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]  Kuang & Gezelter,  J. Chem. Phys. 133, 164101 (2010).
 * [5]  Vardeman, Stocker & Gezelter, J. Chem. Theory Comput. 7, 834 (2011).
 */
 
/**
 * @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:	1687
Committed:	Sat Mar 10 16:00:24 2012 UTC (13 years, 7 months ago) by gezelter
File size:	20563 byte(s)
Log Message:	Fixes for LLVM compilation Bug fix for Gay Berne potential with switching functions
#	User	Rev	Content
1	gezelter	507	/*
2	gezelter	246	* Copyright (c) 2005 The University of Notre Dame. All Rights Reserved.
3	tim	92	*
4	gezelter	246	* 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	gezelter	1390	* 1. Redistributions of source code must retain the above copyright
10	gezelter	246	* notice, this list of conditions and the following disclaimer.
11			*
12	gezelter	1390	* 2. Redistributions in binary form must reproduce the above copyright
13	gezelter	246	* 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	gezelter	1390	*
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	gezelter	1687	* [4] Kuang & Gezelter, J. Chem. Phys. 133, 164101 (2010).
40			* [5] Vardeman, Stocker & Gezelter, J. Chem. Theory Comput. 7, 834 (2011).
41	tim	92	*/
42	gezelter	246
43	tim	92	/**
44			* @file Quaternion.hpp
45			* @author Teng Lin
46			* @date 10/11/2004
47			* @version 1.0
48			*/
49
50			#ifndef MATH_QUATERNION_HPP
51			#define MATH_QUATERNION_HPP
52
53	cli2	1360	#include "math/Vector3.hpp"
54	tim	110	#include "math/SquareMatrix.hpp"
55	cli2	1360	#define ISZERO(a,eps) ( (a)>-(eps) && (a)<(eps) )
56			const RealType tiny=1.0e-6;
57	tim	93
58	gezelter	1390	namespace OpenMD{
59	tim	92
60	gezelter	507	/**
61			* @class Quaternion Quaternion.hpp "math/Quaternion.hpp"
62			* Quaternion is a sort of a higher-level complex number.
63			* It is defined as Q = w + xi + yj + z*k,
64	tim	963	* where w, x, y, and z are numbers of type T (e.g. RealType), and
65	gezelter	507	* ii = -1; jj = -1; k*k = -1;
66			* ij = k; jk = i; k*i = j;
67			*/
68			template<typename Real>
69			class Quaternion : public Vector<Real, 4> {
70	cli2	1360
71	gezelter	507	public:
72			Quaternion() : Vector<Real, 4>() {}
73	tim	92
74	gezelter	507	/** Constructs and initializes a Quaternion from w, x, y, z values */
75			Quaternion(Real w, Real x, Real y, Real z) {
76			this->data_[0] = w;
77			this->data_[1] = x;
78			this->data_[2] = y;
79			this->data_[3] = z;
80			}
81	tim	93
82	gezelter	507	/** Constructs and initializes a Quaternion from a Vector<Real,4> */
83			Quaternion(const Vector<Real,4>& v)
84			: Vector<Real, 4>(v){
85	cli2	1360	}
86	tim	92
87	gezelter	507	/** copy assignment */
88			Quaternion& operator =(const Vector<Real, 4>& v){
89			if (this == & v)
90			return *this;
91	cli2	1360
92	gezelter	507	Vector<Real, 4>::operator=(v);
93	cli2	1360
94	gezelter	507	return *this;
95			}
96	cli2	1360
97	gezelter	507	/**
98			* Returns the value of the first element of this quaternion.
99			* @return the value of the first element of this quaternion
100			*/
101			Real w() const {
102			return this->data_[0];
103			}
104	tim	93
105	gezelter	507	/**
106			* Returns the reference of the first element of this quaternion.
107			* @return the reference of the first element of this quaternion
108			*/
109			Real& w() {
110			return this->data_[0];
111			}
112	tim	93
113	gezelter	507	/**
114			* Returns the value of the first element of this quaternion.
115			* @return the value of the first element of this quaternion
116			*/
117			Real x() const {
118			return this->data_[1];
119			}
120	tim	93
121	gezelter	507	/**
122			* Returns the reference of the second element of this quaternion.
