
QUATERNION(9.2) QUATERNION(9.2)
NAME
qtom, mtoq, qadd, qsub, qneg, qmul, qdiv, qunit, qinv, qlen, slerp, qmid, qsqrt  Quater
nion 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 noncommutative extension field of the Real numbers, designed to do
for rotations in 3space what the complex numbers do for rotations in 2space. Quater
nions 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)=(rsv.w, rw+vs+vxw), where
. and x are the ordinary vector dot and cross products. The multiplicative inverse of a
nonzero quaternion (r,v) is (r,v)/(r2v.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 0 about axis A (where A is a unit vector) can be represented by the
unit quaternion q=(cos 0/2, Asin 0/2). The same rotation is represented by q; a rotation
by 0 about A is the same as a rotation by 0 about A. The quaternion q transforms points
by (0,x',y',z') = q1(0,x,y,z)q. Quaternion multiplication composes rotations. The ori
entation of an object in 3space 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 quater
nions:
mtoq Convert a rotation matrix (see tstack(9.2)) to a unit quaternion.
qtom Convert a unit quaternion to a rotation matrix.
slerp Spherical lerp. Interpolate between two orientations. The rotation that carries q
to r is q1r, so slerp(q, r, t) is q(q1r)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) 
