Logo Search packages:      
Sourcecode: python-scientific version File versions  Download package

Transformation.py

# This module defines classes that represent coordinate translations,
# rotations, and combinations of translation and rotation.
#
# Written by: Konrad Hinsen <hinsen@llb.saclay.cea.fr>
# Contributions from Pierre Legrand <pierre.legrand@synchrotron-soleil.fr>
# Last revision: 2004-10-19
# 

import TensorModule, VectorModule
import Numeric
from math import atan2

#
# Abstract base class
#
00016 class Transformation:

    """Linear coordinate transformation.

    Transformation objects represent linear coordinate transformations
    in a 3D space. They can be applied to vectors, returning another vector.
    If 't' is a transformation and 'v' is a vector, 't(v)' returns
    the transformed vector.

    Transformations support composition: if 't1' and 't2' are transformation
    objects, 't1*t2' is another transformation object which corresponds
    to applying t1 *after* t2.

    This class is an abstract base class. Instances can only be created
    of concrete subclasses, i.e. translations or rotations.
    """

    def rotation(self):
        "Returns the rotational component."
        pass

00037     def translation(self):
        """Returns the translational component. In the case of a mixed
        rotation/translation, this translation is executed
        *after* the rotation."""
        pass

    def inverse(self):
        "Returns the inverse transformation."
        pass

00047     def screwMotion(self):
        """Returns the four parameters '(reference, direction, angle,
        distance)' of a screw-like motion that is equivalent to the
        transformation. The screw motion consists of a displacement
        of 'distance' (a float) along 'direction' (a normalized vector)
        plus a rotation of 'angle' radians around an axis pointing along
        'direction' and passing through the point 'reference' (a vector).
        """
        pass
#
# Pure translation
#
00059 class Translation(Transformation):

    """Translational transformation.

    This is a subclass of Transformation.

    Constructor: Translation(|vector|), where |vector| is the displacement
    vector.
    """

    def __init__(self, vector):
      self.vector = vector

    is_translation = 1

    def __mul__(self, other):
      if hasattr(other, 'is_translation'):
          return Translation(self.vector + other.vector)
      elif hasattr(other, 'is_rotation'):
          return RotationTranslation(other.tensor, self.vector)
      elif hasattr(other, 'is_rotation_translation'):
          return RotationTranslation(other.tensor, other.vector+self.vector)
      else:
          raise ValueError, 'incompatible object'

    def __call__(self, vector):
      return self.vector + vector

    def displacement(self):
        "Returns the displacement vector."
      return self.vector

    def rotation(self):
      return Rotation(VectorModule.ez, 0.)

00094     def translation(self):
      return self

    def inverse(self):
      return Translation(-self.vector)

00100     def screwMotion(self):
      l = self.vector.length()
      if l == 0.:
          return VectorModule.Vector(0.,0.,0.), \
                   VectorModule.Vector(0.,0.,1.), 0., 0.
      else:
          return VectorModule.Vector(0.,0.,0.), self.vector/l, 0., l

#
# Pure rotation
#
00111 class Rotation(Transformation):

    """Rotational transformation.

    This is a subclass of Transformation.

    Constructor:

    - Rotation(|tensor|), where |tensor| is a tensor object containing
      the rotation matrix.

    - Rotation(|axis|, |angle|), where |axis| is a vector and |angle|
      a number (the angle in radians).
    """

    def __init__(self, *args):
      if len(args) == 1:
          self.tensor = args[0]
          if not TensorModule.isTensor(self.tensor):
            self.tensor = TensorModule.Tensor(self.tensor)
      elif len(args) == 2:
          axis, angle = args
          axis = axis.normal()
          projector = axis.dyadicProduct(axis)
          self.tensor = projector - \
                    Numeric.sin(angle)*TensorModule.epsilon*axis + \
                    Numeric.cos(angle)*(TensorModule.delta-projector)
      else:
          raise TypeError, 'one or two arguments required'

    is_rotation = 1

    def __mul__(self, other):
      if hasattr(other, 'is_rotation'):
          return Rotation(self.tensor.dot(other.tensor))
      elif hasattr(other, 'is_translation'):
          return RotationTranslation(self.tensor, self.tensor*other.vector)
      elif hasattr(other, 'is_rotation_translation'):
          return RotationTranslation(self.tensor.dot(other.tensor),
                               self.tensor*other.vector)
      else:
          raise ValueError, 'incompatible object'

    def __call__(self,other):
        if hasattr(other,'is_vector'):
           return self.tensor*other
        elif hasattr(other, 'is_tensor') and other.rank == 2:
           _rinv=self.tensor.inverse()
           return _rinv.dot(other.dot(self.tensor))
        elif hasattr(other, 'is_tensor') and other.rank == 1:
           return self.tensor.dot(other)
        else:
            raise ValueError, 'incompatible object'