123			* @return the reference of the second element of this quaternion
124			*/
125			Real& x() {
126			return this->data_[1];
127			}
128	tim	93
129	gezelter	507	/**
130			* Returns the value of the thirf element of this quaternion.
131			* @return the value of the third element of this quaternion
132			*/
133			Real y() const {
134			return this->data_[2];
135			}
136	tim	93
137	gezelter	507	/**
138			* Returns the reference of the third element of this quaternion.
139			* @return the reference of the third element of this quaternion
140			*/
141			Real& y() {
142			return this->data_[2];
143			}
144	tim	93
145	gezelter	507	/**
146			* Returns the value of the fourth element of this quaternion.
147			* @return the value of the fourth element of this quaternion
148			*/
149			Real z() const {
150			return this->data_[3];
151			}
152			/**
153			* Returns the reference of the fourth element of this quaternion.
154			* @return the reference of the fourth element of this quaternion
155			*/
156			Real& z() {
157			return this->data_[3];
158			}
159	tim	93
160	gezelter	507	/**
161			* Tests if this quaternion is equal to other quaternion
162			* @return true if equal, otherwise return false
163			* @param q quaternion to be compared
164			*/
165			inline bool operator ==(const Quaternion<Real>& q) {
166	tim	110
167	gezelter	507	for (unsigned int i = 0; i < 4; i ++) {
168			if (!equal(this->data_[i], q[i])) {
169			return false;
170			}
171			}
172	tim	110
173	gezelter	507	return true;
174			}
175	tim	110
176	gezelter	507	/**
177			* Returns the inverse of this quaternion
178			* @return inverse
179			* @note since quaternion is a complex number, the inverse of quaternion
180			* q = w + xi + yj+ zk is inv_q = (w -xi - yj - zk)/(\|q\|^2)
181			*/
182			Quaternion<Real> inverse() {
183			Quaternion<Real> q;
184			Real d = this->lengthSquare();
185	tim	93
186	gezelter	507	q.w() = w() / d;
187			q.x() = -x() / d;
188			q.y() = -y() / d;
189			q.z() = -z() / d;
190	tim	93
191	gezelter	507	return q;
192			}
193	tim	93
194	gezelter	507	/**
195			* Sets the value to the multiplication of itself and another quaternion
196			* @param q the other quaternion
197			*/
198			void mul(const Quaternion<Real>& q) {
199			Quaternion<Real> tmp(*this);
200	tim	93
201	gezelter	507	this->data_[0] = (tmp[0]q[0]) -(tmp[1]q[1]) - (tmp[2]q[2]) - (tmp[3]q[3]);
202			this->data_[1] = (tmp[0]q[1]) + (tmp[1]q[0]) + (tmp[2]q[3]) - (tmp[3]q[2]);
203			this->data_[2] = (tmp[0]q[2]) + (tmp[2]q[0]) + (tmp[3]q[1]) - (tmp[1]q[3]);
204			this->data_[3] = (tmp[0]q[3]) + (tmp[3]q[0]) + (tmp[1]q[2]) - (tmp[2]q[1]);
205			}
206	tim	93
207	gezelter	507	void mul(const Real& s) {
208			this->data_[0] *= s;
209			this->data_[1] *= s;
210			this->data_[2] *= s;
211			this->data_[3] *= s;
212			}
213	tim	93
214	gezelter	507	/** Set the value of this quaternion to the division of itself by another quaternion */
215			void div(Quaternion<Real>& q) {
216			mul(q.inverse());
217			}
218	tim	110
219	gezelter	507	void div(const Real& s) {
220			this->data_[0] /= s;
221			this->data_[1] /= s;
222			this->data_[2] /= s;
