e3e58 100644 --- a/lib/damask/orientation.py +++ b/lib/damask/orientation.py @@ -30,12 +30,19 @@ class Quaternion: """ Orientation represented as unit quaternion. - All methods and naming conventions based on http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions. + All methods and naming conventions based on Rowenhorst_etal2015 + Convention 1: coordinate frames are right-handed + Convention 2: a rotation angle ω is taken to be positive for a counterclockwise rotation + when viewing from the end point of the rotation axis unit vector towards the origin + Convention 3: rotations will be interpreted in the passive sense + Convention 4: Euler angle triplets are implemented using the Bunge convention, + with the angular ranges as [0, 2π],[0, π],[0, 2π] + Convention 5: the rotation angle ω is limited to the interval [0, π] w is the real part, (x, y, z) are the imaginary parts. - Representation of rotation is in ACTIVE form! - (Derived directly or through angleAxis, Euler angles, or active matrix) - Vector "a" (defined in coordinate system "A") is actively rotated to new coordinates "b". + + Vector "a" (defined in coordinate system "A") is passively rotated + resulting in new coordinates "b" when expressed in system "B". b = Q * a b = np.dot(Q.asMatrix(),a) """ @@ -309,10 +316,12 @@ class Quaternion: return np.outer([i for i in self],[i for i in self]) def asMatrix(self): - return np.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)], - [ 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)], - [ 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)]]) + qbarhalf = 0.5*(self.w**2 - self.x**2 - self.y**2 - self.z**2) + return 2.0*np.array( + [[ qbarhalf + self.x**2 , self.x*self.y - self.w*self.z, self.x*self.z + self.w*self.y], + [ self.x*self.y + self.w*self.z, qbarhalf + self.y**2 , self.y*self.z - self.w*self.x], + [ self.x*self.z - self.w*self.y, self.y*self.z + self.w*self.x, qbarhalf + self.z**2 ], + ]) def asAngleAxis(self, degrees = False): @@ -335,52 +344,28 @@ class Quaternion: return np.inf*np.ones(3) if self.w == 0.0 else np.array([self.x, self.y, self.z])/self.w def asEulers(self, - type = "bunge", degrees = False, - standardRange = False): + ): """ Orientation as Bunge-Euler angles. - Conversion of ACTIVE rotation to Euler angles taken from: - Melcher, A.; Unser, A.; Reichhardt, M.; Nestler, B.; Poetschke, M.; Selzer, M. - Conversion of EBSD data by a quaternion based algorithm to be used for grain structure simulations - Technische Mechanik 30 (2010) pp 401--413. """ - angles = [0.0,0.0,0.0] - if type.lower() == 'bunge' or type.lower() == 'zxz': - if abs(self.x) < 1e-4 and abs(self.y) < 1e-4: - x = self.w**2 - self.z**2 - y = 2.*self.w*self.z - angles[0] = math.atan2(y,x) - elif abs(self.w) < 1e-4 and abs(self.z) < 1e-4: - x = self.x**2 - self.y**2 - y = 2.*self.x*self.y - angles[0] = math.atan2(y,x) - angles[1] = math.pi - else: - chi = math.sqrt((self.w**2 + self.z**2)*(self.x**2 + self.y**2)) + q03 = self.w**2+self.z**2 + q12 = self.x**2+self.y**2 + chi = np.sqrt(q03*q12) + + if abs(chi) < 1e-10 and abs(q12) < 1e-10: + eulers = np.array([math.atan2(-2*self.w*self.z,self.w**2-self.z**2),0,0]) + elif abs(chi) < 1e-10 and abs(q03) < 1e-10: + eulers = np.array([math.atan2( 2*self.x*self.y,self.x**2-self.y**2),np.pi,0]) + else: + eulers = np.array([math.atan2((self.x*self.z-self.w*self.y)/chi,(-self.w*self.x-self.y*self.z)/chi), + math.atan2(2*chi,q03-q12), + math.atan2((self.w*self.y+self.x*self.z)/chi,( self.y*self.z-self.w*self.x)/chi), + ]) - x = (self.w * self.x - self.y * self.z)/2./chi - y = (self.w * self.y + self.x * self.z)/2./chi - angles[0] = math.atan2(y,x) - - x = self.w**2 + self.z**2 - (self.x**2 + self.y**2) - y = 2.