QUATERNION(2) QUATERNION(2)
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. Quater-
nions have a real component r and an imaginary vector compo-
nent 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)/(r2-v.v).
The following routines do arithmetic on quaternions, repre-
sented as
typedef struct Quaternion Quaternion;
struct Quaternion{
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QUATERNION(2) QUATERNION(2)
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 vec-
tor) can be represented by the unit quaternion q=(cos θ/2,
Asin θ/2). The same rotation is represented by -q; a rota-
tion 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 rota-
tions. The orientation of an object in 3-space can be rep-
resented 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 orienta-
tions. 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
/sys/src/libgeometry/quaternion.c
SEE ALSO
matrix(2), qball(2)
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