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                 CQRlib -- ANSI C API for Quaternion Rotations

                                 Release 1.1.4
                                  29 Apr 2018
              © 2008, 2009, 2010, 2014, 2018 Herbert J. Bernstein
                            yayahjb at gmail dot com
                You may distribute the CQRlib API under the LGPL

   The 1.1.4 release is a documentation change to reflect a move of the
   source to github. The 1.1.3 release parenthesized uses of *this that
   caused errors from OSX clang. Thanks to Zack Settel for reporting the
   problem. The 1.1.2 release improved the portability of the code for Visual
   Studio. The 1.1.1 release relaxed some of the test constraints and
   parametrized the tests against DBL_EPSILON and added the Dist and Distsq
   functions. The 1.1 release added functions for log, exp, power and root,
   added a macro form of the norm and fixed the macro for inverse. The 1.0.6
   release fixed an error in the CQRHLERPDist definition and comments. The
   1.0.5 release added SLERP/HLERP support in C++ and C, moved from the
   vector project. The 1.0.4 release added a version of L. Andrews adaptation
   to a C++ template. The 1.0.3 release changed from use of a FAR macro to
   use of a CQR_FAR macro to avoid name conflicts. the macros for malloc,
   free, memmove and memset were also changed. The 1.0.2 release of 14 June
   2009 corrected the Makefile for case-sensitive file systems and to include
   -lm in loading. Release 1.0.1 of 23 February 2009 was a minor
   documentation update to the original 1.0 release of 22 February 2009.

   CQRlib is an ANSI C implementation of a utility library for quaternion
   arithmetic and quaternion rotation math. See
     * "Quaternions and spatial rotation", Wikipedia
       http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation
     * K. Shoemake, "Quaternions", Department of Computer Science, University
       of Pennsylvania, Philadelphia, PA 19104,
       ftp://ftp.cis.upenn.edu/pub/graphics/shoemake/quatut.ps.Z
     * K. Shoemake, "Animating rotation with quaternion curves", ACM SIGGRAPH
       Computer Graphics, Vol 19, No. 3, pp 245--254, 1985.

   Work supported in part by NIH NIGMS under grant 1R15GM078077-01 and DOE
   under grant ER63601-1021466-0009501. Any opinions, findings, and
   conclusions or recommendations expressed in this material are those of the
   author(s) and do not necessarily reflect the views of the funding
   agencies.

    Installation

   The CQRlib package is available at https://github.com/yayahjb/cqrlib.git.
   A source zip file is available at
   https://github.com/yayahjb/cqrlib/archive/master.zip

   When the source is downloaded and unpacked, you should have a directory
   cqrlib or master. To build you may need to install the libtool-bin
   package. To see the current settings for a build execute

   make

   which should give the following information:

 PLEASE READ README_CQRlib.txt and lgpl.txt
 
  Before making the CQRlib library and example programs, check
  that the chosen settings are correct
 
  The current C and C++ compile commands are:
 
    libtool --mode=compile gcc -g -O2  -Wall -ansi -pedantic -I.  -c
    libtool --mode=compile g++ -g -O2  -Wall -ansi -pedantic -DCQR_NOCCODE=1 -I.  -c
 
  The current library C and C++ link commands are:
 
    libtool --mode=link  gcc -version-info 3:0:1 -rpath /home/yaya/lib
    libtool --mode=link g++ -version-info 3:0:1 -rpath /home/yaya/lib
 
  The current C library local, dynamic and static build commands are:
 
    libtool --mode=link gcc -g -O2  -Wall -ansi -pedantic -I.
    libtool --mode=link gcc -g -O2  -Wall -ansi -pedantic -dynamic -I /home/yaya/include -L/home/yaya/lib
    libtool --mode=link gcc -g -O2  -Wall -ansi -pedantic -static -I /home/yaya/include -L/home/yaya/lib
 
