src/math/Quaternion.hpp

/*
 * Copyright (c) 2005 The University of Notre Dame. All Rights Reserved.
 *
 * The University of Notre Dame grants you ("Licensee") a
 * non-exclusive, royalty free, license to use, modify and
 * redistribute this software in source and binary code form, provided
 * that the following conditions are met:
 *
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer.
 *
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in the
 *    documentation and/or other materials provided with the
 *    distribution.
 *
 * This software is provided "AS IS," without a warranty of any
 * kind. All express or implied conditions, representations and
 * warranties, including any implied warranty of merchantability,
 * fitness for a particular purpose or non-infringement, are hereby
 * excluded.  The University of Notre Dame and its licensors shall not
 * be liable for any damages suffered by licensee as a result of
 * using, modifying or distributing the software or its
 * derivatives. In no event will the University of Notre Dame or its
 * licensors be liable for any lost revenue, profit or data, or for
 * direct, indirect, special, consequential, incidental or punitive
 * damages, however caused and regardless of the theory of liability,
 * arising out of the use of or inability to use software, even if the
 * University of Notre Dame has been advised of the possibility of
 * such damages.
 *
 * SUPPORT OPEN SCIENCE!  If you use OpenMD or its source code in your
 * research, please cite the appropriate papers when you publish your
 * work.  Good starting points are:
 *                                                                      
 * [1]  Meineke, et al., J. Comp. Chem. 26, 252-271 (2005).             
 * [2]  Fennell & Gezelter, J. Chem. Phys. 124, 234104 (2006).          
 * [3]  Sun, Lin & Gezelter, J. Chem. Phys. 128, 24107 (2008).          
 * [4]  Vardeman & Gezelter, in progress (2009).                        
 */
 
/**
 * @file Quaternion.hpp
 * @author Teng Lin
 * @date 10/11/2004
 * @version 1.0
 */

#ifndef MATH_QUATERNION_HPP
#define MATH_QUATERNION_HPP

#include "math/Vector3.hpp"
#include "math/SquareMatrix.hpp"
#define ISZERO(a,eps) ( (a)>-(eps) && (a)<(eps) )
const RealType tiny=1.0e-6;     

namespace OpenMD{

  /**
   * @class Quaternion Quaternion.hpp "math/Quaternion.hpp"
   * Quaternion is a sort of a higher-level complex number.
   * It is defined as Q = w + x*i + y*j + z*k,
   * where w, x, y, and z are numbers of type T (e.g. RealType), and
   * i*i = -1; j*j = -1; k*k = -1;
   * i*j = k; j*k = i; k*i = j;
   */
  template<typename Real>
  class Quaternion : public Vector<Real, 4> {

  public:
    Quaternion() : Vector<Real, 4>() {}

    /** Constructs and initializes a Quaternion from w, x, y, z values */     
    Quaternion(Real w, Real x, Real y, Real z) {
      this->data_[0] = w;
      this->data_[1] = x;
      this->data_[2] = y;
      this->data_[3] = z;                
    }
            
    /** Constructs and initializes a Quaternion from a  Vector<Real,4> */     
    Quaternion(const Vector<Real,4>& v) 
      : Vector<Real, 4>(v){
    }

    /** copy assignment */
    Quaternion& operator =(const Vector<Real, 4>& v){
      if (this == & v)
        return *this;
      
      Vector<Real, 4>::operator=(v);
      
      return *this;
    }
    
    /**
     * Returns the value of the first element of this quaternion.
     * @return the value of the first element of this quaternion
     */
    Real w() const {
      return this->data_[0];
    }

    /**
     * Returns the reference of the first element of this quaternion.
     * @return the reference of the first element of this quaternion
     */
    Real& w() {
      return this->data_[0];    
    }

    /**
     * Returns the value of the first element of this quaternion.
     * @return the value of the first element of this quaternion
     */
    Real x() const {
      return this->data_[1];
    }

    /**
     * Returns the reference of the second element of this quaternion.
     * @return the reference of the second element of this quaternion
     */
    Real& x() {
      return this->data_[1];    
    }

    /**
     * Returns the value of the thirf element of this quaternion.
     * @return the value of the third element of this quaternion
     */
    Real y() const {
      return this->data_[2];
    }

    /**
     * Returns the reference of the third element of this quaternion.
     * @return the reference of the third element of this quaternion
     */           
    Real& y() {
      return this->data_[2];    
    }

    /**
     * Returns the value of the fourth element of this quaternion.
     * @return the value of the fourth element of this quaternion
     */
    Real z() const {
      return this->data_[3];
    }
    /**
     * Returns the reference of the fourth element of this quaternion.
     * @return the reference of the fourth element of this quaternion
     */
    Real& z() {
      return this->data_[3];    
    }

    /**
     * Tests if this quaternion is equal to other quaternion
     * @return true if equal, otherwise return false
     * @param q quaternion to be compared
     */
    inline bool operator ==(const Quaternion<Real>& q) {

      for (unsigned int i = 0; i < 4; i ++) {
        if (!equal(this->data_[i], q[i])) {
          return false;
        }
      }
                
      return true;
    }
            
    /**
     * Returns the inverse of this quaternion
     * @return inverse
     * @note since quaternion is a complex number, the inverse of quaternion
     * q = w + xi + yj+ zk is inv_q = (w -xi - yj - zk)/(|q|^2)
     */
    Quaternion<Real> inverse() {
      Quaternion<Real> q;
      Real d = this->lengthSquare();
                
      q.w() = w() / d;
      q.x() = -x() / d;
      q.y() = -y() / d;
      q.z() = -z() / d;
                
      return q;
    }

    /**
     * Sets the value to the multiplication of itself and another quaternion
     * @param q the other quaternion
     */
    void mul(const Quaternion<Real>& q) {
      Quaternion<Real> tmp(*this);

      this->data_[0] = (tmp[0]*q[0]) -(tmp[1]*q[1]) - (tmp[2]*q[2]) - (tmp[3]*q[3]);
      this->data_[1] = (tmp[0]*q[1]) + (tmp[1]*q[0]) + (tmp[2]*q[3]) - (tmp[3]*q[2]);
      this->data_[2] = (tmp[0]*q[2]) + (tmp[2]*q[0]) + (tmp[3]*q[1]) - (tmp[1]*q[3]);
      this->data_[3] = (tmp[0]*q[3]) + (tmp[3]*q[0]) + (tmp[1]*q[2]) - (tmp[2]*q[1]);                
    }

    void mul(const Real& s) {
      this->data_[0] *= s;
      this->data_[1] *= s;
      this->data_[2] *= s;
      this->data_[3] *= s;
    }