00165     def axisAndAngle(self):
        """Returns the axis (a normalized vector) and angle (a float,
        in radians). The angle is in the interval (-pi, pi]"""
      as = -self.tensor.asymmetricalPart()
      axis = VectorModule.Vector(as[1,2], as[2,0], as[0,1])
      sine = axis.length()
      if abs(sine) > 1.e-10:
          axis = axis/sine
          projector = axis.dyadicProduct(axis)
          cosine = (self.tensor-projector).trace()/(3.-axis*axis)
          angle = angleFromSineAndCosine(sine, cosine)
      else:
          t = 0.5*(self.tensor+TensorModule.delta)
          i = Numeric.argmax(t.diagonal().array)
          axis = (t[i]/Numeric.sqrt(t[i,i])).asVector()
          angle = 0.
          if t.trace() < 2.:
            angle = Numeric.pi
      return axis, angle

00185     def threeAngles(self, e1, e2, e3, tolerance=1e-7):
        """Find three angles a1, a2, a3 such that
        Rotation(a1*|e1|)*Rotation(a2*|e2|)*Rotation(a3*|e3|)
        is equal to the rotation object. |e1|, |e2|, and
        |e3| are non-zero vectors. The result is a list of the two
        possible solutions."""

        # Written by Pierre Legrand (pierre.legrand@synchrotron-soleil.fr)
        #
        # Basically this is a reimplementation of the David
        # Thomas's algorithm [1] described by Gerard Bricogne in [2]:
        #
        # [1] "Modern Equations of Diffractometry. Goniometry." D.J. Thomas
        # Acta Cryst. (1990) A46 Page 321-343.
        #
        # [2] "The ECC Cooperative Programming Workshop on Position-Sensitive
        # Detector Software." G. Bricogne,
        # Computational aspect of Protein Crystal Data Analysis,
        # Proceedings of the Daresbury Study Weekend (23-24/01/1987)
        # Page 122-126

        e1 = e1.normal()
        e2 = e2.normal()
        e3 = e3.normal()

        # We are searching for the three angles a1, a2, a3
        # If 2 consecutive axes are parallel: decomposition is not meaningful
        if (e1.cross(e2)).length() < tolerance or \
           (e2.cross(e3)).length() < tolerance :
            raise ValueError, 'Consecutive parallel axes. Too many solutions'
        w = self(e3)
        
        # Solve the equation : _a.cosx + _b.sinx = _c
        _a = e1*e3 - (e1*e2)*(e2*e3)
        _b = e1*(e2.cross(e3))
        _c = e1*w - (e1*e2)*(e2*e3)
        _norm = (_a**2 + _b**2)**0.5
        
        # Checking for possible errors in initial Rot matrix
        if _norm == 0:
            raise ValueError, 'FAILURE 1, norm = 0'
        if abs(_c/_norm) > 1+tolerance:
            raise ValueError, 'FAILURE 2' + \
                 'malformed rotation Tensor (non orthogonal?) %.8f' \
                 % (_c/_norm)
        #if _c/_norm > 1: raise ValueError, 'Step1: No solution'
        _th = angleFromSineAndCosine(_b/_norm, _a/_norm)
        _xmth = Numeric.arccos(_c/_norm)

        # a2a and a2b are the two possible solutions to the equation.
        a2a = mod_angle((_th + _xmth), 2*Numeric.pi)
        a2b = mod_angle((_th - _xmth), 2*Numeric.pi)
        
        solutions = []
        # for each solution, find the two other angles (a1, a3).
        for a2 in (a2a, a2b):
            R2 = Rotation(e2, a2)
            v =  R2(e3)
            v1 = v - (v*e1)*e1
            w1 = w - (w*e1)*e1
            norm = ((v1*v1)*(w1*w1))**0.5
            if norm == 0: 
                # in that case rotation 1 and 3 are about the same axis
                # so any solution for rotation 1 is OK
                a1 = 0.
            else:
                cosa1 = (v1*w1)/norm
                sina1 = v1*(w1.cross(e1))/norm
                a1 = mod_angle(angleFromSineAndCosine(sina1, cosa1),
                               2*Numeric.pi)
                