  The current C++ template local, dynamic and static build commands are:
 
    libtool --mode=link g++ -g -O2  -Wall -ansi -pedantic -DCQR_NOCCODE=1 -I.
    libtool --mode=link g++ -g -O2  -Wall -ansi -pedantic -DCQR_NOCCODE=1 -dynamic -I /home/yaya/include -L/home/yaya/lib
    libtool --mode=link g++ -g -O2  -Wall -ansi -pedantic -DCQR_NOCCODE=1 -static -I /home/yaya/include -L/home/yaya/lib
 
  Before installing the CQRlib library and example programs, check
  that the install directory and install commands are correct:
 
  The current values are :
 
    /home/yaya
    libtool --mode=install cp
    
 
  To compile the CQRlib library and example programs type:
 
    make clean
    make all
 
  To run a set of tests type:
 
    make tests
 
  To clean up the directories type:
 
    make clean
 
  To install the library and binaries type:
 
    make install

   If these settings need to be changed, edit Makefile. On some systems, e.g.
   Mac OS X, the default libtool is not appropriate. In that case you should
   install a recent version of libtool. The CQRlib kit has been tested with
   libtool versions 1.3.5, 1.5.4 and 2.4.6. If the system libtool is not to
   be used, define the variable LIBTOOL to be the path to the libtool
   executable, e.g. in bash

   export LIBTOOL=$HOME/bin/libtool

   of in the Makefie

   LIBTOOL = $(HOME)/bin/libtool

   If you need to include local header files using #include "..." instead of
   #include <...>, define the variable USE_LOCAL_HEADERS. This definition is
   forced if _MSC_VER is defined, meaning that local headers will
   automatically be used for Visual Studio.

    Synopsis

   #include <cqrlib.h>

     /* CQRCreateQuaternion -- create a quaternion = w +ix+jy+kz */
    
     int CQRCreateQuaternion(CQRQuaternionHandle * quaternion, double w, double x, double y, double z);
    
     /* CQRCreateEmptyQuaternion -- create a quaternion = 0 +i0+j0+k0 */
    
     int CQRCreateEmptyQuaternion(CQRQuaternionHandle * quaternion) ;
    
     /* CQRFreeQuaternion -- free a quaternion  */
    
     int CQRFreeQuaternion(CQRQuaternionHandle * quaternion);       
    
     /* CQRSetQuaternion -- create an existing quaternion = w +ix+jy+kz */
    
     int CQRSetQuaternion( CQRQuaternionHandle quaternion, double w, double x, double y, double z);

     /* CQRGetQuaternionW -- get the w component of a quaternion */
    
     int CQRGetQuaternionW( double CQR_FAR * qw, CQRQuaternionHandle q );
    
     /* CQRGetQuaternionX -- get the x component of a quaternion */
    
     int CQRGetQuaternionX( double CQR_FAR * qx, CQRQuaternionHandle q );
    
     /* CQRGetQuaternionY -- get the y component of a quaternion */
    
     int CQRGetQuaternionY( double CQR_FAR * qy, CQRQuaternionHandle q );
    
     /* CQRGetQuaternionZ -- get the z component of a quaternion */
    
     int CQRGetQuaternionZ( double CQR_FAR * qz, CQRQuaternionHandle q );
    
     /* CQRGetQuaternionIm -- get the imaginary component of a quaternion */
    
     int CQRGetQuaternionIm( CQRQuaternionHandle quaternion, CQRQuaternionHandle q );
    
     /* CQRGetQuaternionAxis -- get the axis for the polar representation of a quaternion */
    
     int CQRGetQuaternionAxis( CQRQuaternionHandle quaternion, CQRQuaternionHandle q );
    
     /* CQRGetQuaternionAngle -- get the angular component of the polar representation
      of aquaternion */
    
     int CQRGetQuaternionAngle( double CQR_FAR * angle, CQRQuaternionHandle q );
    
