Path: utzoo!utgpu!news-server.csri.toronto.edu!rpi!zaphod.mps.ohio-state.edu!samsung!umich!sharkey!bnlux1.bnl.gov!bstewart
From: bstewart@bnlux1.bnl.gov (Bruce Stewart)
Newsgroups: comp.graphics.visualization
Subject: Re: surface rendering of fuzzy cloud
Message-ID: <1991May1.205240.28361@bnlux1.bnl.gov>
Date: 1 May 91 20:52:40 GMT
References: <2594@amethyst.math.arizona.edu> <73521@eerie.acsu.Buffalo.EDU> <1991Apr30.182711.2223@fido.wpd.sgi.com>
Distribution: usa
Organization: Brookhaven National Laboratory, Upton, NY 11973
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In article <1991Apr30.182711.2223@fido.wpd.sgi.com> robert@sgi.com writes:
>In article <73521@eerie.acsu.Buffalo.EDU>, okeefe@cs.Buffalo.EDU (Paul O'Keefe) writes:
>|> 
>|> In article <2594@amethyst.math.arizona.edu>, winfree@.math.arizona.edu (Art Winfree) writes:
>|> |> 
>|> |> Does anybody know how to render the surface of a fuzzy 3D cloud of
>|> |> data points? The "cloud" is actually supposed to be a 2D surface, but
>
>For some time I've wanted to construct and visualize the surface of a
>chaotic attractor, given an unordered list of points on the surface.
>

   IMHO this is an excellent example of a problem which needs some
thought as to how the problem should be posed before attempting to
find a solution.

   "The surface of a chaotic attractor" may or may not be a well-defined
concept, depending on the attractor. In some cases, notably systems
with very strong dissipation (= area or volume contraction rate), a
chaotic attractor may be very close to a branched manifold; the
Lorenz system and the Roessler band are examples. In other cases,
fractal structure is a prominent feature of an attractor; Ueda has
made high resolution pictures of some in this category, see for
example D. Ruelle's article in the Math. Intelligencer 1980,
pp.126-137; the Japanese attractor illustrated therein is in some sense
equivalent to a disk, but I doubt if any general algorithm would
produce anything close to a circle as an outer boundary of this
object, and I would mistrust any algorithm which did give such
an answer.

   (I am assuming that the outer boundary - whatever that is - of
this two-dimensional attractor is analogous to what is sought
by the poster for a three-dimensional attractor.)

   Better to ask why one would want to look at the "surface of
a chaotic attractor."  One might want to give a simple characterization
of the ensemble of phase space points which may be visited, for
example to give a convenient engineering bound on the maximum
displacement (or velocity or energy or strain or ...) which
can be expected.  This might lead to a formulation which
would point to an algorithm or class of algorithms. However,
approaching this via any finite sample of points would be very
dangerous, since even simple attractors like the Lorenz attractor
visit their outer limits extremely rarely.

   If one wants to "get an idea of the shape and structure" of
an attractor, the right way to proceed is to understand the
dynamics: find the unstable periodic points or unstable periodic
orbits of lowest period, and construct their unstable manifolds
(outsets). This is not always a cookbook procedure: for example
the Lorenz attrator has unstable periodic orbits AND unstable
fixed points; the two spiral points are not within the attractor,
but they are inside any simply connected hull, and they are
probably the objects whose outsets are most informative. On
the other hand, this approach is not (as it might seem)
limited to chaos techno-freaks with specialized software;
one purpose of my interactive software for Iris workstations
(and the accompanying documentation and tutorial exercises)
is to show that fixed points and invariant manifolds can be
constructed via primitive phase-space-geometric operations.
(End of advertisement.)

   I apologize for all this babble, which may be of little or
no intrinsic interest to many in this newsgroup. It was merely
intended to put some kind of fig leaf on the following opinion:
Looking for the surface of a chaotic attractor is probably
a misguided enterprise, and I would be wary of any conlcusions
reached in the process of trying to accomplish this.

   It appears that the question originally posted by Art Winfree
IS based on a well-posed problem, but it might have been
useful to give a more complete description of how the problem
arose and how a solution is judged good or not.

-Bruce Stewart