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)