     /* CQRLog -- get the natural logarithm of a quaternion */
    
     int CQRLog( CQRQuaternionHandle quaternion, CQRQuaternionHandle q );
    
     /* CQRExp -- get the exponential (exp) of a quaternion */
    
     int CQRExp( CQRQuaternionHandle quaternion, CQRQuaternionHandle q );
    
     /* CQRQuaternionPower -- take a quarernion to a quaternion power */
    
     int CQRQuaternionPower( CQRQuaternionHandle quaternion, CQRQuaternionHandle q, CQRQuaternionHandle p);
    
     /* CQRDoublePower -- take a quarernion to a double power */
    
     int CQRDoublePower( CQRQuaternionHandle quaternion, CQRQuaternionHandle q, double p);
    
     /* CQRIntegerPower -- take a quaternion to an integer power */
    
     int CQRIntegerPower( CQRQuaternionHandle quaternion, CQRQuaternionHandle q, int p);
    
     /* CQRIntegerRoot -- take the given integer root  of a quaternion, returning
      the indicated mth choice from among multiple roots.
      For reals the cycle runs through first the i-based
      roots, then the j-based roots and then the k-based roots,
      out of the infinite number of possible roots of reals. */
    
     int CQRIntegerRoot( CQRQuaternionHandle quaternion, CQRQuaternionHandle q, int r, int m);

     /*  CQRAdd -- add a quaternion (q1) to a quaternion (q2) */
    
     int CQRAdd (CQRQuaternionHandle quaternion,  CQRQuaternionHandle q1, CQRQuaternionHandle q2 );
    
     /*  CQRSubtract -- subtract a quaternion (q2) from a quaternion (q1)  */
    
     int CQRSubtract (CQRQuaternionHandle quaternion,  CQRQuaternionHandle q1, CQRQuaternionHandle q2 );
    
     /*  CQRMultiply -- multiply a quaternion (q1) by quaternion (q2)  */
    
     int CQRMultiply (CQRQuaternionHandle quaternion,  CQRQuaternionHandle q1, CQRQuaternionHandle q2 );
    
     /*  CQRDot -- dot product of quaternion (q1) by quaternion (q2) as 4-vectors  */
    
     int CQRDot (double CQR_FAR * dotprod,  CQRQuaternionHandle q1, CQRQuaternionHandle q2 );   

     /*  CQRDivide -- Divide a quaternion (q1) by quaternion (q2)  */
    
     int CQRDivide (CQRQuaternionHandle quaternion,  CQRQuaternionHandle q1, CQRQuaternionHandle q2 );

     /*  CQRScalarMultiply -- multiply a quaternion (q) by scalar (s)  */
    
     int CQRScalarMultiply (CQRQuaternionHandle quaternion,  CQRQuaternionHandle q, double s );

     /*  CQREqual -- return 0 if quaternion q1 == q2  */
    
     int CQREqual (CQRQuaternionHandle q1, CQRQuaternionHandle q2 );
    
     /*  CQRConjugate -- Form the conjugate of a quaternion qconj */

     int CQRConjugate (CQRQuaternionHandle qconjugate, CQRQuaternionHandle quaternion);
    
     /*  CQRNormsq -- Form the normsquared of a quaternion */
    
     int CQRNormsq (double * normsq, CQRQuaternionHandle quaternion ) ;
    
     /*  CQRNorm -- Form the norm of a quaternion */
    
     int CQRNorm (double * norm, CQRQuaternionHandle quaternion ) ;

     /*  CQRDistsq -- Form the distance squared between two quaternions */
    
     int CQRDistsq (double CQR_FAR * distsq, CQRQuaternionHandle q1, CQRQuaternionHandle q2) ;
    
     /*  CQRDist -- Form the distance between two quaternions */
    
     int CQRDist (double CQR_FAR * dist, CQRQuaternionHandle q1, CQRQuaternionHandle q2 ) ;
    
