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CQRlib -- ANSI C API for Quaternion Rotations Release 1.1.2 21 December 2010 (c) 2008, 2009, 2010 Herbert J. Bernstein yaya at bernstein-plus-sons dot com You may distribute the CQRlib API under the LGPL 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 www.sourceforge.net/projects/cqrlib. A source tarball is available at downloads.sourceforge.net/cqrlib/CQRlib-1.1.1.tar.gz. Later tarballs may be available. When the source tarball is downloaded and unpacked, you should have a directory CQRlib-1.1.1 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: /Users/yaya/bin/libtool --mode=compile gcc -g -O2 -Wall -ansi -pedantic -I. -c /Users/yaya/bin/libtool --mode=compile g++ -g -O2 -Wall -ansi -pedantic -DCQR_NOCCODE=1 -I. -c The current library C and C++ link commands are: /Users/yaya/bin/libtool --mode=link gcc -version-info 3:0:1 -rpath /Users/yaya/lib /Users/yaya/bin/libtool --mode=link g++ -version-info 3:0:1 -rpath /Users/yaya/lib The current C library local, dynamic and static build commands are: /Users/yaya/bin/libtool --mode=link gcc -g -O2 -Wall -ansi -pedantic -I. /Users/yaya/bin/libtool --mode=link gcc -g -O2 -Wall -ansi -pedantic -dynamic -I /Users/yaya/include -L/Users/yaya/lib /Users/yaya/bin/libtool --mode=link gcc -g -O2 -Wall -ansi -pedantic -static -I /Users/yaya/include -L/Users/yaya/lib The current C++ template local, dynamic and static build commands are: /Users/yaya/bin/libtool --mode=link g++ -g -O2 -Wall -ansi -pedantic -DCQR_NOCCODE=1 -I. /Users/yaya/bin/libtool --mode=link g++ -g -O2 -Wall -ansi -pedantic -DCQR_NOCCODE=1 -dynamic -I /Users/yaya/include -L/Users/yaya/lib /Users/yaya/bin/libtool --mode=link g++ -g -O2 -Wall -ansi -pedantic -DCQR_NOCCODE=1 -static -I /Users/yaya/include -L/Users/yaya/lib Before installing the CQRlib library and example programs, check that the install directory and install commands are correct: The current values are : /usr/local /usr/local/bin/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 and 1.5.4. 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 21 December 2010 yaya at bernstein-plus-sons dot com |