```! quaternions.cpl -- Copyright 2017 Paolo Luchini
! http://CPLcode.net/Applications/Numerical/Quaternions/
!
! CPL library providing a QUATERNION type and operations upon it.
!
!( Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
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
```