*chi - angles[1] = math.atan2(y,x) - - x = (self.w * self.x + self.y * self.z)/2./chi - y = (self.z * self.x - self.y * self.w)/2./chi - angles[2] = math.atan2(y,x) - - if standardRange: - angles[0] %= 2*math.pi - if angles[1] < 0.0: - angles[1] += math.pi - angles[2] *= -1.0 - angles[2] %= 2*math.pi - - return np.degrees(angles) if degrees else angles + return np.degrees(eulers) if degrees else eulers # # Static constructors @@ -408,7 +393,7 @@ class Quaternion: halfangle = math.atan(np.linalg.norm(rodrigues)) c = math.cos(halfangle) w = c - x,y,z = c*rodrigues + x,y,z = rodrigues/c return cls([w,x,y,z]) @@ -431,24 +416,19 @@ class Quaternion: @classmethod def fromEulers(cls, eulers, - type = 'Bunge', degrees = False): if not isinstance(eulers, np.ndarray): eulers = np.array(eulers,dtype='d') eulers = np.radians(eulers) if degrees else eulers - c = np.cos(0.5 * eulers) - s = np.sin(0.5 * eulers) + sigma = 0.5*(eulers[0]+eulers[2]) + delta = 0.5*(eulers[0]-eulers[2]) + c = np.cos(0.5*eulers[1]) + s = np.sin(0.5*eulers[1]) - if type.lower() == 'bunge' or type.lower() == 'zxz': - w = c[0] * c[1] * c[2] - s[0] * c[1] * s[2] - x = c[0] * s[1] * c[2] + s[0] * s[1] * s[2] - y = - c[0] * s[1] * s[2] + s[0] * s[1] * c[2] - z = c[0] * c[1] * s[2] + s[0] * c[1] * c[2] - else: - w = c[0] * c[1] * c[2] - s[0] * s[1] * s[2] - x = s[0] * s[1] * c[2] + c[0] * c[1] * s[2] - y = s[0] * c[1] * c[2] + c[0] * s[1] * s[2] - z = c[0] * s[1] * c[2] - s[0] * c[1] * s[2] + w = c * np.cos(sigma) + x = -s * np.cos(delta) + y = -s * np.sin(delta) + z = -c * np.sin(sigma) return cls([w,x,y,z]) @@ -460,49 +440,16 @@ class Quaternion: if m.shape != (3,3) and np.prod(m.shape) == 9: m = m.reshape(3,3) - tr = np.trace(m) - if tr > 1e-8: - s = math.sqrt(tr + 1.0)*2.0 + w = 0.5*math.sqrt(1.+m[0,0]+m[1,1]+m[2,2]) + x = 0.5*math.sqrt(1.+m[0,0]-m[1,1]-m[2,2]) + y = 0.5*math.sqrt(1.-m[0,0]+m[1,1]-m[2,2]) + z = 0.5*math.sqrt(1.-m[0,0]-m[1,1]+m[2,2]) - return cls( - [ s*0.25, - (m[2,1] - m[1,2])/s, - (m[0,2] - m[2,0])/s, - (m[1,0] - m[0,1])/s, - ]) + x *= -1 if m[2,1] < m[1,2] else 1 + y *= -1 if m[0,2] < m[2,0] else 1 + z *= -1 if m[1,0] < m[0,1] else 1 - elif m[0,0] > m[1,1] and m[0,0] > m[2,2]: - t = m[0,0] - m[1,1] - m[2,2] + 1.0 - s = 2.0*math.sqrt(t) - - return cls( - [ (m[2,1] - m[1,2])/s, - s*0.25, - (m[0,1] + m[1,0])/s, - (m[2,0] + m[0,2])/s, - ]) - - elif m[1,1] > m[2,2]: - t = -m[0,0] + m[1,1] - m[2,2] + 1.0 - s = 2.0*math.sqrt(t) - - 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, - ]) + return cls( np.array([w,x,y,z])/math.sqrt(w**2 + x**2 + y**2 + z**2)) @classmethod @@ -829,7 +776,7 @@ class Orientation: 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) + self.quaternion = Quaternion.fromEulers(Eulers,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 @@ -855,16 +802,15 @@ class Orientation: 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))) ) + 'Bunge Eulers / deg: %s' % ('\t'.join(map(str,self.asEulers(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) + ): + return self.quaternion.asEulers(degrees) eulers = property(asEulers) def asRodrigues(self):