     /*  CQRInverse -- Form the inverse of a quaternion */
    
     int CQRInverse (CQRQuaternionHandle inversequaternion, CQRQuaternionHandle quaternion );
    
     /* CQRRotateByQuaternion -- Rotate a vector by a Quaternion, w = qvq* */
    
     int CQRRotateByQuaternion(double * w, CQRQuaternionHandle rotquaternion, double * v);       
    
     /* CQRAxis2Quaternion -- Form the quaternion for a rotation around axis v  by angle theta */
    
     int CQRAxis2Quaternion (CQRQuaternionHandle rotquaternion, double * v, double theta);
    
     /* CQRMatrix2Quaterion -- Form the quaternion from a 3x3 rotation matrix R */
    
     int CQRMatrix2Quaternion (CQRQuaternionHandle rotquaternion, double R[3][3]);
    
     /* CQRQuaternion2Matrix -- Form the 3x3 rotation matrix from a quaternion */
    
     int CQRQuaternion2Matrix (double R[3][3], CQRQuaternionHandle rotquaternion);
    
     /* CQRQuaternion2Angles -- Convert a Quaternion into Euler Angles for Rz(Ry(Rx))) convention */
    
     int CQRQuaternion2Angles (double * RotX, double * RotY, double * RotZ, CQRQuaternionHandle rotquaternion);
    
     /* CQRAngles2Quaternion -- Convert Euler Angles for Rz(Ry(Rx))) convention into a quaternion */
    
     int CQRAngles2Quaternion (CQRQuaternionHandle rotquaternion, double RotX, double RotY, double RotZ );

     /* Represent a 3-vector as a quaternion with w=0 */
    
     int CQRPoint2Quaternion( CQRQuaternionHandle quaternion, double v[3] );
    
     /*  SLERP -- Spherical Linear Interpolation   */
    
     int CQRSLERP (CQRQuaternionHandle quaternion, const CQRQuaternionHandle q1, const CQRQuaternionHandle q2,
                   const double w1, const double w2);
    
     /*  HLERP -- Hemispherical Linear Interpolation   */
    
     int CQRHLERP (CQRQuaternionHandle quaternion, const CQRQuaternionHandle q1, const CQRQuaternionHandle q2,
                   const double w1, const double w2);
    
     /*  CQRSLERPDist -- Spherical Linear Interpolation distance */
    
     int CQRSLERPDist (double CQR_FAR * dist, const CQRQuaternionHandle q1, const CQRQuaternionHandle q2);
    
     /*  CQRHLERPDist -- Hemispherical Linear Interpolation distance */
    
     int CQRHLERPDist (double CQR_FAR * dist, const CQRQuaternionHandle q1, const CQRQuaternionHandle q2);


   and for C++

 template< typename DistanceType=double, typename VectorType=double[3], typename MatrixType=double[9] >
 class CPPQR
 {

 public:

      /* Constructors  */
          inline CPPQR( );  // default constructor
          inline CPPQR( const CPPQR& q ); // copy constructor
          inline CPPQR( const DistanceType& wi, const DistanceType& xi, const DistanceType& yi, const DistanceType& zi );

      /* Set -- set the values of an existing quaternion = w +ix+jy+kz */
          inline void Set ( const DistanceType& wi, const DistanceType& xi, const DistanceType& yi, const DistanceType& zi );

      /* Accessors */
          inline DistanceType GetW( void ) const;
          inline DistanceType GetX( void ) const;
          inline DistanceType GetY( void ) const;
          inline DistanceType GetZ( void ) const;
          inline CPPQR GetIm( void ) const;
          inline CPPQR GetAxis( void ) const;
          inline double GetAngle( void ) const;
         
