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.TH QUATERNION 2

.SH NAME

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

.SH SYNOPSIS

.B

#include <draw.h>

.br

.B

#include <geometry.h>

.PP

.B

Quaternion qadd(Quaternion q, Quaternion r)

.PP

.B

Quaternion qsub(Quaternion q, Quaternion r)

.PP

.B

Quaternion qneg(Quaternion q)

.PP

.B

Quaternion qmul(Quaternion q, Quaternion r)

.PP

.B

Quaternion qdiv(Quaternion q, Quaternion r)

.PP

.B

Quaternion qinv(Quaternion q)

.PP

.B

double qlen(Quaternion p)

.PP

.B

Quaternion qunit(Quaternion q)

.PP

.B

void qtom(Matrix m, Quaternion q)

.PP

.B

Quaternion mtoq(Matrix mat)

.PP

.B

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

.PP

.B

Quaternion qmid(Quaternion q, Quaternion r)

.PP

.B

Quaternion qsqrt(Quaternion q)

.SH DESCRIPTION

The Quaternions are a non-commutative extension field of the Real numbers, designed

to do for rotations in 3-space what the complex numbers do for rotations in 2-space.

Quaternions have a real component

.I r

and an imaginary vector component \fIv\fP=(\fIi\fP,\fIj\fP,\fIk\fP).

Quaternions add componentwise and multiply according to the rule

(\fIr\fP,\fIv\fP)(\fIs\fP,\fIw\fP)=(\fIrs\fP-\fIv\fP\v'-.3m'.\v'.3m'\fIw\fP, \fIrw\fP+\fIvs\fP+\fIv\fP×\fIw\fP),

where \v'-.3m'.\v'.3m' and × are the ordinary vector dot and cross products.

The multiplicative inverse of a non-zero quaternion (\fIr\fP,\fIv\fP)

is (\fIr\fP,\fI-v\fP)/(\fIr\^\fP\u\s-22\s+2\d-\fIv\fP\v'-.3m'.\v'.3m'\fIv\fP).

.PP

The following routines do arithmetic on quaternions, represented as

.IP

.EX

.ta 6n

typedef struct Quaternion Quaternion;

struct Quaternion{

	double r, i, j, k;

};

.EE

.TF qunit

.TP

Name

Description

.TP

.B qadd

Add two quaternions.

.TP

.B qsub

Subtract two quaternions.

.TP

.B qneg

Negate a quaternion.

.TP

.B qmul

Multiply two quaternions.

.TP

.B qdiv

Divide two quaternions.

.TP

.B qinv

Return the multiplicative inverse of a quaternion.

.TP

.B qlen

Return

.BR sqrt(q.r*q.r+q.i*q.i+q.j*q.j+q.k*q.k) ,

the length of a quaternion.

.TP

.B qunit

Return a unit quaternion 

.RI ( length=1 )

with components proportional to

.IR q 's.

.PD

.PP

A rotation by angle \fIθ\fP about axis

.I A

(where

.I A

is a unit vector) can be represented by

the unit quaternion \fIq\fP=(cos \fIθ\fP/2, \fIA\fPsin \fIθ\fP/2).

The same rotation is represented by \(mi\fIq\fP; a rotation by \(mi\fIθ\fP about \(mi\fIA\fP is the same as a rotation by \fIθ\fP about \fIA\fP.

The quaternion \fIq\fP transforms points by

(0,\fIx',y',z'\fP) = \%\fIq\fP\u\s-2-1\s+2\d(0,\fIx,y,z\fP)\fIq\fP.

Quaternion multiplication composes rotations.

The orientation of an object in 3-space can be represented by a quaternion

giving its rotation relative to some `standard' orientation.

.PP

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

.TF slerp

.TP

.B mtoq

Convert a rotation matrix (see

.IR matrix (2))

to a unit quaternion.

.TP

.B qtom

Convert a unit quaternion to a rotation matrix.

.TP

.B slerp

Spherical lerp.  Interpolate between two orientations.

The rotation that carries

.I q

to

.I r

is \%\fIq\fP\u\s-2-1\s+2\d\fIr\fP, so

.B slerp(q, r, t)

is \fIq\fP(\fIq\fP\u\s-2-1\s+2\d\fIr\fP)\u\s-2\fIt\fP\s+2\d.

.TP

.B qmid

.B slerp(q, r, .5)

.TP

.B qsqrt

The square root of

.IR q .

This is just a rotation about the same axis by half the angle.

.PD

.SH SOURCE

.B /sys/src/libgeometry/quaternion.c

.SH SEE ALSO

.IR matrix (2),

.IR qball (2)