    /** Set the value of this quaternion to the division of itself by another quaternion */
    void div(Quaternion<Real>& q) {
      mul(q.inverse());
    }

    void div(const Real& s) {
      this->data_[0] /= s;
      this->data_[1] /= s;
      this->data_[2] /= s;
      this->data_[3] /= s;
    }
            
    Quaternion<Real>& operator *=(const Quaternion<Real>& q) {
      mul(q);
      return *this;
    }

    Quaternion<Real>& operator *=(const Real& s) {
      mul(s);
      return *this;
    }
            
    Quaternion<Real>& operator /=(Quaternion<Real>& q) {                
      *this *= q.inverse();
      return *this;
    }

    Quaternion<Real>& operator /=(const Real& s) {
      div(s);
      return *this;
    }            
    /**
     * Returns the conjugate quaternion of this quaternion
     * @return the conjugate quaternion of this quaternion
     */
    Quaternion<Real> conjugate() const {
      return Quaternion<Real>(w(), -x(), -y(), -z());            
    }


    /**
       return rotation angle from -PI to PI 
    */
    inline Real get_rotation_angle() const{
      if( w < (Real)0.0 )
        return 2.0*atan2(-sqrt( x()*x() + y()*y() + z()*z() ), -w() );
      else
        return 2.0*atan2( sqrt( x()*x() + y()*y() + z()*z() ),  w() );
    }

    /**
       create a unit quaternion from axis angle representation
    */
    Quaternion<Real> fromAxisAngle(const Vector3<Real>& axis, 
                                   const Real& angle){
      Vector3<Real> v(axis);
      v.normalize();
      Real half_angle = angle*0.5;
      Real sin_a = sin(half_angle);
      *this = Quaternion<Real>(cos(half_angle), 
                               v.x()*sin_a, 
                               v.y()*sin_a, 
                               v.z()*sin_a);
      return *this;
    }
    
    /**
       convert a quaternion to axis angle representation, 
       preserve the axis direction and angle from -PI to +PI
    */
    void toAxisAngle(Vector3<Real>& axis, Real& angle)const {
      Real vl = sqrt( x()*x() + y()*y() + z()*z() );
      if( vl > tiny ) {
        Real ivl = 1.0/vl;
        axis.x() = x() * ivl;
        axis.y() = y() * ivl;
        axis.z() = z() * ivl;

        if( w() < 0 )
          angle = 2.0*atan2(-vl, -w()); //-PI,0 
        else
          angle = 2.0*atan2( vl,  w()); //0,PI 
      } else {
        axis = Vector3<Real>(0.0,0.0,0.0);
        angle = 0.0;
      }
    }

    /**
       shortest arc quaternion rotate one vector to another by shortest path.
       create rotation from -> to, for any length vectors.
    */
    Quaternion<Real> fromShortestArc(const Vector3d& from, 
                                     const Vector3d& to ) {
      
      Vector3d c( cross(from,to) );
      *this = Quaternion<Real>(dot(from,to), 
                               c.x(), 
                               c.y(),
                               c.z());

      this->normalize();    // if "from" or "to" not unit, normalize quat
      w += 1.0f;            // reducing angle to halfangle
      if( w <= 1e-6 ) {     // angle close to PI
        if( ( from.z()*from.z() ) > ( from.x()*from.x() ) ) {
          this->data_[0] =  w;    
          this->data_[1] =  0.0;       //cross(from , Vector3d(1,0,0))
          this->data_[2] =  from.z();
          this->data_[3] = -from.y();
        } else {
          this->data_[0] =  w;
          this->data_[1] =  from.y();  //cross(from, Vector3d(0,0,1))
          this->data_[2] = -from.x();
          this->data_[3] =  0.0;
        }
      }
      this->normalize(); 
    }

    Real ComputeTwist(const Quaternion& q) {
      return  (Real)2.0 * atan2(q.z(), q.w());
    }

    void RemoveTwist(Quaternion& q) {
      Real t = ComputeTwist(q);
      Quaternion rt = fromAxisAngle(V3Z, t);
      
      q *= rt.inverse();
    }

    void getTwistSwingAxisAngle(Real& twistAngle, Real& swingAngle, 
                                Vector3<Real>& swingAxis) {
      
      twistAngle = (Real)2.0 * atan2(z(), w());
      Quaternion rt, rs;
      rt.fromAxisAngle(V3Z, twistAngle);
      rs = *this * rt.inverse();
      
      Real vl = sqrt( rs.x()*rs.x() + rs.y()*rs.y() + rs.z()*rs.z() );
      if( vl > tiny ) {
        Real ivl = 1.0 / vl;
        swingAxis.x() = rs.x() * ivl;
        swingAxis.y() = rs.y() * ivl;
        swingAxis.z() = rs.z() * ivl;

        if( rs.w() < 0.0 )
          swingAngle = 2.0*atan2(-vl, -rs.w()); //-PI,0 
        else
          swingAngle = 2.0*atan2( vl,  rs.w()); //0,PI 
      } else {
        swingAxis = Vector3<Real>(1.0,0.0,0.0);
        swingAngle = 0.0;
      }           
    }