      /* Operators */
          inline CPPQR operator+ ( const CPPQR& q ) const;
          inline CPPQR& operator+= ( const CPPQR& q );
          inline CPPQR& operator-= ( const CPPQR& q );
          inline CPPQR operator- ( const CPPQR& q ) const;
          inline CPPQR operator* ( const CPPQR& q ) const;
          inline CPPQR operator/ ( const CPPQR& q2 ) const;
          inline CPPQR operator* ( const DistanceType& d ) const;
          inline CPPQR operator/ ( const DistanceType& d ) const;
          inline CPPQR Conjugate ( void ) const;
          inline CPPQR& operator= ( const CPPQR& q );
          inline bool operator== ( const CPPQR& q ) const;
          inline bool operator!= ( const CPPQR& q ) const;
          inline VectorType& operator* ( const VectorType& v );
          DistanceType operator[] ( const int k ) const;

      /* Dot -- Dot product of 2 quaternions as 4-vectors */
          inline DistanceType Dot( const CPPQR& q) const;

      /* Normsq -- Form the normsquared of a quaternion */
          inline DistanceType Normsq ( void ) const;

      /* Norm -- Form the norm of a quaternion */
          inline DistanceType Norm ( void ) const;

      /* Distsq -- Form the distance squared from a quaternion */
          inline DistanceType Distsq ( const CPPQR& q ) const;

      /* Dist -- Form the distance from a quaternion */
          inline DistanceType Dist ( const CPPQR& q ) const;

      /* Inverse -- Form the inverse of a quaternion */
          inline CPPQR Inverse ( void ) const;
         
      /* log -- Get the natural logarithm of a quaternion */
          inline CPPQR log( void ) const;
         
      /* exp -- Get the exponential of a quaternion */
          inline CPPQR exp( void ) const;
         
      /* pow -- Take a power of a quaternion */
          template
          inline CPPQR pow( const powertype p) const;
          inline CPPQR pow( const int p) const;
         
      /* root -- Take an integer root of a quaternion */
          inline CPPQR root( const int r, const int m) const;

      /* RotateByQuaternion -- Rotate a vector by a Quaternion, w = qvq* */
          inline void RotateByQuaternion(VectorType &w, const VectorType v );
          inline VectorType& RotateByQuaternion( const VectorType v );

      /* Axis2Quaternion -- Form the quaternion for a rotation around axis v  by angle theta */
          static inline CPPQR Axis2Quaternion ( const DistanceType& angle, const VectorType v );
          static inline CPPQR Axis2Quaternion ( const VectorType v, const DistanceType& angle  );

      /* Matrix2Quaterion -- Form the quaternion from a 3x3 rotation matrix R */
          static inline void Matrix2Quaternion ( CPPQR& rotquaternion, const MatrixType m );
          static inline void Matrix2Quaternion ( CPPQR& rotquaternion, const DistanceType R[3][3] );

      /* Quaternion2Matrix -- Form the 3x3 rotation matrix from a quaternion */   
          static inline void Quaternion2Matrix( MatrixType& m, const CPPQR q );
          static inline void Quaternion2Matrix( DistanceType m[3][3], const CPPQR q );

      /* Get a unit quaternion from a general one */
          inline CPPQR UnitQ( void ) const;

      /* Quaternion2Angles -- Convert a Quaternion into Euler Angles for Rz(Ry(Rx))) convention */ 
          inline bool Quaternion2Angles ( DistanceType& rotX, DistanceType& rotY, DistanceType& rotZ ) const;

      /* Angles2Quaternion -- Convert Euler Angles for Rz(Ry(Rx))) convention into a quaternion */
          static inline CPPQR Angles2Quaternion ( const DistanceType& rotX, const DistanceType& rotY, const DistanceType& rotZ );
          static inline CPPQR Point2Quaternion( const DistanceType v[3] );

      /*  SLERP -- Spherical Linear Interpolation  */
          inline CPPQR SLERP (const CPPQR& q, DistanceType w1, DistanceType w2) const;

      /*  HLERP -- Hemispherical Linear Interpolation */
          inline CPPQR HLERP (const CPPQR& q, DistanceType w1, DistanceType w2) const;

