;+ ; NAME: ; QTMULT ; ; 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: ; Multiply quaternions ; ; MAJOR TOPICS: ; Geometry ; ; CALLING SEQUENCE: ; QRESULT = QTMULT(Q1, [/INV1,] Q2, [/INV2]) ; ; DESCRIPTION: ; ; The function QTMULT performs multiplication of quaternions. ; Quaternion multiplication is not component-by-component, but ; rather represents the composition of two rotations, namely Q2 ; followed by Q1. ; ; More than one multiplication can be performed at one time if Q1 ; and Q2 are 4xN arrays. In that case both input arrays must be of ; the same dimension. ; ; If INV1 is set, then the inverse of Q1 is used. This is a ; convenience, to avoid the call QTINV(Q1). Of course, INV2 can ; be set to use the inverse of Q2. ; ; Note that quaternion multiplication is not commutative. ; ; 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: ; ; Q1 - array of one or more unit quaternions, the first operand in ; the multiplication. For a single quaternion, Q1 should be a ; 4-vector. For N quaternions, Q1 should be a 4xN array. ; If INV1 is set, then the inverse of Q1 is used. ; ; Q2 - same as Q1, for the second operand. ; If INV2 is set, then the inverse of Q2 is used. ; ; RETURNS: ; ; The resulting multiplied unit quaternions. For a single inputs, ; returns a 4-vector. For N input quaternions, returns N ; quaternions as a 4xN array. ; ; ; KEYWORD PARAMETERS: ; ; INV1 - if set, use QTINV(Q1) in place of Q1. ; ; INV2 - if set, use QTINV(Q2) in place of Q2. ; ; EXAMPLE: ; ; Q1 = qtcompose([0,0,1], 32d*!dpi/180d) ; Q2 = qtcompose([1,0,0], 116d*!dpi/180d) ; ; IDL> print, qtmult(q1, q2) ; 0.81519615 0.23375373 0.14606554 0.50939109 ; ; Form a rotation quaternion of 32 degrees around the Z axis, and ; 116 degrees around the X axis, then multiply the two quaternions. ; ; SEE ALSO ; QTANG, QTAXIS, QTCOMPOSE, QTERP, QTEXP, QTFIND, QTINV, QTLOG, ; QTMAT, QTMULT, QTMULTN, QTPOW, QTVROT ; ; MODIFICATION HISTORY: ; Written, July 2001, CM ; Documented, Dec 2001, CM ; Documentation, allow 1xN or Nx1 multiplies, 27 Jan 2002, CM ; Usage message, error checking, 15 Mar 2002, CM ; Add the INV1 and INV2 keywords, 30 Aug 2007, CM ; ; \$Id: qtmult.pro,v 1.8 2007/09/03 07:18:25 craigm Exp \$ ; ;- ; Copyright (C) 2001, 2002, 2007, 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 qtmult, aqt, bqt, inv1=inverse1, inv2=inverse2 ; THIS ROUTINE MULTIPLIES QUATERNIONS ; CQT CORRESPONDS TO THE ROTATION AQT FOLLOWED BY BQT ; ASSUMING S/C COORDINATES ARE INITIALLY ALIGN WITH INERTIAL COORD. ; THEN ROTATION AQT DESCRIBES ROTATION SUCH THAT THE SUBROUTINE ; QTXRA GIVES THE INERTIAL COORDINATES OF THE S/C X-AXIS ; THE FIRST 3 COMPONENTS OF AQT GIVE THE EIGENAXIS EXPRESSED ; IN S/C COORDINATES BEFORE THE ROTATION (=INTERTIAL COORD.). ; THE BQT ROTATION FOLLOWS THE AQT ROTATION. CQT THEN DESCRIBES ; THIS COMBINATION SUCH THAT QTXRA GIVES THE INERTIAL COORDINATES ; OF THE S/C X-AXIS AFTER BOTH ROTATIONS. ; THE FIRST 3 COMPONENTS OF BQT GIVE THE EIGENAXIS EXPRESSED ; IN S/C COORDINATES AFTER THE AQT ROTATION. if n_params() EQ 0 then begin info = 1 USAGE: message, 'USAGE:', /info message, 'QNEW = QTMULT(Q1, Q2)', info=info return, 0 endif sz1 = size(aqt) sz2 = size(bqt) if sz1(0) LT 1 OR sz2(0) LT 1 then \$ message, 'ERROR: Q1 and Q2 must be quaternions' if sz1(1) NE 4 OR sz2(1) NE 4 then \$ message, 'ERROR: Q1 and Q2 must be quaternions' n1 = n_elements(aqt)/4 n2 = n_elements(bqt)/4 if n1 NE n2 AND n1 NE 1 AND n2 NE 1 then \$ message, 'ERROR: Q1 and Q2 must both have the same number of quaternions' nq = n1>n2 cqt = make_array(value=aqt(0)*bqt(0)*0, dimension=[4,nq]) if n1 GT 1 then begin aqt0 = aqt(0,*) & aqt1 = aqt(1,*) & aqt2 = aqt(2,*) & aqt3 = aqt(3,*) endif else begin aqt0 = aqt(0) & aqt1 = aqt(1) & aqt2 = aqt(2) & aqt3 = aqt(3) endelse if n2 GT 1 then begin bqt0 = bqt(0,*) & bqt1 = bqt(1,*) & bqt2 = bqt(2,*) & bqt3 = bqt(3,*) endif else begin bqt0 = bqt(0) & bqt1 = bqt(1) & bqt2 = bqt(2) & bqt3 = bqt(3) endelse if keyword_set(inverse1) then begin aqt0 = -aqt0 & aqt1 = -aqt1 & aqt2 = -aqt2 endif if keyword_set(inverse2) then begin bqt0 = -bqt0 & bqt1 = -bqt1 & bqt2 = -bqt2 endif CQT(0,0) = AQT0*BQT3 + AQT1*BQT2 - AQT2*BQT1 + AQT3*BQT0 CQT(1,0) =-AQT0*BQT2 + AQT1*BQT3 + AQT2*BQT0 + AQT3*BQT1 CQT(2,0) = AQT0*BQT1 - AQT1*BQT0 + AQT2*BQT3 + AQT3*BQT2 CQT(3,0) =-AQT0*BQT0 - AQT1*BQT1 - AQT2*BQT2 + AQT3*BQT3 return, cqt end