            R3 = Rotation(e2, -1*a2)*Rotation(e1, -1*a1)*self
            # u = normalized test vector perpendicular to e3
            # if e2 and e3 are // we have an exception before.
            # if we take u = e1^e3 then it will not work for
            # Euler and Kappa axes.
            u = (e2.cross(e3)).normal()
            cosa3 = u*R3(u)
            sina3 = u*(R3(u).cross(e3))
            a3 =  mod_angle(angleFromSineAndCosine(sina3, cosa3),
                            2*Numeric.pi)
            
            solutions.append(Numeric.array([a1, a2, a3]))
            
        # Gives the closest solution to 0,0,0 first
        if Numeric.add.reduce(solutions[0]**2) > \
               Numeric.add.reduce(solutions[1]**2):
            solutions = [solutions[1], solutions[0]]
        return solutions

    def asQuaternion(self):
        "Returns a quaternion representing the same rotation."
        from Quaternion import Quaternion
        axis, angle = self.axisAndAngle()
        sin_angle_2 = Numeric.sin(0.5*angle)
        cos_angle_2 = Numeric.cos(0.5*angle)
        return Quaternion(cos_angle_2, sin_angle_2*axis[0],
                          sin_angle_2*axis[1], sin_angle_2*axis[2])

    def rotation(self):
      return self

00287     def translation(self):
      return Translation(VectorModule.Vector(0.,0.,0.))

    def inverse(self):
      return Rotation(self.tensor.transpose())

00293     def screwMotion(self):
      axis, angle = self.axisAndAngle()
      return VectorModule.Vector(0., 0., 0.), axis, angle, 0.

#
# Combined translation and rotation
#
00300 class RotationTranslation(Transformation):

    """Combined translational and rotational transformation.

    This is a subclass of Transformation.

    Objects of this class are not created directly, but can be the
    result of a composition of rotations and translations.
    """

    def __init__(self, tensor, vector):
      self.tensor = tensor
      self.vector = vector

    is_rotation_translation = 1

    def __mul__(self, other):
      if hasattr(other, 'is_rotation'):
          return RotationTranslation(self.tensor.dot(other.tensor),
                               self.vector)
      elif hasattr(other, 'is_translation'):
          return RotationTranslation(self.tensor,
                               self.tensor*other.vector+self.vector)
      elif hasattr(other, 'is_rotation_translation'):
          return RotationTranslation(self.tensor.dot(other.tensor),
                               self.tensor*other.vector+self.vector)
      else:
          raise ValueError, 'incompatible object'

    def __call__(self, vector):
      return self.tensor*vector + self.vector

    def rotation(self):
      return Rotation(self.tensor)

00335     def translation(self):
      return Translation(self.vector)

    def inverse(self):
      return Rotation(self.tensor.transpose())*Translation(-self.vector)

00341     def screwMotion(self):
      import LinearAlgebra
      axis, angle = self.rotation().axisAndAngle()
      d = self.vector*axis
      x = d*axis-self.vector
      r0 = Numeric.dot(LinearAlgebra.generalized_inverse(
                          self.tensor.array-Numeric.identity(3)), x.array)
      return VectorModule.Vector(r0), axis, angle, d

# Utility functions

def angleFromSineAndCosine(y, x):
    return atan2(y, x)

def mod_angle(angle, mod):
    return (angle + mod/2.) % mod - mod/2


# Test code

if __name__ == '__main__':

    t = Translation(VectorModule.Vector(1,0,0))
    r = Rotation(VectorModule.ex+VectorModule.ey, Numeric.pi)
    q = r.asQuaternion()
    angles = r.threeAngles(VectorModule.Vector(1., 0., 0.),
                           VectorModule.Vector(0., 1., 0.),
                           VectorModule.Vector(0., 0., 1.))
    print angles

Generated by  Doxygen 1.6.0   Back to index