Quaternion math routines. More...

Collaboration diagram for Quaternion Math:

Functions
void	quat_mat2quat (quat_t quat, const mat_t mat)
	Convert Matrix to Quaternion. More...

void	quat_quat2mat (mat_t mat, const quat_t quat)
	Convert Quaternion to Matrix. More...

double	quat_distance (const quat_t q1, const quat_t q2)
	Gives the euclidean distance between two quaternions. More...

void	quat_double (quat_t qout, const quat_t q1, const quat_t q2)
	Gives the quaternion point representing twice the rotation from q1 to q2. More...

void	quat_bisect (quat_t qout, const quat_t q1, const quat_t q2)
	Gives the bisector of quaternions q1 and q2. More...

void	quat_slerp (quat_t qout, const quat_t q1, const quat_t q2, double f)
	Do Spherical Linear Interpolation between two unit quaternions by the given factor. More...

void	quat_sberp (quat_t qout, const quat_t q1, const quat_t qa, const quat_t qb, const quat_t q2, double f)
	Spherical Bezier Interpolate between four quaternions by amount f. These are intended to be used as start and stop quaternions along with two control quaternions chosen to match spline segments with first order continuity. More...

void	quat_make_nearest (quat_t q1, const quat_t q2)
	Set the quaternion q1 to the quaternion which yields the smallest rotation from q2 (of the two versions of q1 which produce the same orientation). More...

void	quat_print (const char *title, const quat_t quat)

void	quat_exp (quat_t out, const quat_t in)
	Exponentiate a quaternion, assuming that the scalar part is 0. Code by Ken Shoemake. More...

void	quat_log (quat_t out, const quat_t in)
	Take the natural logarithm of a unit quaternion. Code by Ken Shoemake. More...

Detailed Description

Quaternion math routines.

Unit Quaternions: Q = [ r, a ] where r = cos(theta/2) = rotation amount |a| = sin(theta/2) = rotation axis

If a = 0 we have the reals; if one coord is zero we have complex numbers (2D rotations).

[r, a][s, b] = [rs - a.b, rb + sa + axb]

-1

[r, a] = (r - a) / (r^2 + a.a)

Powers of quaternions yield incremental rotations, e.g. Q^3 is rotated three times as far as Q.

Some operations on quaternions: -1 [0, P'] = Q [0, P]Q Rotate a point P by quaternion Q -1 a slerp(Q, R, a) = Q(Q R) Spherical linear interp: 0 < a < 1

bisect(P, Q) = (P + Q) / |P + Q| Great circle bisector

Additions inspired by "Quaternion Calculus For Animation" by Ken Shoemake, SIGGRAPH '89 course notes for "Math for SIGGRAPH", May 1989.

Function Documentation

◆ quat_mat2quat()

void quat_mat2quat	(	quat_t	quat,
		const mat_t	mat
	)

Convert Matrix to Quaternion.

◆ quat_quat2mat()

void quat_quat2mat	(	mat_t	mat,
		const quat_t	quat
	)

Convert Quaternion to Matrix.

◆ quat_distance()

double quat_distance	(	const quat_t	q1,
		const quat_t	q2
	)

Gives the euclidean distance between two quaternions.

◆ quat_double()

void quat_double	(	quat_t	qout,
		const quat_t	q1,
		const quat_t	q2
	)

Gives the quaternion point representing twice the rotation from q1 to q2.

Needed for patching Bezier curves together. A rather poor name admittedly.

◆ quat_bisect()

void quat_bisect	(	quat_t	qout,
		const quat_t	q1,
		const quat_t	q2
	)

Gives the bisector of quaternions q1 and q2.

(Could be done with quat_slerp and factor 0.5) [I believe they must be unit quaternions this to work]

◆ quat_slerp()

void quat_slerp	(	quat_t	qout,
		const quat_t	q1,
		const quat_t	q2,
		double	f
	)

Do Spherical Linear Interpolation between two unit quaternions by the given factor.

As f goes from 0 to 1, qout goes from q1 to q2. Code based on code by Ken Shoemake

◆ quat_sberp()

void quat_sberp	(	quat_t	qout,
		const quat_t	q1,
		const quat_t	qa,
		const quat_t	qb,
		const quat_t	q2,
		double	f
	)

Spherical Bezier Interpolate between four quaternions by amount f. These are intended to be used as start and stop quaternions along with two control quaternions chosen to match spline segments with first order continuity.

Uses the method of successive bisection.

◆ quat_make_nearest()

void quat_make_nearest	(	quat_t	q1,
		const quat_t	q2
	)

Set the quaternion q1 to the quaternion which yields the smallest rotation from q2 (of the two versions of q1 which produce the same orientation).

Note that smallest euclidean distance implies smallest great circle distance as well (since surface is convex).

◆ quat_print()

void quat_print	(	const char *	title,
		const quat_t	quat
	)

◆ quat_exp()

void quat_exp	(	quat_t	out,
		const quat_t	in
	)

Exponentiate a quaternion, assuming that the scalar part is 0. Code by Ken Shoemake.

◆ quat_log()

void quat_log	(	quat_t	out,
		const quat_t	in
	)

Take the natural logarithm of a unit quaternion. Code by Ken Shoemake.

Functions