    Vector3<Real> rotate(const Vector3<Real>& v) {

      Quaternion<Real> q(v.x() * w() + v.z() * y() - v.y() * z(),
                         v.y() * w() + v.x() * z() - v.z() * x(),
                         v.z() * w() + v.y() * x() - v.x() * y(),
                         v.x() * x() + v.y() * y() + v.z() * z());

      return Vector3<Real>(w()*q.x() + x()*q.w() + y()*q.z() - z()*q.y(),
                           w()*q.y() + y()*q.w() + z()*q.x() - x()*q.z(),
                           w()*q.z() + z()*q.w() + x()*q.y() - y()*q.x())*
        ( 1.0/this->lengthSquare() );      
    }   

    Quaternion<Real>& align (const Vector3<Real>& V1,
                             const Vector3<Real>& V2) {

      // If V1 and V2 are not parallel, the axis of rotation is the unit-length
      // vector U = Cross(V1,V2)/Length(Cross(V1,V2)).  The angle of rotation,
      // A, is the angle between V1 and V2.  The quaternion for the rotation is
      // q = cos(A/2) + sin(A/2)*(ux*i+uy*j+uz*k) where U = (ux,uy,uz).
      //
      // (1) Rather than extract A = acos(Dot(V1,V2)), multiply by 1/2, then
      //     compute sin(A/2) and cos(A/2), we reduce the computational costs
      //     by computing the bisector B = (V1+V2)/Length(V1+V2), so cos(A/2) =
      //     Dot(V1,B).
      //
      // (2) The rotation axis is U = Cross(V1,B)/Length(Cross(V1,B)), but
      //     Length(Cross(V1,B)) = Length(V1)*Length(B)*sin(A/2) = sin(A/2), in
      //     which case sin(A/2)*(ux*i+uy*j+uz*k) = (cx*i+cy*j+cz*k) where
      //     C = Cross(V1,B).
      //
      // If V1 = V2, then B = V1, cos(A/2) = 1, and U = (0,0,0).  If V1 = -V2,
      // then B = 0.  This can happen even if V1 is approximately -V2 using
      // floating point arithmetic, since Vector3::Normalize checks for
      // closeness to zero and returns the zero vector accordingly.  The test
      // for exactly zero is usually not recommend for floating point
      // arithmetic, but the implementation of Vector3::Normalize guarantees
      // the comparison is robust.  In this case, the A = pi and any axis
      // perpendicular to V1 may be used as the rotation axis.

      Vector3<Real> Bisector = V1 + V2;
      Bisector.normalize();

      Real CosHalfAngle = dot(V1,Bisector);

      this->data_[0] = CosHalfAngle;

      if (CosHalfAngle != (Real)0.0) {
        Vector3<Real> Cross = cross(V1, Bisector);
        this->data_[1] = Cross.x();
        this->data_[2] = Cross.y();
        this->data_[3] = Cross.z();
      } else {
        Real InvLength;
        if (fabs(V1[0]) >= fabs(V1[1])) {
          // V1.x or V1.z is the largest magnitude component
          InvLength = (Real)1.0/sqrt(V1[0]*V1[0] + V1[2]*V1[2]);

          this->data_[1] = -V1[2]*InvLength;
          this->data_[2] = (Real)0.0;
          this->data_[3] = +V1[0]*InvLength;
        } else {
          // V1.y or V1.z is the largest magnitude component
          InvLength = (Real)1.0 / sqrt(V1[1]*V1[1] + V1[2]*V1[2]);
          
          this->data_[1] = (Real)0.0;
          this->data_[2] = +V1[2]*InvLength;
          this->data_[3] = -V1[1]*InvLength;
        }
      }
      return *this;
    }

    void toTwistSwing ( Real& tw, Real& sx, Real& sy ) {
      
      // First test if the swing is in the singularity:

      if ( ISZERO(z(),tiny) && ISZERO(w(),tiny) ) { sx=sy=M_PI; tw=0; return; }

      // Decompose into twist-swing by solving the equation:
      //
      //                       Qtwist(t*2) * Qswing(s*2) = q
      //
      // note: (x,y) is the normalized swing axis (x*x+y*y=1)
      //
      //          ( Ct 0 0 St ) * ( Cs xSs ySs 0 ) = ( qw qx qy qz )
      //  ( CtCs  xSsCt-yStSs  xStSs+ySsCt  StCs ) = ( qw qx qy qz )      (1)
      // From (1): CtCs / StCs = qw/qz => Ct/St = qw/qz => tan(t) = qz/qw (2)
      //
      // The swing rotation/2 s comes from:
      //
      // From (1): (CtCs)^2 + (StCs)^2 = qw^2 + qz^2 =>  
      //                                       Cs = sqrt ( qw^2 + qz^2 ) (3)
      //
      // From (1): (xSsCt-yStSs)^2 + (xStSs+ySsCt)^2 = qx^2 + qy^2 => 
      //                                       Ss = sqrt ( qx^2 + qy^2 ) (4)
      // From (1):  |SsCt -StSs| |x| = |qx|
      //            |StSs +SsCt| |y|   |qy|                              (5)

      Real qw, qx, qy, qz;
      
      if ( w()<0 ) {
        qw=-w(); 
        qx=-x(); 
        qy=-y(); 
        qz=-z();
      } else {
        qw=w(); 
        qx=x(); 
        qy=y(); 
        qz=z();
      }
      
      Real t = atan2 ( qz, qw ); // from (2)
      Real s = atan2( sqrt(qx*qx+qy*qy), sqrt(qz*qz+qw*qw) ); // from (3)
                                                              // and (4)

      Real x=0.0, y=0.0, sins=sin(s);

      if ( !ISZERO(sins,tiny) ) {
        Real sint = sin(t);
        Real cost = cos(t);
        
