# FreeBSD Manual Pages

```QUATERNION(3)		   Library Functions Manual		 QUATERNION(3)

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

SYNOPSIS
#include	<draw.h>

#include	<geometry.h>

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+vAw),
where . and A 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

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 _I, about axis A (where A is a unit vector)  can  be
represented  by	the unit quaternion q=(cos _I,/2, Asin _I,/2).  The same
rotation	is represented by -q; a	rotation by -_I,	about -A is  the  same
as  a  rotation	by  _I, 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 to	a unit quaternion.

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
/src/libgeometry/quaternion.c