223			this->data_[3] /= s;
224			}
225	tim	93
226	gezelter	507	Quaternion<Real>& operator *=(const Quaternion<Real>& q) {
227			mul(q);
228			return *this;
229			}
230	tim	110
231	gezelter	507	Quaternion<Real>& operator *=(const Real& s) {
232			mul(s);
233			return *this;
234			}
235	tim	93
236	gezelter	507	Quaternion<Real>& operator /=(Quaternion<Real>& q) {
237			this = q.inverse();
238			return *this;
239			}
240	tim	110
241	gezelter	507	Quaternion<Real>& operator /=(const Real& s) {
242			div(s);
243			return *this;
244			}
245			/**
246			* Returns the conjugate quaternion of this quaternion
247			* @return the conjugate quaternion of this quaternion
248			*/
249	cli2	1360	Quaternion<Real> conjugate() const {
250	gezelter	507	return Quaternion<Real>(w(), -x(), -y(), -z());
251			}
252	tim	93
253	cli2	1360
254	gezelter	507	/**
255	cli2	1360	return rotation angle from -PI to PI
256			*/
257			inline Real get_rotation_angle() const{
258	gezelter	1687	if( w() < (Real)0.0 )
259	cli2	1360	return 2.0atan2(-sqrt( x()x() + y()y() + z()z() ), -w() );
260			else
261			return 2.0atan2( sqrt( x()x() + y()y() + z()z() ), w() );
262			}
263
264			/**
265			create a unit quaternion from axis angle representation
266			*/
267			Quaternion<Real> fromAxisAngle(const Vector3<Real>& axis,
268			const Real& angle){
269			Vector3<Real> v(axis);
270			v.normalize();
271			Real half_angle = angle*0.5;
272			Real sin_a = sin(half_angle);
273			*this = Quaternion<Real>(cos(half_angle),
274			v.x()*sin_a,
275			v.y()*sin_a,
276			v.z()*sin_a);
277	gezelter	1390	return *this;
278	cli2	1360	}
279
280			/**
281			convert a quaternion to axis angle representation,
282			preserve the axis direction and angle from -PI to +PI
283			*/
284			void toAxisAngle(Vector3<Real>& axis, Real& angle)const {
285			Real vl = sqrt( x()x() + y()y() + z()*z() );
286			if( vl > tiny ) {
287			Real ivl = 1.0/vl;
288			axis.x() = x() * ivl;
289			axis.y() = y() * ivl;
290			axis.z() = z() * ivl;
291
292			if( w() < 0 )
293			angle = 2.0*atan2(-vl, -w()); //-PI,0
294			else
295			angle = 2.0*atan2( vl, w()); //0,PI
296			} else {
297			axis = Vector3<Real>(0.0,0.0,0.0);
298			angle = 0.0;
299			}
300			}
301
302			/**
303			shortest arc quaternion rotate one vector to another by shortest path.
304			create rotation from -> to, for any length vectors.
305			*/
306			Quaternion<Real> fromShortestArc(const Vector3d& from,
307			const Vector3d& to ) {
308
309			Vector3d c( cross(from,to) );
310			*this = Quaternion<Real>(dot(from,to),
311			c.x(),
312			c.y(),
313			c.z());
314
315			this->normalize(); // if "from" or "to" not unit, normalize quat
316	gezelter	1687	w() += 1.0f; // reducing angle to halfangle
317			if( w() <= 1e-6 ) { // angle close to PI
318	cli2	1360	if( ( from.z()from.z() ) > ( from.x()from.x() ) ) {
319	gezelter	1687	this->data_[0] = w();
320	cli2	1360	this->data_[1] = 0.0; //cross(from , Vector3d(1,0,0))
321			this->data_[2] = from.z();
322			this->data_[3] = -from.y();
323			} else {
324	gezelter	1687	this->data_[0] = w();
325	cli2	1360	this->data_[1] = from.y(); //cross(from, Vector3d(0,0,1))