        // by solving the linear system in (5):
        y = (-qx*sint + qy*cost)/sins;
        x = ( qx*cost + qy*sint)/sins;
      }

      tw = (Real)2.0*t;
      sx = (Real)2.0*x*s;
      sy = (Real)2.0*y*s;
    }

    void toSwingTwist(Real& sx, Real& sy, Real& tw ) {

      // Decompose q into swing-twist using a similar development as
      // in function toTwistSwing

      if ( ISZERO(z(),tiny) && ISZERO(w(),tiny) ) { sx=sy=M_PI; tw=0; }
      
      Real qw, qx, qy, qz;
      if ( w() < 0 ){
        qw=-w(); 
        qx=-x(); 
        qy=-y(); 
        qz=-z();
      } else {
        qw=w(); 
        qx=x(); 
        qy=y(); 
        qz=z(); 
      }

      // Get the twist t:
      Real t = 2.0 * atan2(qz,qw);
      
      Real bet = atan2( sqrt(qx*qx+qy*qy), sqrt(qz*qz+qw*qw) );
      Real gam = t/2.0;
      Real sinc = ISZERO(bet,tiny)? 1.0 : sin(bet)/bet;
      Real singam = sin(gam);
      Real cosgam = cos(gam);

      sx = Real( (2.0/sinc) * (cosgam*qx - singam*qy) );
      sy = Real( (2.0/sinc) * (singam*qx + cosgam*qy) );
      tw = Real( t );
    }
      
    
    /**
     * Returns the corresponding rotation matrix (3x3)
     * @return a 3x3 rotation matrix
     */
    SquareMatrix<Real, 3> toRotationMatrix3() {
      SquareMatrix<Real, 3> rotMat3;
      
      Real w2;
      Real x2;
      Real y2;
      Real z2;

      if (!this->isNormalized())
        this->normalize();
                
      w2 = w() * w();
      x2 = x() * x();
      y2 = y() * y();
      z2 = z() * z();

      rotMat3(0, 0) = w2 + x2 - y2 - z2;
      rotMat3(0, 1) = 2.0 * ( x() * y() + w() * z() );
      rotMat3(0, 2) = 2.0 * ( x() * z() - w() * y() );

      rotMat3(1, 0) = 2.0 * ( x() * y() - w() * z() );
      rotMat3(1, 1) = w2 - x2 + y2 - z2;
      rotMat3(1, 2) = 2.0 * ( y() * z() + w() * x() );

      rotMat3(2, 0) = 2.0 * ( x() * z() + w() * y() );
      rotMat3(2, 1) = 2.0 * ( y() * z() - w() * x() );
      rotMat3(2, 2) = w2 - x2 -y2 +z2;

      return rotMat3;
    }

  };//end Quaternion


    /**
     * Returns the vaule of scalar multiplication of this quaterion q (q * s). 
     * @return  the vaule of scalar multiplication of this vector
     * @param q the source quaternion
     * @param s the scalar value
     */
  template<typename Real, unsigned int Dim>                 
  Quaternion<Real> operator * ( const Quaternion<Real>& q, Real s) {       
    Quaternion<Real> result(q);
    result.mul(s);
    return result;           
  }
    
  /**
   * Returns the vaule of scalar multiplication of this quaterion q (q * s). 
   * @return  the vaule of scalar multiplication of this vector
   * @param s the scalar value
   * @param q the source quaternion
   */  
  template<typename Real, unsigned int Dim>
  Quaternion<Real> operator * ( const Real& s, const Quaternion<Real>& q ) {
    Quaternion<Real> result(q);
    result.mul(s);
    return result;           
  }    

  /**
   * Returns the multiplication of two quaternion
   * @return the multiplication of two quaternion
   * @param q1 the first quaternion
   * @param q2 the second quaternion
   */
  template<typename Real>
  inline Quaternion<Real> operator *(const Quaternion<Real>& q1, const Quaternion<Real>& q2) {
    Quaternion<Real> result(q1);
    result *= q2;
    return result;
  }

  /**
   * Returns the division of two quaternion
   * @param q1 divisor
   * @param q2 dividen
   */

  template<typename Real>
  inline Quaternion<Real> operator /( Quaternion<Real>& q1,  Quaternion<Real>& q2) {
    return q1 * q2.inverse();
  }

  /**
   * Returns the value of the division of a scalar by a quaternion
   * @return the value of the division of a scalar by a quaternion
   * @param s scalar
   * @param q quaternion
   * @note for a quaternion q, 1/q = q.inverse()
   */
  template<typename Real>
  Quaternion<Real> operator /(const Real& s,  Quaternion<Real>& q) {

    Quaternion<Real> x;
    x = q.inverse();
    x *= s;
    return x;
  }
    
  template <class T>
  inline bool operator==(const Quaternion<T>& lhs, const Quaternion<T>& rhs) {
    return equal(lhs[0] ,rhs[0]) && equal(lhs[1] , rhs[1]) && equal(lhs[2], rhs[2]) && equal(lhs[3], rhs[3]);
  }
    
