;+ ; NAME: ; QTVROT ; ; 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: ; Apply quaternion rotation to a 3-vector ; ; MAJOR TOPICS: ; Geometry ; ; CALLING SEQUENCE: ; VNEW = QTVROT(V, Q, [/INVERT]) ; ; DESCRIPTION: ; ; The function QTVROT applies a quaternion rotation (or its inverse) ; to a 3-vector V to produce a new vector VNEW. ; ; If both V and VNEW are vector components measured in the same ; inertial coordinate system, then VNEW returns the components of ; the vector V rotated by quaternion Q. I.e., the AXES stay fixed ; and the VECTOR rotates. Replace Q by QTINV(Q) in the case of ; /INVERT. ; ; If V are components of a vector measured in the "body" coordinate ; frame, and Q represents the orientation of the body frame ; w.r.t. the inertial frame, then VNEW are the components of the ; same vector in the inertial frame. I.e., the VECTOR stays fixed ; and the AXES rotate. For /INVERT, the coordinate transformation ; is from inertial frame to body frame. ; ; If either Q is a single quaternion, or V is a single 3-vector, ; then QTVROT will expand the single to the number of elements of ; the other operand. Otherwise, the number of quaternions and ; vectors must be equal. ; ; 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: ; ; V - array of one or more 3-vectors. For a single vector, V should ; be a 3-vector. For N vectors, V should be a 3xN array. ; ; 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 resulting rotated vectors. For single inputs, returns a ; 3-vector. For N inputs, returns N vectors as a 3xN array. ; ; ; KEYWORD PARAMETERS: ; ; INVERT - if set, then the antirotation represented by QTINV(Q) is ; performed. ; ; ; EXAMPLE: ; ; Q1 = qtcompose([0,0,1], 32d*!dpi/180d) ; Q2 = qtcompose([1,0,0], 116d*!dpi/180d) ; Q = qtmult(Q1, Q2) ; ; V = [[1d,0,0],[0,1,0],[0,0,1]] ; ; IDL> print, qtvrot(v, q) ; 0.84804810 0.52991926 0.0000000 ; 0.23230132 -0.37175982 0.89879405 ; 0.47628828 -0.76222058 -0.43837115 ; ; ; SEE ALSO ; QTANG, QTAXIS, QTCOMPOSE, QTERP, QTEXP, QTFIND, QTINV, QTLOG, ; QTMAT, QTMULT, QTPOW, QTVROT ; ; MODIFICATION HISTORY: ; Written, July 2001, CM ; Documented, Dec 2001, CM ; Small changes, 28 Jan 2002, CM ; Usage message, error checking, 15 Mar 2002, CM ; ; \$Id: qtvrot.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. ;- ;; QVROT ;; ;; The FORWARD (default) transform: ;; ;; * takes a vector vin (components given in inertial coordinates) and ;; returns the components of the rotated vector vout (components ;; given in inertial coordinates) -- ie, the AXES stay fixed and the ;; VECTOR rotates; OR, equivalently, ;; ;; * takes a fixed vector vin (components given in body coordinates) ;; and returns the components of the vector in inertial coordinates, ;; where the body system is described by quaternion q -- ie, the ;; VECTOR stays fixed and the AXES rotate. ;; ;; ;; The INVERSE transform (gotten by setting /INVERT): ;; ;; * takes a vector vin (components given in inertial coordinates) and ;; returns the components of the anti-rotated vector vout ;; (components given in inertial coordinates) -- ie, the AXES stay ;; fixed and the VECTOR rotates. Anti-rotated here means rotated in ;; the opposite direction of q; OR, equivalently, ;; ;; * takes a fixed vector vin (components given in inertial ;; coordinates) and returns the components of the vector in body ;; coordinates, where the body system is described by quaternion q ;; -- ie, the VECTOR stays fixed and the AXES rotate. ;; function qtvrot, vin, q, invert=invert if n_params() EQ 0 then begin info = 1 USAGE: message, 'USAGE:', /info message, 'VNEW = QTVROT(V, Q)', info=info return, 0 endif nq = n_elements(q)/4 nv = n_elements(vin)/3 if nq LT 1 OR nv LT 1 then goto, USAGE if n_elements(q) GT 4 AND n_elements(vin) GT 3 then begin if n_elements(q)/4 NE n_elements(vin)/3 then begin message, 'ERROR: incompatible number of quaternions & vectors' return, -1L end vout = vin*q(0)*0. nq = n_elements(q)/4 nv = nq endif else if n_elements(q) GT 4 then begin nq = n_elements(q)/4 nv = 1L vout = vin(*) # (fltarr(nq)+1) * q(0)*0. endif else begin nq = 1L nv = n_elements(vin)/3 vout = vin*q(0)*0. endelse vout = reform(vout, 3, max([nv,nq]), /overwrite) q1 = q(0,*) & q2 = q(1,*) & q3 = q(2,*) & q4 = q(3,*) if n_elements(q1) EQ 1 then begin q1 = q1(0) & q2 = q2(0) & q3 = q3(0) & q4 = q4(0) endif else begin q1 = q1(*) & q2 = q2(*) & q3 = q3(*) & q4 = q4(*) endelse v0 = vin(0,*) & v1 = vin(1,*) & v2 = vin(2,*) if n_elements(v0) EQ 1 then begin v0 = v0(0) & v1 = v1(0) & v2 = v2(0) endif else begin v0 = v0(*) & v1 = v1(*) & v2 = v2(*) endelse if NOT keyword_set(INVERT) then begin ;; FORWARD TRANSFORMATION VOUT(0,*)=((Q1*Q1-Q2*Q2-Q3*Q3+Q4*Q4)*V0 \$ + 2.D0*(Q1*Q2-Q3*Q4)*V1 \$ + 2.D0*(Q1*Q3+Q2*Q4)*V2) VOUT(1,*)=(2.D0*(Q1*Q2+Q3*Q4)*V0 \$ + (-Q1*Q1+Q2*Q2-Q3*Q3+Q4*Q4)*V1 \$ + 2.D0*(Q2*Q3-Q1*Q4)*V2) VOUT(2,*)=(2.D0*(Q1*Q3-Q2*Q4)*V0 \$ + 2.D0*(Q2*Q3+Q1*Q4)*V1 \$ + (-Q1*Q1-Q2*Q2+Q3*Q3+Q4*Q4)*V2) endif else begin ;; INVERSE TRANSFORMATION VOUT(0,*)=((Q1*Q1-Q2*Q2-Q3*Q3+Q4*Q4)*V0 \$ + 2.D0*(Q1*Q2+Q3*Q4)*V1 \$ + 2.D0*(Q1*Q3-Q2*Q4)*V2) VOUT(1,*)=(2.D0*(Q1*Q2-Q3*Q4)*V0 \$ + (-Q1*Q1+Q2*Q2-Q3*Q3+Q4*Q4)*V1 \$ + 2.D0*(Q2*Q3+Q1*Q4)*V2) VOUT(2,*)=(2.D0*(Q1*Q3+Q2*Q4)*V0 \$ + 2.D0*(Q2*Q3-Q1*Q4)*V1 \$ + (-Q1*Q1-Q2*Q2+Q3*Q3+Q4*Q4)*V2) endelse vout = vout return, vout end