Alejandro Saez (cano++at++krusty.engr.sgi.com)
Tue, 15 Sep 1998 18:10:12 -0500
> > One final word about splines. If you are using splines for motion
description,
> > most splines, unlike linear interpolation, are not uniformly spaced, that
is if
> > you evaluate the spline at 4 different places but using the same delta u,
you
> > won't get the same delta distance ...
>
> Yes, that is a problem. My original intention was to allow splines to be
> "deparameterized", by subdividing them into nearly linear segments and
> calculating the distance travelled, as you suggest. But if you do still
> have those papers about redefining a spline for constant speed, I would
> be interested in reading them....
>
> Thanks,
> Geoff
>
> --
> Geoff Levner -- geoff++at++mclink.it OR glevner++at++hotmail.com
> ACS Studio, Via Aurelia 58, 00165 Rome, Italy
>
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>-- End of excerpt from Geoff Levner
Geoff,
It took me quite a time to find the bookmarks (both in my head and in my
browser). But here it is.
The basic problem when using splines to do motion planning is that splines are
used to define positions but they don't say anything about speed. Michael
Boccara's tip is lacking that too, it ensures that initial speed and position
will be preserved but doesn't say anything about in-between speeds (or
positions) and still doesn't solve the problem of knowing which u value give us
a delta X traveled distance in delta t time (the scenario when doing real time
motion simulation). What's true, though, is you could use Micheal's approach
over tiny sub-segments of the original spline.
The solution is reparameterization by arc length which means to guarantee that
the variance in t is directly related to the distance traveled along the curve.
A general description on this subject plus other motion planing subjects such
as how to describe the speed behaviour when interpolating positions can be
found at:
http://www.cis.ohio-state.edu/~parent/book/outline.html
This is an online book called COMPUTER ANIMATION: ALGORITHMS AND TECHNIQUES by
a Rick Parent. Check out chapter 4: Aids to Motion Specification. It's only a
glimpse at the matter and makes a lot of references to printed books. The good
news are that reparameterization is actually an integral (arc length) which is
(can be) solved by Gaussian quadrature as the book suggets, and Cambridge
University Press has made on-line availabe it's numerical recepies book
covering (among others) Gaussian Quadrature!! Actually, this is the reading
Parent recomends using to get details on the subject. The url for this book is:
That's all. Hope this helps... and tell me how it works :), I never had the
time to implement it and went for the linear approach.
--
------------------------------------------------------------------------
Alejandro Saez
Software Engineer
Silicon Chile S.A.
Avda. Santa Maria 2560
E-mail: asaez++at++silicon.cl Providencia
Phone: +56 (2) 203 3371 Ext. 107 Santiago
Fax: +56 (2) 203 3370 Chile
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