  typedef Quaternion<RealType> Quat4d;
}
#endif //MATH_QUATERNION_HPP 
Revision:	1390
Committed:	Wed Nov 25 20:02:06 2009 UTC (15 years, 5 months ago) by gezelter
Original Path:	*trunk/src/math/Quaternion.hpp*
File size:	20487 byte(s)
Log Message:	Almost all of the changes necessary to create OpenMD out of our old project (OOPSE-4)
#	Content
1	/*
2	* Copyright (c) 2005 The University of Notre Dame. All Rights Reserved.
3	*
4	* The University of Notre Dame grants you ("Licensee") a
5	* non-exclusive, royalty free, license to use, modify and
6	* redistribute this software in source and binary code form, provided
7	* that the following conditions are met:
8	*
9	* 1. Redistributions of source code must retain the above copyright
10	* notice, this list of conditions and the following disclaimer.
11	*
12	* 2. Redistributions in binary form must reproduce the above copyright
13	* notice, this list of conditions and the following disclaimer in the
14	* documentation and/or other materials provided with the
15	* distribution.
16	*
17	* This software is provided "AS IS," without a warranty of any
18	* kind. All express or implied conditions, representations and
19	* warranties, including any implied warranty of merchantability,
20	* fitness for a particular purpose or non-infringement, are hereby
21	* excluded. The University of Notre Dame and its licensors shall not
22	* be liable for any damages suffered by licensee as a result of
23	* using, modifying or distributing the software or its
24	* derivatives. In no event will the University of Notre Dame or its
25	* licensors be liable for any lost revenue, profit or data, or for
26	* direct, indirect, special, consequential, incidental or punitive
27	* damages, however caused and regardless of the theory of liability,
28	* arising out of the use of or inability to use software, even if the
29	* University of Notre Dame has been advised of the possibility of
30	* such damages.
31	*
32	* SUPPORT OPEN SCIENCE! If you use OpenMD or its source code in your
33	* research, please cite the appropriate papers when you publish your
34	* work. Good starting points are:
35	*
36	* [1] Meineke, et al., J. Comp. Chem. 26, 252-271 (2005).
37	* [2] Fennell & Gezelter, J. Chem. Phys. 124, 234104 (2006).
38	* [3] Sun, Lin & Gezelter, J. Chem. Phys. 128, 24107 (2008).
39	* [4] Vardeman & Gezelter, in progress (2009).
40	*/
41
42	/**
43	* @file Quaternion.hpp
44	* @author Teng Lin
45	* @date 10/11/2004
46	* @version 1.0
47	*/
48
49	#ifndef MATH_QUATERNION_HPP
50	#define MATH_QUATERNION_HPP
51
52	#include "math/Vector3.hpp"
53	#include "math/SquareMatrix.hpp"
54	#define ISZERO(a,eps) ( (a)>-(eps) && (a)<(eps) )
55	const RealType tiny=1.0e-6;
56
57	namespace OpenMD{
58
59	/**
60	* @class Quaternion Quaternion.hpp "math/Quaternion.hpp"
61	* Quaternion is a sort of a higher-level complex number.
62	* It is defined as Q = w + xi + yj + z*k,
63	* where w, x, y, and z are numbers of type T (e.g. RealType), and
64	* ii = -1; jj = -1; k*k = -1;
65	* ij = k; jk = i; k*i = j;
66	*/
67	template<typename Real>
68	class Quaternion : public Vector<Real, 4> {
69
70	public:
71	Quaternion() : Vector<Real, 4>() {}
72
73	/** Constructs and initializes a Quaternion from w, x, y, z values */
74	Quaternion(Real w, Real x, Real y, Real z) {
75	this->data_[0] = w;
76	this->data_[1] = x;
77	this->data_[2] = y;
78	this->data_[3] = z;
79	}
80
81	/** Constructs and initializes a Quaternion from a Vector<Real,4> */
82	Quaternion(const Vector<Real,4>& v)
83	: Vector<Real, 4>(v){
84	}
85
86	/** copy assignment */
87	Quaternion& operator =(const Vector<Real, 4>& v){
88	if (this == & v)
89	return *this;
90
91	Vector<Real, 4>::operator=(v);
92
93	return *this;
94	}
95
96	/**
97	* Returns the value of the first element of this quaternion.
98	* @return the value of the first element of this quaternion
99	*/
100	Real w() const {
101	return this->data_[0];
102	}
103
104	/**
105	* Returns the reference of the first element of this quaternion.
106	* @return the reference of the first element of this quaternion
107	*/
108	Real& w() {
109	return this->data_[0];
110	}
111
112	/**
113	* Returns the value of the first element of this quaternion.
114	* @return the value of the first element of this quaternion
115	*/
116	Real x() const {
117	return this->data_[1];
118	}
119
120	/**
121	* Returns the reference of the second element of this quaternion.
122	* @return the reference of the second element of this quaternion
123	*/
124	Real& x() {
125	return this->data_[1];
126	}
127
128	/**
129	* Returns the value of the thirf element of this quaternion.
130	* @return the value of the third element of this quaternion
131	*/
132	Real y() const {
133	return this->data_[2];
134	}
135
136	/**
137	* Returns the reference of the third element of this quaternion.
