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

#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)

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+vxw), where . and x are the ordinary vector dot and cross products. The multiplicative 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 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(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 q–1r, so slerp(q, r, t) is q(q–1r)t.
qmid    slerp(q, r, .5)
The square root of q. This is just a rotation about the same axis by half the angle.


matrix(2), qball(2)
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