! quaternions.cpl -- Copyright 2017 Paolo Luchini
! http://CPLcode.net/Applications/Numerical/Quaternions/
!
! CPL library providing a QUATERNION type and operations upon it.
!
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  of this software and associated documentation files (the "Software"), to deal
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SOFTWARE. !)

USE rbmat
TYPE QUATERNION=STRUCTURED ARRAY(REAL,IMAG,JMAG,KMAG) OF REAL

INLINE QUATERNION FUNCTION CONJG(QUATERNION x)
  RESULT.REAL=x.REAL
  RESULT.IMAG=-x.IMAG
  RESULT.JMAG=-x.JMAG
  RESULT.KMAG=-x.KMAG
END CONJG

QUATERNION FUNCTION qmult(QUATERNION a,b)
  RESULT.REAL=a.REAL∗b.REAL-a.IMAG∗b.IMAG-a.JMAG∗b.JMAG-a.KMAG∗b.KMAG
  RESULT.IMAG=a.IMAG∗b.REAL+a.REAL∗b.IMAG+a.JMAG∗b.KMAG-a.KMAG∗b.JMAG
  RESULT.JMAG=a.JMAG∗b.REAL+a.REAL∗b.JMAG+a.KMAG∗b.IMAG-a.IMAG∗b.KMAG
  RESULT.KMAG=a.KMAG∗b.REAL+a.REAL∗b.KMAG+a.IMAG∗b.JMAG-a.JMAG∗b.IMAG
END qmult

INLINE QUATERNION FUNCTION INV(QUATERNION x)=CONJG(x)/NORM(x)

! notice: vector-space operations such as addition and multiplication by 
! a scalar are provided by rbmat.cpl insofar as a QUATERNION is also a vector
OPERATOR QUATERNION x ∗ QUATERNION y = qmult(x,y)
OPERATOR QUATERNION x / QUATERNION y = qmult(x,INV(y))
OPERATOR REAL x / QUATERNION y = x∗INV(y)

QUATERNION FUNCTION EXP(QUATERNION x)
  REAL m=EXP(x.REAL)
  s=SQRT(x.IMAG^2+x.JMAG^2+x.KMAG^2)
  RESULT.REAL=m∗COS(s)
  IF s>1E-8 THEN m=m∗SIN(s)/s
  RESULT.IMAG=m∗x.IMAG
  RESULT.JMAG=m∗x.JMAG
  RESULT.KMAG=m∗x.KMAG
END EXP

QUATERNION FUNCTION LOG(QUATERNION x)
  REAL m=x.IMAG^2+x.JMAG^2+x.KMAG^2
  RESULT.REAL=0.5∗LOG(m+x.REAL^2)
  IF m>1E-16 THEN m=atan2(SQRT(m),x.REAL)/SQRT(m) ELSE m=1/x.REAL
  RESULT.IMAG=m∗x.IMAG
  RESULT.JMAG=m∗x.JMAG
  RESULT.KMAG=m∗x.KMAG
END LOG

INLINE QUATERNION FUNCTION POWER(QUATERNION x; REAL y)=EXP[LOG(x)∗y]

ARRAY(1..3,1..3) OF REAL FUNCTION rotation(QUATERNION x)
  u=x/ABS(x)
  RESULT(1,1)=1-2∗(u.JMAG^2+u.KMAG^2)
  RESULT(1,2)=2∗(u.IMAG∗u.JMAG-u.REAL∗u.KMAG)
  RESULT(1,3)=2∗(u.IMAG∗u.KMAG+u.REAL∗u.JMAG)
  RESULT(2,1)=2∗(u.IMAG∗u.JMAG+u.REAL∗u.KMAG)
  RESULT(2,2)=1-2∗(u.IMAG^2+u.KMAG^2)
  RESULT(2,3)=2∗(u.JMAG∗u.KMAG-u.REAL∗u.IMAG)
  RESULT(3,1)=2∗(u.IMAG∗u.KMAG-u.REAL∗u.JMAG)
  RESULT(3,2)=2∗(u.JMAG∗u.KMAG+u.REAL∗u.IMAG)
  RESULT(3,3)=1-2∗(u.IMAG^2+u.JMAG^2)
END rotation