;+ ; 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 \$ ; ;- ; Copyright (C) 2001, 2002, Craig Markwardt ; This software is provided as is without any warranty whatsoever. ; Permission to use, copy, modify, and distribute modified or ; unmodified copies is granted, provided this copyright and disclaimer ; are included unchanged. ;- function qtlog, q if n_params() EQ 0 then begin info = 1 USAGE: message, 'USAGE:', /info message, 'QNEW = QTLOG(Q)', info=info return, 0 endif nq = n_elements(q)/4 if nq LT 1 then goto, USAGE v = q(0:2,*) sinth = sqrt(total(v^2,1)) wh = where(sinth NE 0, ct) if ct GT 0 then v(*,wh) = v(*,wh)/rebin(reform(sinth(wh),1,ct),3,ct) costh = q(3,*) q1 = q q1(3,*) = 0 q1(0,0) = rebin(reform(atan(sinth,costh),1,nq),3,nq) * v return, q1 end