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 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 numbers, 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+v×w), where . and × are the ordinary vector dot and cross products. The multiplicative inverse of a non-zero quaternion (r,v) is (r,-v)/(r-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 rotations. 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 matrix((3))) 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 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

SEE ALSO
matrix((3)), qball((3))

BUGS
To avoid name conflicts with NetBSD, qdiv is a preprocessor macro defined as p9qdiv; see intro((3)).