      /*  SLERPDist -- Spherical Linear Interpolation distance */
          inline DistanceType SLERPDist (const CPPQR& q) const;

      /*  HLERPDist -- Hemispherical Linear Interpolation distance */
          inline DistanceType HLERPDist (const CPPQR& q) const;

         
     

 }; // end class CPPQR



    Description

   The cqrlib.h header file defines the CQRQuaternionHandle type as a pointer
   to a struct of the CQRQuaternion type:

     typedef struct {
         double w;
         double x;
         double y;
         double z; } CQRQuaternion;

   representing w + xi +yj + zk. A quaternion may be declared directly using
   the CQRQuaternion type or dynamically allocated by CQRCreateQuaternion or
   CQRCreateEmptyQuaternion, in which case it is a user responsibility to
   eventually free the allocated memory with CQRFreeQuaternion. The
   components of an existing quaternion may be set by CQRSetQuaternion.

   The rules of quaternion arithmetic are applied:

   -1 = i*i = j*j = k*k, i*j=k=-j*i, j*k=i=-j*k, k*i=j=-i*k

   by CQRAdd, CQRSubtract, CQRMultiply and CQRDivide. CQRScalarMultiply
   multiplies a quaternion by a scalar.

   CQREqual returns 0 if quaternion q1 == q2, component by component.
   CQRConjugate computes a quaternion with the same scalar component and the
   negative of the vector component. CQRNormsq computes the sum of the
   squares of the components. CQRInverse computes the inverse of a non-zero
   quaternion.

   The functions CQRGetQuaternionW, CQRGetQuaternionX, CQRGetQuaternionY and
   CQRGetQuaternionZ extract the 4 components of a quaternion. The function
   CQRGetQuaternionIm extract the imaginary part of a quaternion as a
   quaternion with w=0. The function CQRQGetQuaternion extracts the imaginary
   part and normalizes it to a unit vector. The function
   CQRGetQuaternionAngle extracts the angle for the polar representation of a
   quaternion as an exponential (see below).

   In handling rotations, a right-handed system is assumed.
   CQRRotateByQuaternion rotates a vector by a quaternion, w = qvq*.
   CQRAxis2Quaternion forms the quaternion for a rotation around axis v by
   angle theta. CQRMatrix2Quaterion forms the quaternion equivalent a 3x3
   rotation matrix R. CQRQuaternion2Matrix forms a 3x3 rotation matrix from a
   quaternion. CQRQuaternion2Angles converts a quaternion into Euler Angles
   for the Rz(Ry(Rx))) convention. CQRAngles2Quaternion convert Euler angles
   for the Rz(Ry(Rx))) convention into a quaternion.

   The logarithm of a quaternion in CQRLog is based on the polar
   representation

   q = r*cos(theta) + r*sin(theta) [ i*axis_x + j*axis_y +k*axis_z]
   = r*exp(theta*[ i*axis_x + j*axis_y +k*axis_z])

   with a unit axis. Then the natural logarithm is given by

   log(q) = log(r) + theta*[ i*axis_x + j*axis_y +k*axis_z])

   Note than any integer multiple of 2*PI could have been added to theta, so
   the logarithm is multivalued. The code only returns one of these values.
   The exponential in CQRExp is created by reversing the transformation.
   Taking a quaternion to a quaternion power is done by taking the log,
   multiplying by the power and then taking the exponential. Only one
   representative power is returned by CQRQuaternionPower. CQRDoublePower
   takes a quaternion to a double power by the same log-multiply-exp
   approach. CQRIntegerPower applies positive and negative integer powers by
   multiplication withou taking any logs or exponentials. CQRIntegerRoot
   applies the log-multiply-exp approach for integer roots. The second
   integer argument allow selection of one of the multiple roots. For roots
   of quaternions with a non-zero imaginary part, there are r roots, so m =
   0, 1, 2, ... r-1 are meaningful. For roots of reals, there can be
   infinitely many alternate roots. In the case, m will cycle first through
   the i-based roots, then the j-based roots and then the k-based roots.

