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QTLOG

QTLOG

Name


  QTLOG

Author


  Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
  craigm@lheamail.gsfc.nasa.gov
  UPDATED VERSIONs can be found on my WEB PAGE:
      http://cow.physics.wisc.edu/~craigm/idl/idl.html

Purpose


  Compute "logarithm" of a unit quaternion

Major Topics


  Geometry

Calling Sequence


  QLOG = QTLOG(Q)

Description



  The function QTLOG computes the "logarithm" of a unit quaternion.
  The logarithm of a quaternion is defined for any unit quaternion,
  such that the expression
    QTLOG([VAXIS*SIN(PHI/2), COS(PHI/2)]
  becomes
    [VAXIS * PHI/2, 0]
  where VAXIS is the unit vector rotation axis and PHI is the
  rotation angle. Note that the output quaternion is not a *unit*
  quaternion. The output of QTLOG is also commonly known as an
  *axial vector*, for a rotation axis VAXIS and rotation angle
  PHI/2.
  Typically the output to QTLOG is eventually exponentiated with the
  QTEXP function, and the identity QTEXP(QTLOG(Q)) is the same as Q.
  Conventions for storing quaternions vary in the literature and from
  library to library. This library uses the convention that the
  first three components of each quaternion are the 3-vector axis of
  rotation, and the 4th component is the rotation angle. Expressed
  in formulae, a single quaternion is given by:
    Q(0:2) = [VX, VY, VZ]*SIN(PHI/2)
    Q(3) = COS(PHI/2)
  where PHI is the rotation angle, and VAXIS = [VX, VY, VZ] is the
  rotation eigen axis expressed as a unit vector. This library
  accepts quaternions of both signs, but by preference returns
  quaternions with a positive 4th component.

Inputs



  Q - array of one or more unit quaternions. For a single
      quaternion, Q should be a 4-vector. For N quaternions, Q
      should be a 4xN array.

Returns



  The non-unit quaternion logarithm(s). For a single input
  quaternion, returns a 4-vector of the form [VX, VY, VZ, 0]. For N
  input quaternions, returns N quaternions of the same form as a 4xN
  array.

Keyword Parameters



  NONE

Example



  IDL> q = qtlog(qtcompose([0d,1,0], !dpi/4))
  IDL> print, qtexp(2 * q)
          0.0000000 0.70710678 0.0000000 0.70710678
  First, computes the logarithm Q of the quaternion composed of a
  rotation of !dpi/4 radians around the axis [0,1,0]. Second,
  computes the exponentiation of 2*Q. This is the same as raising
  the original quaternion to the second power.

See Also


  QTANG, QTAXIS, QTCOMPOSE, QTERP, QTEXP, QTFIND, QTINV, QTLOG,
  QTMAT, QTMULT, QTPOW, QTVROT

Modification History


  Written, July 2001, CM
  Documented, Dec 2001, CM
  Documentation clarified & corrected, 27 Jan 2002, CM
  Usage message, error checking, 15 Mar 2002, CM
  $Id: qtlog.pro,v 1.6 2002/05/09 23:03:27 craigm Exp $



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