@c DO NOT EDIT!  Generated automatically by munge-texi.

@c Copyright (C) 1996, 1997, 2007 John W. Eaton
@c This is part of the Octave manual.
@c For copying conditions, see the file gpl.texi.

@node Quaternions
@chapter Quaternions

Quaternions are hypercomplex numbers used to represent spatial
rotations in three dimensions.  This set of routines provides a useful
basis for working with quaternions in Octave.  A tutorial is in the
Octave source, scripts/quaternion/quaternion.ps.

These functions were written by A. S. Hodel, Associate Professor,
Auburn University.

@anchor{doc-quaternion}
@deftypefn {Function File} {[@var{a}, @var{b}, @var{c}, @var{d}] =} quaternion (w)
@deftypefnx {Function File} {[@var{vv}, @var{theta}] =} quaternion (w)
@deftypefnx {Function File} {@var{w} =} quaternion (@var{a}, @var{b}, @var{c}, @var{d})
@deftypefnx {Function File} {@var{w} =} quaternion (@var{vv}, @var{theta})
Construct or extract a quaternion

@example
w = a*i + b*j + c*k + d
@end example

@noindent
from given data.
@end deftypefn


@anchor{doc-qconj}
@deftypefn {Function File} {} qconj (@var{q})
Conjugate of a quaternion.

@example
q = [w, x, y, z] = w*i + x*j + y*k + z
qconj (q) = -w*i -x*j -y*k + z
@end example
@end deftypefn


@anchor{doc-qderiv}
@deftypefn {Function File} {} qderiv (omega)
Derivative of a quaternion.

Let Q be a quaternion to transform a vector from a fixed frame to
a rotating frame.  If the rotating frame is rotating about the 
[x, y, z] axes at angular rates [wx, wy, wz], then the derivative
of Q is given by

@example
Q' = qderivmat (omega) * Q
@end example

If the passive convention is used (rotate the frame, not the vector),
then

@example
Q' = -qderivmat (omega) * Q
@end example
@end deftypefn


@anchor{doc-qderivmat}
@deftypefn {Function File} {} qderivmat (@var{omega})
Derivative of a quaternion.

Let Q be a quaternion to transform a vector from a fixed frame to
a rotating frame.  If the rotating frame is rotating about the 
[x, y, z] axes at angular rates [wx, wy, wz], then the derivative
of Q is given by

@example
Q' = qderivmat (omega) * Q
@end example

If the passive convention is used (rotate the frame, not the vector),
then

@example
Q' = -qderivmat (omega) * Q.
@end example
@end deftypefn


@anchor{doc-qinv}
@deftypefn {Function File} {} qinv (@var{q})
Return the inverse of a quaternion.

@example
q = [w, x, y, z] = w*i + x*j + y*k + z
qmult (q, qinv (q)) = 1 = [0 0 0 1]
@end example
@end deftypefn


@anchor{doc-qmult}
@deftypefn {Function File} {} qmult (@var{a}, @var{b})
Multiply two quaternions.

@example
[w, x, y, z] = w*i + x*j + y*k + z
@end example

@noindent
identities:

@example
i^2 = j^2 = k^2 = -1
ij = k                 jk = i
ki = j                 kj = -i
ji = -k                ik = -j
@end example
@end deftypefn


@anchor{doc-qtrans}
@deftypefn {Function File} {} qtrans (@var{v}, @var{q})
Transform the unit quaternion @var{v} by the unit quaternion @var{q}.
Returns @code{@var{v} = @var{q}*@var{v}/@var{q}}.
@end deftypefn


@anchor{doc-qtransv}
@deftypefn {Function File} {} qtransv (@var{v}, @var{q})
Transform the 3-D vector @var{v} by the unit quaternion @var{q}.
Return a column vector.

@example
vi = (2*real(q)^2 - 1)*vb + 2*imag(q)*(imag(q)'*vb) 
   + 2*real(q)*cross(imag(q),vb)
@end example

@noindent
Where imag(q) is a column vector of length 3.
@end deftypefn


@anchor{doc-qtransvmat}
@deftypefn {Function File} {} qtransvmat (@var{qib})
Construct a 3x3 transformation matrix from quaternion @var{qib} that
is equivalent to rotation of th radians about axis @var{vv}, where
@code{[@var{vv}, @var{th}] = quaternion (@var{qib})}.
@end deftypefn


@anchor{doc-qcoordinate_plot}
@deftypefn {Function File} {} qcoordinate_plot (@var{qf}, @var{qb}, @var{qv})
Plot in the current figure a set of coordinate axes as viewed from 
the orientation specified by quaternion @var{qv}.  Inertial axes are
also plotted:

@table @var
@item qf
Quaternion from reference (x,y,z) to inertial.
@item qb
Quaternion from reference to body.
@item qv
Quaternion from reference to view angle.
@end table
@end deftypefn