138	* @return the reference of the third element of this quaternion
139	*/
140	Real& y() {
141	return this->data_[2];
142	}
143
144	/**
145	* Returns the value of the fourth element of this quaternion.
146	* @return the value of the fourth element of this quaternion
147	*/
148	Real z() const {
149	return this->data_[3];
150	}
151	/**
152	* Returns the reference of the fourth element of this quaternion.
153	* @return the reference of the fourth element of this quaternion
154	*/
155	Real& z() {
156	return this->data_[3];
157	}
158
159	/**
160	* Tests if this quaternion is equal to other quaternion
161	* @return true if equal, otherwise return false
162	* @param q quaternion to be compared
163	*/
164	inline bool operator ==(const Quaternion<Real>& q) {
165
166	for (unsigned int i = 0; i < 4; i ++) {
167	if (!equal(this->data_[i], q[i])) {
168	return false;
169	}
170	}
171
172	return true;
173	}
174
175	/**
176	* Returns the inverse of this quaternion
177	* @return inverse
178	* @note since quaternion is a complex number, the inverse of quaternion
179	* q = w + xi + yj+ zk is inv_q = (w -xi - yj - zk)/(\|q\|^2)
180	*/
181	Quaternion<Real> inverse() {
182	Quaternion<Real> q;
183	Real d = this->lengthSquare();
184
185	q.w() = w() / d;
186	q.x() = -x() / d;
187	q.y() = -y() / d;
188	q.z() = -z() / d;
189
190	return q;
191	}
192
193	/**
194	* Sets the value to the multiplication of itself and another quaternion
195	* @param q the other quaternion
196	*/
197	void mul(const Quaternion<Real>& q) {
198	Quaternion<Real> tmp(*this);
199
200	this->data_[0] = (tmp[0]q[0]) -(tmp[1]q[1]) - (tmp[2]q[2]) - (tmp[3]q[3]);
201	this->data_[1] = (tmp[0]q[1]) + (tmp[1]q[0]) + (tmp[2]q[3]) - (tmp[3]q[2]);
202	this->data_[2] = (tmp[0]q[2]) + (tmp[2]q[0]) + (tmp[3]q[1]) - (tmp[1]q[3]);
203	this->data_[3] = (tmp[0]q[3]) + (tmp[3]q[0]) + (tmp[1]q[2]) - (tmp[2]q[1]);
204	}
205
206	void mul(const Real& s) {
207	this->data_[0] *= s;
208	this->data_[1] *= s;
209	this->data_[2] *= s;
210	this->data_[3] *= s;
211	}
212
213	/** Set the value of this quaternion to the division of itself by another quaternion */
214	void div(Quaternion<Real>& q) {
215	mul(q.inverse());
216	}
217
218	void div(const Real& s) {
219	this->data_[0] /= s;
220	this->data_[1] /= s;
221	this->data_[2] /= s;
222	this->data_[3] /= s;
223	}
224
225	Quaternion<Real>& operator *=(const Quaternion<Real>& q) {
226	mul(q);
227	return *this;
228	}
229
230	Quaternion<Real>& operator *=(const Real& s) {
231	mul(s);
232	return *this;
233	}
234
235	Quaternion<Real>& operator /=(Quaternion<Real>& q) {
236	this = q.inverse();
237	return *this;
238	}
239
240	Quaternion<Real>& operator /=(const Real& s) {
241	div(s);
242	return *this;
243	}
244	/**
245	* Returns the conjugate quaternion of this quaternion
246	* @return the conjugate quaternion of this quaternion
247	*/
248	Quaternion<Real> conjugate() const {
249	return Quaternion<Real>(w(), -x(), -y(), -z());
250	}
251
252
253	/**
254	return rotation angle from -PI to PI
255	*/
256	inline Real get_rotation_angle() const{
257	if( w < (Real)0.0 )
258	return 2.0atan2(-sqrt( x()x() + y()y() + z()z() ), -w() );
259	else
260	return 2.0atan2( sqrt( x()x() + y()y() + z()z() ), w() );
261	}
262
263	/**
264	create a unit quaternion from axis angle representation
265	*/
266	Quaternion<Real> fromAxisAngle(const Vector3<Real>& axis,
267	const Real& angle){
268	Vector3<Real> v(axis);
269	v.normalize();
270	Real half_angle = angle*0.5;
271	Real sin_a = sin(half_angle);
272	*this = Quaternion<Real>(cos(half_angle),
273	v.x()*sin_a,
274	v.y()*sin_a,
275	v.z()*sin_a);
276	return *this;
277	}
278
279	/**
280	convert a quaternion to axis angle representation,
281	preserve the axis direction and angle from -PI to +PI
282	*/
283	void toAxisAngle(Vector3<Real>& axis, Real& angle)const {
284	Real vl = sqrt( x()x() + y()y() + z()*z() );
285	if( vl > tiny ) {
286	Real ivl = 1.0/vl;
287	axis.x() = x() * ivl;
288	axis.y() = y() * ivl;
289	axis.z() = z() * ivl;
290
291	if( w() < 0 )
292	angle = 2.0*atan2(-vl, -w()); //-PI,0
293	else
294	angle = 2.0*atan2( vl, w()); //0,PI
295	} else {
296	axis = Vector3<Real>(0.0,0.0,0.0);
297	angle = 0.0;
298	}
299	}
300
301	/**
302	shortest arc quaternion rotate one vector to another by shortest path.
303	create rotation from -> to, for any length vectors.
304	*/
305	Quaternion<Real> fromShortestArc(const Vector3d& from,
306	const Vector3d& to ) {
307
308	Vector3d c( cross(from,to) );
309	*this = Quaternion<Real>(dot(from,to),
310	c.x(),
311	c.y(),
312	c.z());
313
314	this->normalize(); // if "from" or "to" not unit, normalize quat
315	w += 1.0f; // reducing angle to halfangle
316	if( w <= 1e-6 ) { // angle close to PI
317	if( ( from.z()from.z() ) > ( from.x()from.x() ) ) {
318	this->data_[0] = w;