326			this->data_[2] = -from.x();
327			this->data_[3] = 0.0;
328			}
329			}
330			this->normalize();
331			}
332
333			Real ComputeTwist(const Quaternion& q) {
334			return (Real)2.0 * atan2(q.z(), q.w());
335			}
336
337			void RemoveTwist(Quaternion& q) {
338			Real t = ComputeTwist(q);
339			Quaternion rt = fromAxisAngle(V3Z, t);
340
341			q *= rt.inverse();
342			}
343
344			void getTwistSwingAxisAngle(Real& twistAngle, Real& swingAngle,
345			Vector3<Real>& swingAxis) {
346
347			twistAngle = (Real)2.0 * atan2(z(), w());
348			Quaternion rt, rs;
349			rt.fromAxisAngle(V3Z, twistAngle);
350			rs = this rt.inverse();
351
352			Real vl = sqrt( rs.x()rs.x() + rs.y()rs.y() + rs.z()*rs.z() );
353			if( vl > tiny ) {
354			Real ivl = 1.0 / vl;
355			swingAxis.x() = rs.x() * ivl;
356			swingAxis.y() = rs.y() * ivl;
357			swingAxis.z() = rs.z() * ivl;
358
359			if( rs.w() < 0.0 )
360			swingAngle = 2.0*atan2(-vl, -rs.w()); //-PI,0
361			else
362			swingAngle = 2.0*atan2( vl, rs.w()); //0,PI
363			} else {
364			swingAxis = Vector3<Real>(1.0,0.0,0.0);
365			swingAngle = 0.0;
366			}
367			}
368
369
370			Vector3<Real> rotate(const Vector3<Real>& v) {
371
372			Quaternion<Real> q(v.x() * w() + v.z() * y() - v.y() * z(),
373			v.y() * w() + v.x() * z() - v.z() * x(),
374			v.z() * w() + v.y() * x() - v.x() * y(),
375			v.x() * x() + v.y() * y() + v.z() * z());
376
377			return Vector3<Real>(w()q.x() + x()q.w() + y()q.z() - z()q.y(),
378			w()q.y() + y()q.w() + z()q.x() - x()q.z(),
379			w()q.z() + z()q.w() + x()q.y() - y()q.x())*
380			( 1.0/this->lengthSquare() );
381			}
382
383			Quaternion<Real>& align (const Vector3<Real>& V1,
384			const Vector3<Real>& V2) {
385
386			// If V1 and V2 are not parallel, the axis of rotation is the unit-length
387			// vector U = Cross(V1,V2)/Length(Cross(V1,V2)). The angle of rotation,
388			// A, is the angle between V1 and V2. The quaternion for the rotation is
389			// q = cos(A/2) + sin(A/2)(uxi+uyj+uzk) where U = (ux,uy,uz).
390			//
391			// (1) Rather than extract A = acos(Dot(V1,V2)), multiply by 1/2, then
392			// compute sin(A/2) and cos(A/2), we reduce the computational costs
393			// by computing the bisector B = (V1+V2)/Length(V1+V2), so cos(A/2) =
394			// Dot(V1,B).
395			//
396			// (2) The rotation axis is U = Cross(V1,B)/Length(Cross(V1,B)), but
397			// Length(Cross(V1,B)) = Length(V1)Length(B)sin(A/2) = sin(A/2), in
398			// which case sin(A/2)(uxi+uyj+uzk) = (cxi+cyj+cz*k) where
399			// C = Cross(V1,B).
400			//
401			// If V1 = V2, then B = V1, cos(A/2) = 1, and U = (0,0,0). If V1 = -V2,
402			// then B = 0. This can happen even if V1 is approximately -V2 using
403			// floating point arithmetic, since Vector3::Normalize checks for
404			// closeness to zero and returns the zero vector accordingly. The test
405			// for exactly zero is usually not recommend for floating point
406			// arithmetic, but the implementation of Vector3::Normalize guarantees
407			// the comparison is robust. In this case, the A = pi and any axis
408			// perpendicular to V1 may be used as the rotation axis.
