QUATERNION(9.2) 														   QUATERNION(9.2)

NAME

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

SYNOPSIS

       #include <libg.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.  Quaternions have a real component r and an imaginary vector component v=(i,j,k).  Quaternions add	component-
       wise  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 0 about axis A (where A is a unit vector) can be represented by the unit quaternion q=(cos 0/2, Asin  0/2).	 The  same
       rotation  is  represented  by  -q; a rotation by -0 about -A is the same as a rotation by 0 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 tstack(9.2)) 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)

       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

       tstack(9.2), qball(9.2)

																   QUATERNION(9.2)