;+ ; NAME: ; QTEXP ; ; 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 "exponentiation" of a non-unit quaternion ; ; MAJOR TOPICS: ; Geometry ; ; CALLING SEQUENCE: ; Q = QTEXP(QLOG) ; ; DESCRIPTION: ; ; The function QTEXP computes the "exponentiation" of a quaternion. ; ; The exponential is only defined for a non-unit quaternion with a ; *zero* rotation angle. Specifically, the expression ; ; QTEXP([VAXIS * PHI/2, 0]) ; ; becomes ; ; [VAXIS*SIN(PHI/2), COS(PHI/2)] ; ; where VAXIS is the unit vector rotation axis and PHI is the ; rotation angle. Note that since VAXIS is a unit vector, the ; product VAXIS*PHI can have an arbitrary direction and magnitude. ; ; Typically the input to QTEXP is found by taking the logarithm of a ; unit quaternion using QTLOG, 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: ; ; QLOG - a non-unit quaternion of the form [VX, VY, VZ, 0]; or, ; N quaternions of the same form, as a 4xN array. ; ; ; RETURNS: ; ; The exponentiated unit quaternion(s). For a single input ; quaternion, returns a 4-vector; for N input quaternions, returns a ; 4xN array. ; ; ; KEYWORD PARAMETERS: ; ; NONE ; ; EXAMPLE: ; ; IDL> print, qtlog(qtcompose([0d,1,0], !dpi/4)) ; 0.0000000 0.39269908 0.0000000 0.0000000 ; ; Prints the logarithm of the quaternion composed of a rotation of ; !dpi/4 radians around the axis [0,1,0] ; ; ; 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 corrected, 27 Jan 2002, CM ; Usage message, error checking, 15 Mar 2002, CM ; ; \$Id: qtexp.pro,v 1.7 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 qtexp, q if n_params() EQ 0 then begin info = 1 USAGE: message, 'USAGE:', /info message, 'QNEW = QTEXP(Q)', info=info return, 0 endif nq = n_elements(q)/4 if nq LT 1 then goto, USAGE v = q(0:2,*) th = sqrt(total(v^2,1)) wh = where(th NE 0, ct) if ct GT 0 then v(*,wh) = v(*,wh)/rebin(reform(th(wh),1,ct),3,ct) q1 = q q1(3,*) = cos(th) q1(0:2,*) = rebin(reform([sin(th)],1,nq),3,nq)*v return, q1 end