409
410			Vector3<Real> Bisector = V1 + V2;
411			Bisector.normalize();
412
413			Real CosHalfAngle = dot(V1,Bisector);
414
415			this->data_[0] = CosHalfAngle;
416
417			if (CosHalfAngle != (Real)0.0) {
418			Vector3<Real> Cross = cross(V1, Bisector);
419			this->data_[1] = Cross.x();
420			this->data_[2] = Cross.y();
421			this->data_[3] = Cross.z();
422			} else {
423			Real InvLength;
424			if (fabs(V1[0]) >= fabs(V1[1])) {
425			// V1.x or V1.z is the largest magnitude component
426			InvLength = (Real)1.0/sqrt(V1[0]V1[0] + V1[2]V1[2]);
427
428			this->data_[1] = -V1[2]*InvLength;
429			this->data_[2] = (Real)0.0;
430			this->data_[3] = +V1[0]*InvLength;
431			} else {
432			// V1.y or V1.z is the largest magnitude component
433			InvLength = (Real)1.0 / sqrt(V1[1]V1[1] + V1[2]V1[2]);
434
435			this->data_[1] = (Real)0.0;
436			this->data_[2] = +V1[2]*InvLength;
437			this->data_[3] = -V1[1]*InvLength;
438			}
439			}
440			return *this;
441			}
442
443			void toTwistSwing ( Real& tw, Real& sx, Real& sy ) {
444
445			// First test if the swing is in the singularity:
446
447			if ( ISZERO(z(),tiny) && ISZERO(w(),tiny) ) { sx=sy=M_PI; tw=0; return; }
448
449			// Decompose into twist-swing by solving the equation:
450			//
451			// Qtwist(t2) Qswing(s*2) = q
452			//
453			// note: (x,y) is the normalized swing axis (xx+yy=1)
454			//
455			// ( Ct 0 0 St ) * ( Cs xSs ySs 0 ) = ( qw qx qy qz )
456			// ( CtCs xSsCt-yStSs xStSs+ySsCt StCs ) = ( qw qx qy qz ) (1)
457			// From (1): CtCs / StCs = qw/qz => Ct/St = qw/qz => tan(t) = qz/qw (2)
458			//
459			// The swing rotation/2 s comes from:
460			//
461			// From (1): (CtCs)^2 + (StCs)^2 = qw^2 + qz^2 =>
462			// Cs = sqrt ( qw^2 + qz^2 ) (3)
463			//
464			// From (1): (xSsCt-yStSs)^2 + (xStSs+ySsCt)^2 = qx^2 + qy^2 =>
465			// Ss = sqrt ( qx^2 + qy^2 ) (4)
466			// From (1): \|SsCt -StSs\| \|x\| = \|qx\|
467			// \|StSs +SsCt\| \|y\| \|qy\| (5)
468
469			Real qw, qx, qy, qz;
470
471			if ( w()<0 ) {
472			qw=-w();
473			qx=-x();
474			qy=-y();
475			qz=-z();
476			} else {
477			qw=w();
478			qx=x();
479			qy=y();
480			qz=z();
481			}
482
483			Real t = atan2 ( qz, qw ); // from (2)
484			Real s = atan2( sqrt(qxqx+qyqy), sqrt(qzqz+qwqw) ); // from (3)
485			// and (4)
486
487			Real x=0.0, y=0.0, sins=sin(s);
488
489			if ( !ISZERO(sins,tiny) ) {
490			Real sint = sin(t);
491			Real cost = cos(t);
492
493			// by solving the linear system in (5):
494			y = (-qxsint + qycost)/sins;
495			x = ( qxcost + qysint)/sins;
496			}
497
498			tw = (Real)2.0*t;
499			sx = (Real)2.0xs;
500			sy = (Real)2.0ys;
501			}
502
503			void toSwingTwist(Real& sx, Real& sy, Real& tw ) {
504
505			// Decompose q into swing-twist using a similar development as
506			// in function toTwistSwing
507
508			if ( ISZERO(z(),tiny) && ISZERO(w(),tiny) ) { sx=sy=M_PI; tw=0; }
509
510			Real qw, qx, qy, qz;
511			if ( w() < 0 ){
512			qw=-w();
513			qx=-x();
514			qy=-y();
515			qz=-z();
516			} else {
517			qw=w();
518			qx=x();
519			qy=y();
520			qz=z();
521			}
522
523			// Get the twist t:
524			Real t = 2.0 * atan2(qz,qw);
525
526			Real bet = atan2( sqrt(qxqx+qyqy), sqrt(qzqz+qwqw) );
527			Real gam = t/2.0;
528			Real sinc = ISZERO(bet,tiny)? 1.0 : sin(bet)/bet;
529			Real singam = sin(gam);
530			Real cosgam = cos(gam);
531
532			sx = Real( (2.0/sinc) * (cosgamqx - singamqy) );
533			sy = Real( (2.0/sinc) * (singamqx + cosgamqy) );
534			tw = Real( t );
535			}
536
537
538
539			/**
540	gezelter	507	* Returns the corresponding rotation matrix (3x3)
541			* @return a 3x3 rotation matrix
