Merge pull request #161 from simpeg/CoordRotations

coordinate rotations

SimPEG/Utils/__init__.py

@@ -7,4 +7,4 @@ from ipythonutils import easyAnimate as animate
 from CounterUtils import *
 import ModelBuilder
 import SolverUtils
-
+from coordutils import *

SimPEG/Utils/coordutils.py

@@ -0,0 +1,62 @@
+import numpy as np
+from SimPEG.Utils import mkvc
+
+def rotationMatrixFromNormals(v0,v1,tol=1e-20):
+    """
+        Performs the minimum number of rotations to define a rotation from the direction indicated by the vector n0 to the direction indicated by n1.
+        The axis of rotation is n0 x n1
+        https://en.wikipedia.org/wiki/Rodrigues%27_rotation_formula
+
+        :param numpy.array v0: vector of length 3
+        :param numpy.array v1: vector of length 3
+        :param tol = 1e-20: tolerance. If the norm of the cross product between the two vectors is below this, no rotation is performed
+        :rtype: numpy.array, 3x3
+        :return: rotation matrix which rotates the frame so that n0 is aligned with n1
+
+    """
+
+    # ensure both n0, n1 are vectors of length 1
+    assert len(v0) == 3, "Length of n0 should be 3"
+    assert len(v1) == 3, "Length of n1 should be 3"
+
+    # ensure both are true normals
+    n0 = v0*1./np.linalg.norm(v0)
+    n1 = v1*1./np.linalg.norm(v1)
+
+    n0dotn1 = n0.dot(n1) 
+
+    # define the rotation axis, which is the cross product of the two vectors
+    rotAx = np.cross(n0,n1)
+
+    if np.linalg.norm(rotAx) < tol:
+        return np.eye(3,dtype=float)
+
+    rotAx *= 1./np.linalg.norm(rotAx)
+
+    cosT = n0dotn1/(np.linalg.norm(n0)*np.linalg.norm(n1))
+    sinT = np.sqrt(1.-n0dotn1**2)
+
+    ux = np.array([[0., -rotAx[2], rotAx[1]], [rotAx[2], 0., -rotAx[0]], [-rotAx[1], rotAx[0], 0.]],dtype=float)
+
+    return np.eye(3,dtype=float) + sinT*ux + (1.-cosT)*(ux.dot(ux))
+
+
+def rotatePointsFromNormals(XYZ,n0,n1,x0=np.r_[0.,0.,0.]):
+    """
+        rotates a grid so that the vector n0 is aligned with the vector n1
+
+        :param numpy.array n0: vector of length 3, should have norm 1
+        :param numpy.array n1: vector of length 3, should have norm 1
+        :param numpy.array x0: vector of length 3, point about which we perform the rotation 
+        :rtype: numpy.array, 3x3
+        :return: rotation matrix which rotates the frame so that n0 is aligned with n1
+    """
+
+    R = rotationMatrixFromNormals(n0, n1)
+
+    assert XYZ.shape[1] == 3, "Grid XYZ should be 3 wide"
+    assert len(x0) == 3, "x0 should have length 3"
+
+    X0 = np.ones([XYZ.shape[0],1])*mkvc(x0)
+
+    return (XYZ - X0).dot(R.T) + X0 # equivalent to (R*(XYZ - X0)).T + X0

tests/utils/test_coordutils.py

@@ -0,0 +1,45 @@
+import unittest, os
+import numpy as np
+from SimPEG import Utils
+
+tol = 1e-15
+
+class coorUtilsTest(unittest.TestCase):
+
+    def test_rotationMatrixFromNormals(self):
+        v0 = np.random.rand(3)
+        v0 *= 1./np.linalg.norm(v0)
+        v1 = np.random.rand(3)
+        v1 *= 1./np.linalg.norm(v1)
+        Rf = Utils.coordutils.rotationMatrixFromNormals(v0,v1)
+        Ri = Utils.coordutils.rotationMatrixFromNormals(v1,v0)
+
+        self.assertTrue(np.linalg.norm(Utils.mkvc(Rf.dot(v0) - v1)) < tol)
+        self.assertTrue(np.linalg.norm(Utils.mkvc(Ri.dot(v1) - v0)) < tol)
+
+    def test_rotatePointsFromNormals(self):
+        v0 = np.random.rand(3)
+        v0*= 1./np.linalg.norm(v0)
+        v1 = np.random.rand(3) 
+        v1*= 1./np.linalg.norm(v1)   
+
+        v2 = Utils.mkvc(Utils.coordutils.rotatePointsFromNormals(Utils.mkvc(v0,2).T,v0,v1)) 
+        
+        self.assertTrue(np.linalg.norm(v2-v1) < tol)
+
+    def test_rotateMatrixFromNormals(self):
+        n0 = np.random.rand(3)
+        n0*= 1./np.linalg.norm(n0)
+        n1 = np.random.rand(3) 
+        n1*= 1./np.linalg.norm(n1) 
+
+        scale =  np.random.rand(100,1)
+        XYZ0 = scale * n0
+        XYZ1 = scale * n1
+
+        XYZ2 = Utils.coordutils.rotatePointsFromNormals(XYZ0,n0,n1)
+        self.assertTrue(np.linalg.norm(Utils.mkvc(XYZ1) - Utils.mkvc(XYZ2))/np.linalg.norm(Utils.mkvc(XYZ1)) < tol)
+
+if __name__ == '__main__':
+    unittest.main()
+