   The SLERP and HLERP functions combine quaternions by speherical linear
   interpolation. SLERP take two quaternions and two weights and combine them
   following a great circle on the unit quaternion 4-D sphere and linear
   interpolation between the radii. SLERP keeps a quaternion separate from
   the negative of the same quaternion and is not appropriate for quaternions
   representing rotations. Use HLERP to apply SLERP to quaternions
   representing rotations.

   If operating with __cplusplus defined, then the CPPQR template is defined
   allowing the creation of CPPQR quaternion objects. The template has three
   typename arguments: DistanceType, VectorType and MatrixType that default
   to double, double[3] and double[9]. Specializations are provided to
   support a double[3][3] MatrixType.

    Returns

   The CQRlib functions return 0 for normal completion, or the sum of one or
   more of the following non-zero error codes:

     Error Return     Numeric Value    Meaning                                
     CQR_BAD_ARGUMENT    1             /* An argument is not valid */         
     CQR_NO_MEMORY       2             /* A call to allocate memory failed */ 
     CQR_FAILED          4             /* Operation failed */                 

    Examples

   To create a quaternion dynamically from memory, initialized as the x
   vector with a zero scalar value, reporting failure to stderr:

         #include <cqrlib.h>
         #include <stdio.h>
         ...
         CQRQuaternionHandle quathandle;
         ...
         if (CQRCreateQuaternion(&quathandle,0.,1.,0.,0.)) fprintf(stderr," CQRCreateQuaternion failed!!\n");

   To create an x vector quaternion, a y vector quaternion, add then together
   and multiply by a z-vector, and print the result :

         #include <cqrlib.h>
         #include <stdio.h>
         ...
         CQRQuaternion qx, qy, qz, qresult1, qresult2;
         ...
         if (CQRSetQuaternion(&qx,0.,1.,0.,0.)
           ||CQRSetQuaternion(&qy,0.,0.,1.,0.)
           ||CQRSetQuaternion(&qz,0.,0.,0.,1.)) fprintf(stderr," CQRSetQuaternion failed!!\n");
         if (CQRAdd(&qresult1,&qx,&qy)||CQRMultiply(&qresult2,&qresult1,&qz))
           fprintf(stderr," CQR Add or Multiply failed!!\n");
         fprintf(stdout,"Result = ((i+j)*k) = %g %+gi %+gj + %+gk\n",
           qresult2.w, qresult2.x, qresult2.y, qresult2.z);

   The output should be "Result = ((i+j)*k) = 0 +1i -1j +0k".

   To rotate the 3D vector [-1.,0.,1.] 90 degrees clockwise around the vector
   [1.,1.,1.]:

         #include <cqrlib.h>
         #include <math.h>
         #include <stdio.h>
         ...
         double axis[3] = {1.,1.,1.};
         double vector[3] = {-1.,0.,1.};
         double result[3];
         CQRQuaternion rotquat;
        
         double PI;
         PI = 4.*atan2(1.,1.);

         CQRAxis2Quaternion(&rotquat,axis,PI/2);
         CQRRotateByQuaternion(result, &rotquat, vector);
         ...
         fprintf(stdout," [-1.,0.,1.] rotated 90 degrees clockwise"
         " around the vector [1.,1.,1.] = [%g, %g, %g]\n",
         result[0], result[1], result[2]);

   The output should be "[-1.,0.,1.] rotated 90 degrees clockwise around the
   vector [1.,1.,1.] = [0.57735, -1.1547, 0.57735]".

   See the test program CQRlibTest.c.

   For examples of the use of the CPPQR template, see the C++ test program
   CPPQRTest.cpp.

     ----------------------------------------------------------------------

   Updated 29 April 2018
   yayahjb at gmail dot com