542			*/
543			SquareMatrix<Real, 3> toRotationMatrix3() {
544			SquareMatrix<Real, 3> rotMat3;
545	cli2	1360
546	gezelter	507	Real w2;
547			Real x2;
548			Real y2;
549			Real z2;
550	tim	93
551	gezelter	507	if (!this->isNormalized())
552			this->normalize();
553	tim	93
554	gezelter	507	w2 = w() * w();
555			x2 = x() * x();
556			y2 = y() * y();
557			z2 = z() * z();
558	tim	93
559	gezelter	507	rotMat3(0, 0) = w2 + x2 - y2 - z2;
560			rotMat3(0, 1) = 2.0 * ( x() * y() + w() * z() );
561			rotMat3(0, 2) = 2.0 * ( x() * z() - w() * y() );
562	tim	93
563	gezelter	507	rotMat3(1, 0) = 2.0 * ( x() * y() - w() * z() );
564			rotMat3(1, 1) = w2 - x2 + y2 - z2;
565			rotMat3(1, 2) = 2.0 * ( y() * z() + w() * x() );
566	tim	93
567	gezelter	507	rotMat3(2, 0) = 2.0 * ( x() * z() + w() * y() );
568			rotMat3(2, 1) = 2.0 * ( y() * z() - w() * x() );
569			rotMat3(2, 2) = w2 - x2 -y2 +z2;
570	tim	110
571	gezelter	507	return rotMat3;
572			}
573	tim	93
574	gezelter	507	};//end Quaternion
575	tim	93
576	tim	110
577	tim	93	/**
578	tim	110	* Returns the vaule of scalar multiplication of this quaterion q (q * s).
579			* @return the vaule of scalar multiplication of this vector
580			* @param q the source quaternion
581			* @param s the scalar value
582			*/
583	gezelter	507	template<typename Real, unsigned int Dim>
584			Quaternion<Real> operator * ( const Quaternion<Real>& q, Real s) {
585			Quaternion<Real> result(q);
586			result.mul(s);
587			return result;
588			}
589	tim	110
590	gezelter	507	/**
591			* Returns the vaule of scalar multiplication of this quaterion q (q * s).
592			* @return the vaule of scalar multiplication of this vector
593			* @param s the scalar value
594			* @param q the source quaternion
595			*/
596			template<typename Real, unsigned int Dim>
597			Quaternion<Real> operator * ( const Real& s, const Quaternion<Real>& q ) {
598			Quaternion<Real> result(q);
599			result.mul(s);
600			return result;
601			}
602	tim	110
603	gezelter	507	/**
604			* Returns the multiplication of two quaternion
605			* @return the multiplication of two quaternion
606			* @param q1 the first quaternion
607			* @param q2 the second quaternion
608			*/
609			template<typename Real>
610			inline Quaternion<Real> operator *(const Quaternion<Real>& q1, const Quaternion<Real>& q2) {
611			Quaternion<Real> result(q1);
612			result *= q2;
613			return result;
614			}
615	tim	93
616	gezelter	507	/**
617			* Returns the division of two quaternion
618			* @param q1 divisor
619			* @param q2 dividen
620			*/
621	tim	93
622	gezelter	507	template<typename Real>
623			inline Quaternion<Real> operator /( Quaternion<Real>& q1, Quaternion<Real>& q2) {
624			return q1 * q2.inverse();
625			}
626	tim	93
627	gezelter	507	/**
628			* Returns the value of the division of a scalar by a quaternion
629			* @return the value of the division of a scalar by a quaternion
630			* @param s scalar
631			* @param q quaternion
632			* @note for a quaternion q, 1/q = q.inverse()
633			*/
634			template<typename Real>
635			Quaternion<Real> operator /(const Real& s, Quaternion<Real>& q) {
636	tim	93
637	gezelter	507	Quaternion<Real> x;
638			x = q.inverse();
639			x *= s;
640			return x;
641			}
642	tim	110
643	gezelter	507	template <class T>
644			inline bool operator==(const Quaternion<T>& lhs, const Quaternion<T>& rhs) {
645			return equal(lhs[0] ,rhs[0]) && equal(lhs[1] , rhs[1]) && equal(lhs[2], rhs[2]) && equal(lhs[3], rhs[3]);
646			}
647	tim	110
648	tim	963	typedef Quaternion<RealType> Quat4d;
649	tim	92	}
650			#endif //MATH_QUATERNION_HPP