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QTERP

QTERP

## 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

Smoothly interpolate from a grid of quaternions (spline or slerp)

Geometry

## Calling Sequence

QNEW = QTERP(TGRID, QGRID, TNEW, [/SLERP], QDIFF=, [/RESET])

## Description

The function QTERP is used to interplate from a set of known unit
quaternions specified on a grid of independent values, to a new set
of independent values. For example, given a set of quaternions at
specified key times, QTERP can interpolate at any points between
those times. This has applications for computer animation and
spacecraft attitude control.
The "grid" of quaternions can be regularly or irregularly sampled.
The new values can also be regularly or irregularly sampled.
The simplest case comes when one wants to interpolate between two
quaternions Q1 and Q2. In that case the user should specify the
gridded quaterion as QGRID = [[Q1], [Q2]], with grid points at
TGRID = [0d, 1d]. Then the user can sample any intermediate
orientation by specifying TNEW anywhere between 0 and 1.
The user has the option of performing pure spline interpolation of
the quaternion components (the default technique). The resulting
interpolants are normalized to be unit quaternions. This option is
useful for fast interpolation of quaternions, but suffers if the
grid is not well sampled enough. Spline interpolation will not
strictly find the shortest path between two orientations.
The second option is to use Spherical Linear IntERPolation, or
SLERPing, to interpolate between quaternions (by specifying the
SLERP keyword). This technique is guaranteed to find the shortest
path between two orientations, but is somewhat slower than spline
interpolation. This approach involves computing a finite
difference of the data. To avoid repeated computation of the
difference on every call, users can pass a named variable in the
QDIFF keyword. This value can be reset with the RESET keyword.
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.
Users must have the VALUE_LOCATE() function, available either in
IDL 5.3 or later, or from the Markwardt web page.

## Inputs

TGRID - a vector of N independent variable values. In the
simplest case, this can be [0, 1, ...] up to the number of
quaternions in the grid. The grid sampling does not have
to be uniform.
QGRID - an 4xN array of unit quaternions specified on the grid.
TNEW - a vector of M desired independent variable values which
sample the grid specified by TGRID. The desired values do
not have to be uniformly sampled.

## Returns

A 4xM array of unit quaternions, where is M is the number of
desired samples.

## Keyword Parameters

SLERP - if set, then spherical linear interpolation is performed.
The default is to perform spline interpolation on the
quaternion coefficients.
QDIFF - upon return, QDIFF is filled with finite difference values
which can be used to speed computations in subsequent
calls. Users should be aware that QDIFF may be
inadvertently reused from one call to the next. When the
difference data should no longer be reused, the named
variable passed to the QDIFF keyword should be set to a
scalar, or the /RESET keyword should be used.
RESET - if set, then the QDIFF finite difference will be forced to
be recalculated, even if there is already data present and
passed to the QDIFF keyword.

## Example

This example starts with two quaternions representing rotations of
0 degrees and 45 degrees, and forms 1001 quaternions which are
smooth interpolations between 0 and 45 degrees.
;; Create a grid of two quaternions at times 0 and 1
Q0 = qtcompose([1,0,0], 0D) & T0 = 0D
Q1 = qtcompose([1,0,0], !dpi/4) & T1 = 1D
;; Put the grid elements into an array
TGRID = [T0, T1]
QGRID = [[Q0], [Q1]]
;; Make an array of 11 values smoothly varying from 0 to 1
TNEW = dindgen(11)/10d
;; Perform spherical linear interpolation
QNEW = QTERP(TGRID, QGRID, TNEW, /SLERP)
---> (interpolated results in QNEW)

0.0000000 0.0000000 0.0000000 1.0000000
0.039259816 0.0000000 0.0000000 0.99922904
0.078459096 0.0000000 0.0000000 0.99691733
0.11753740 0.0000000 0.0000000 0.99306846
0.15643447 0.0000000 0.0000000 0.98768834
0.19509032 0.0000000 0.0000000 0.98078528
0.23344536 0.0000000 0.0000000 0.97236992
0.27144045 0.0000000 0.0000000 0.96245524
0.30901699 0.0000000 0.0000000 0.95105652
0.34611706 0.0000000 0.0000000 0.93819134
0.38268343 0.0000000 0.0000000 0.92387953

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

## Modification History

Written, July 2001, CM
Documented, Dec 2001, CM
Usage message; check for 0- and 1-length quaternions; handle case
when quaternions are GE 180 degrees apart; handle case of
interpolating beyond end of known grid, 15 Mar 2002, CM
Use simplified QTMULT with /INV, 21 Sep 2007, CM
Added sample output, 29 Sep 2008, CM
Handle pathalogical case when some input quaternions were NAN,
2012-10-10, CM
\$Id: qterp.pro,v 1.9 2012/10/10 23:27:05 cmarkwar Exp \$

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