319	this->data_[1] = 0.0; //cross(from , Vector3d(1,0,0))
320	this->data_[2] = from.z();
321	this->data_[3] = -from.y();
322	} else {
323	this->data_[0] = w;
324	this->data_[1] = from.y(); //cross(from, Vector3d(0,0,1))
325	this->data_[2] = -from.x();
326	this->data_[3] = 0.0;
327	}
328	}
329	this->normalize();
330	}
331
332	Real ComputeTwist(const Quaternion& q) {
333	return (Real)2.0 * atan2(q.z(), q.w());
334	}
335
336	void RemoveTwist(Quaternion& q) {
337	Real t = ComputeTwist(q);
338	Quaternion rt = fromAxisAngle(V3Z, t);
339
340	q *= rt.inverse();
341	}
342
343	void getTwistSwingAxisAngle(Real& twistAngle, Real& swingAngle,
344	Vector3<Real>& swingAxis) {
345
346	twistAngle = (Real)2.0 * atan2(z(), w());
347	Quaternion rt, rs;
348	rt.fromAxisAngle(V3Z, twistAngle);
349	rs = this rt.inverse();
350
351	Real vl = sqrt( rs.x()rs.x() + rs.y()rs.y() + rs.z()*rs.z() );
352	if( vl > tiny ) {
353	Real ivl = 1.0 / vl;
354	swingAxis.x() = rs.x() * ivl;
355	swingAxis.y() = rs.y() * ivl;
356	swingAxis.z() = rs.z() * ivl;
357
358	if( rs.w() < 0.0 )
359	swingAngle = 2.0*atan2(-vl, -rs.w()); //-PI,0
360	else
361	swingAngle = 2.0*atan2( vl, rs.w()); //0,PI
362	} else {
363	swingAxis = Vector3<Real>(1.0,0.0,0.0);
364	swingAngle = 0.0;
365	}
366	}
367
368
369	Vector3<Real> rotate(const Vector3<Real>& v) {
370
371	Quaternion<Real> q(v.x() * w() + v.z() * y() - v.y() * z(),
372	v.y() * w() + v.x() * z() - v.z() * x(),
373	v.z() * w() + v.y() * x() - v.x() * y(),
374	v.x() * x() + v.y() * y() + v.z() * z());
375
376	return Vector3<Real>(w()q.x() + x()q.w() + y()q.z() - z()q.y(),
377	w()q.y() + y()q.w() + z()q.x() - x()q.z(),
378	w()q.z() + z()q.w() + x()q.y() - y()q.x())*
379	( 1.0/this->lengthSquare() );
380	}
381
382	Quaternion<Real>& align (const Vector3<Real>& V1,
383	const Vector3<Real>& V2) {
384
385	// If V1 and V2 are not parallel, the axis of rotation is the unit-length
386	// vector U = Cross(V1,V2)/Length(Cross(V1,V2)). The angle of rotation,
387	// A, is the angle between V1 and V2. The quaternion for the rotation is
388	// q = cos(A/2) + sin(A/2)(uxi+uyj+uzk) where U = (ux,uy,uz).
389	//
390	// (1) Rather than extract A = acos(Dot(V1,V2)), multiply by 1/2, then
391	// compute sin(A/2) and cos(A/2), we reduce the computational costs
392	// by computing the bisector B = (V1+V2)/Length(V1+V2), so cos(A/2) =
393	// Dot(V1,B).
394	//
395	// (2) The rotation axis is U = Cross(V1,B)/Length(Cross(V1,B)), but
396	// Length(Cross(V1,B)) = Length(V1)Length(B)sin(A/2) = sin(A/2), in
397	// which case sin(A/2)(uxi+uyj+uzk) = (cxi+cyj+cz*k) where
398	// C = Cross(V1,B).
399	//
400	// If V1 = V2, then B = V1, cos(A/2) = 1, and U = (0,0,0). If V1 = -V2,
401	// then B = 0. This can happen even if V1 is approximately -V2 using
402	// floating point arithmetic, since Vector3::Normalize checks for
403	// closeness to zero and returns the zero vector accordingly. The test
404	// for exactly zero is usually not recommend for floating point
405	// arithmetic, but the implementation of Vector3::Normalize guarantees
406	// the comparison is robust. In this case, the A = pi and any axis
407	// perpendicular to V1 may be used as the rotation axis.
408
409	Vector3<Real> Bisector = V1 + V2;
410	Bisector.normalize();
411
412	Real CosHalfAngle = dot(V1,Bisector);
413
414	this->data_[0] = CosHalfAngle;
415
416	if (CosHalfAngle != (Real)0.0) {
417	Vector3<Real> Cross = cross(V1, Bisector);
418	this->data_[1] = Cross.x();
419	this->data_[2] = Cross.y();
420	this->data_[3] = Cross.z();
421	} else {
422	Real InvLength;
423	if (fabs(V1[0]) >= fabs(V1[1])) {
424	// V1.x or V1.z is the largest magnitude component
425	InvLength = (Real)1.0/sqrt(V1[0]V1[0] + V1[2]V1[2]);
426
427	this->data_[1] = -V1[2]*InvLength;
428	this->data_[2] = (Real)0.0;
429	this->data_[3] = +V1[0]*InvLength;
430	} else {
431	// V1.y or V1.z is the largest magnitude component
432	InvLength = (Real)1.0 / sqrt(V1[1]V1[1] + V1[2]V1[2]);
433
434	this->data_[1] = (Real)0.0;
435	this->data_[2] = +V1[2]*InvLength;
436	this->data_[3] = -V1[1]*InvLength;
437	}
438	}
439	return *this;
440	}
441
442	void toTwistSwing ( Real& tw, Real& sx, Real& sy ) {
443
444	// First test if the swing is in the singularity:
445
446	if ( ISZERO(z(),tiny) && ISZERO(w(),tiny) ) { sx=sy=M_PI; tw=0; return; }
447
448	// Decompose into twist-swing by solving the equation:
449	//
450	// Qtwist(t2) Qswing(s*2) = q
451	//
452	// note: (x,y) is the normalized swing axis (xx+yy=1)
453	//
454	// ( Ct 0 0 St ) * ( Cs xSs ySs 0 ) = ( qw qx qy qz )
455	// ( CtCs xSsCt-yStSs xStSs+ySsCt StCs ) = ( qw qx qy qz ) (1)
456	// From (1): CtCs / StCs = qw/qz => Ct/St = qw/qz => tan(t) = qz/qw (2)
457	//
458	// The swing rotation/2 s comes from:
459	//
460	// From (1): (CtCs)^2 + (StCs)^2 = qw^2 + qz^2 =>
461	// Cs = sqrt ( qw^2 + qz^2 ) (3)
