qtom, mtoq, qadd, qsub, qneg, qmul, qdiv, qunit, qinv, qlen, slerp,
qmid, qsqrt - Quaternion arithmetic
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 num-
bers, 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+vw), where
. and are the ordinary vector dot and cross products. The multiplica-
tive 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;
double r, i, j, k;
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
qunit Return a unit quaternion (length=1) with components proportional
A rotation by angle about axis A (where A is a unit vector) can be
represented by the unit quaternion q=(cos /2, Asin /2). The same rota-
tion is represented by -q; a rotation by - about -A is the same as a
rotation by 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 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 tstack(9.2)) to a unit quater-
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
qmid slerp(q, r, .5)
qsqrt The square root of q. This is just a rotation about the same
axis by half the angle.