$home/manuals/plan9_2nd/9/quaternion

QUATERNION(9.2)                                                QUATERNION(9.2)



NAME
       qtom,  mtoq,  qadd,  qsub,  qneg, qmul, qdiv, qunit, qinv, qlen, slerp,
       qmid, qsqrt - Quaternion arithmetic

SYNOPSIS
       #include <libg.h>

       #include <geometry.h>

       Quaternion qadd(Quaternion q, Quaternion r)

       Quaternion qsub(Quaternion q, Quaternion r)

       Quaternion qneg(Quaternion q)

       Quaternion qmul(Quaternion q, Quaternion r)

       Quaternion qdiv(Quaternion q, Quaternion r)

       Quaternion qinv(Quaternion q)

       double qlen(Quaternion p)

       Quaternion qunit(Quaternion q)

       void qtom(Matrix m, Quaternion q)

       Quaternion mtoq(Matrix mat)

       Quaternion slerp(Quaternion q, Quaternion r, double a)

       Quaternion qmid(Quaternion q, Quaternion r)

       Quaternion qsqrt(Quaternion q)

DESCRIPTION
       The Quaternions are a non-commutative extension field of the Real  num‐
       bers,  designed to do for rotations in 3-space what the complex numbers
       do for rotations in 2-space.  Quaternions have a real component  r  and
       an imaginary vector component v=(i,j,k).  Quaternions add componentwise
       and multiply according  to  the  rule  (r,v)(s,w)=(rs-v.w,  rw+vs+vÃw),
       where . and à are the ordinary vector dot and cross products.  The mul‐
       tiplicative inverse of a non-zero quaternion (r,v) is (r,-v)/(r2-v.v).

       The following routines do arithmetic on quaternions, represented as

              typedef struct Quaternion Quaternion;
              struct Quaternion{
                    double r, i, j, k;
              };

       Name   Description

       qadd   Add two quaternions.

       qsub   Subtract two quaternions.

       qneg   Negate a quaternion.

       qmul   Multiply two quaternions.

       qdiv   Divide two quaternions.

       qinv   Return the multiplicative inverse of a quaternion.

       qlen   Return sqrt(q.r*q.r+q.i*q.i+q.j*q.j+q.k*q.k), the  length  of  a
              quaternion.

       qunit  Return a unit quaternion (length=1) with components proportional
              to q's.

       A rotation by angle θ about axis A (where A is a unit vector)  can  be
       represented  by  the unit quaternion q=(cos θ/2, Asin θ/2).  The same
       rotation is represented by −q; a rotation by −θ about −A is  the  same
       as  a  rotation  by  θ about A.  The quaternion q transforms points by
       (0,x',y',z') = q-1(0,x,y,z)q.  Quaternion multiplication composes rota‐
       tions.  The orientation of an object in 3-space can be represented by a
       quaternion giving its rotation relative to some `standard' orientation.

       The following routines operate on rotations or orientations represented
       as unit quaternions:

       mtoq   Convert  a  rotation  matrix (see tstack(9.2)) to a unit quater‐
              nion.

       qtom   Convert a unit quaternion to a rotation matrix.

       slerp  Spherical lerp.  Interpolate between two orientations.  The  ro‐
              tation  that  carries  q  to  r  is  q-1r,  so slerp(q, r, t) is
              q(q-1r)t.

       qmid   slerp(q, r, .5)

       qsqrt  The square root of q.  This is just a rotation  about  the  same
              axis by half the angle.

SOURCE
       /sys/src/libgeometry/quaternion.c

SEE ALSO
       tstack(9.2), qball(9.2)



                                                               QUATERNION(9.2)