462	//
463	// From (1): (xSsCt-yStSs)^2 + (xStSs+ySsCt)^2 = qx^2 + qy^2 =>
464	// Ss = sqrt ( qx^2 + qy^2 ) (4)
465	// From (1): \|SsCt -StSs\| \|x\| = \|qx\|
466	// \|StSs +SsCt\| \|y\| \|qy\| (5)
467
468	Real qw, qx, qy, qz;
469
470	if ( w()<0 ) {
471	qw=-w();
472	qx=-x();
473	qy=-y();
474	qz=-z();
475	} else {
476	qw=w();
477	qx=x();
478	qy=y();
479	qz=z();
480	}
481
482	Real t = atan2 ( qz, qw ); // from (2)
483	Real s = atan2( sqrt(qxqx+qyqy), sqrt(qzqz+qwqw) ); // from (3)
484	// and (4)
485
486	Real x=0.0, y=0.0, sins=sin(s);
487
488	if ( !ISZERO(sins,tiny) ) {
489	Real sint = sin(t);
490	Real cost = cos(t);
491
492	// by solving the linear system in (5):
493	y = (-qxsint + qycost)/sins;
494	x = ( qxcost + qysint)/sins;
495	}
496
497	tw = (Real)2.0*t;
498	sx = (Real)2.0xs;
499	sy = (Real)2.0ys;
500	}
501
502	void toSwingTwist(Real& sx, Real& sy, Real& tw ) {
503
504	// Decompose q into swing-twist using a similar development as
505	// in function toTwistSwing
506
507	if ( ISZERO(z(),tiny) && ISZERO(w(),tiny) ) { sx=sy=M_PI; tw=0; }
508
509	Real qw, qx, qy, qz;
510	if ( w() < 0 ){
511	qw=-w();
512	qx=-x();
513	qy=-y();
514	qz=-z();
515	} else {
516	qw=w();
517	qx=x();
518	qy=y();
519	qz=z();
520	}
521
522	// Get the twist t:
523	Real t = 2.0 * atan2(qz,qw);
524
525	Real bet = atan2( sqrt(qxqx+qyqy), sqrt(qzqz+qwqw) );
526	Real gam = t/2.0;
527	Real sinc = ISZERO(bet,tiny)? 1.0 : sin(bet)/bet;
528	Real singam = sin(gam);
529	Real cosgam = cos(gam);
530
531	sx = Real( (2.0/sinc) * (cosgamqx - singamqy) );
532	sy = Real( (2.0/sinc) * (singamqx + cosgamqy) );
533	tw = Real( t );
534	}
535
536
537
538	/**
539	* Returns the corresponding rotation matrix (3x3)
540	* @return a 3x3 rotation matrix
541	*/
542	SquareMatrix<Real, 3> toRotationMatrix3() {
543	SquareMatrix<Real, 3> rotMat3;
544
545	Real w2;
546	Real x2;
547	Real y2;
548	Real z2;
549
550	if (!this->isNormalized())
551	this->normalize();
552
553	w2 = w() * w();
554	x2 = x() * x();
555	y2 = y() * y();
556	z2 = z() * z();
557
558	rotMat3(0, 0) = w2 + x2 - y2 - z2;
559	rotMat3(0, 1) = 2.0 * ( x() * y() + w() * z() );
560	rotMat3(0, 2) = 2.0 * ( x() * z() - w() * y() );
561
562	rotMat3(1, 0) = 2.0 * ( x() * y() - w() * z() );
563	rotMat3(1, 1) = w2 - x2 + y2 - z2;
564	rotMat3(1, 2) = 2.0 * ( y() * z() + w() * x() );
565
566	rotMat3(2, 0) = 2.0 * ( x() * z() + w() * y() );
567	rotMat3(2, 1) = 2.0 * ( y() * z() - w() * x() );
568	rotMat3(2, 2) = w2 - x2 -y2 +z2;
569
570	return rotMat3;
571	}
572
573	};//end Quaternion
574
575
576	/**
577	* Returns the vaule of scalar multiplication of this quaterion q (q * s).
578	* @return the vaule of scalar multiplication of this vector
579	* @param q the source quaternion
580	* @param s the scalar value
581	*/
582	template<typename Real, unsigned int Dim>
583	Quaternion<Real> operator * ( const Quaternion<Real>& q, Real s) {
584	Quaternion<Real> result(q);
585	result.mul(s);
586	return result;
587	}
588
589	/**
590	* Returns the vaule of scalar multiplication of this quaterion q (q * s).
591	* @return the vaule of scalar multiplication of this vector
592	* @param s the scalar value
593	* @param q the source quaternion
594	*/
595	template<typename Real, unsigned int Dim>
596	Quaternion<Real> operator * ( const Real& s, const Quaternion<Real>& q ) {
597	Quaternion<Real> result(q);
598	result.mul(s);
599	return result;
600	}
601
602	/**
603	* Returns the multiplication of two quaternion
604	* @return the multiplication of two quaternion
605	* @param q1 the first quaternion
606	* @param q2 the second quaternion
607	*/
608	template<typename Real>
609	inline Quaternion<Real> operator *(const Quaternion<Real>& q1, const Quaternion<Real>& q2) {
610	Quaternion<Real> result(q1);
611	result *= q2;
612	return result;
613	}
614
615	/**
616	* Returns the division of two quaternion
617	* @param q1 divisor
618	* @param q2 dividen
619	*/
620
621	template<typename Real>
622	inline Quaternion<Real> operator /( Quaternion<Real>& q1, Quaternion<Real>& q2) {
623	return q1 * q2.inverse();
624	}
625
626	/**
627	* Returns the value of the division of a scalar by a quaternion
628	* @return the value of the division of a scalar by a quaternion
629	* @param s scalar
630	* @param q quaternion
631	* @note for a quaternion q, 1/q = q.inverse()
632	*/
633	template<typename Real>
634	Quaternion<Real> operator /(const Real& s, Quaternion<Real>& q) {
635
636	Quaternion<Real> x;
637	x = q.inverse();
638	x *= s;
639	return x;
640	}
641
642	template <class T>
643	inline bool operator==(const Quaternion<T>& lhs, const Quaternion<T>& rhs) {
644	return equal(lhs[0] ,rhs[0]) && equal(lhs[1] , rhs[1]) && equal(lhs[2], rhs[2]) && equal(lhs[3], rhs[3]);
645	}
646
647	typedef Quaternion<RealType> Quat4d;
648	}
649	#endif //MATH_QUATERNION_HPP