[roseman at dimension4.math.uiowa.edu: higher dimensional geoms]

[mbp]

forwarded to software for logging purposes:

Date: Mon, 5 Oct 92 19:11:22 -0600
From: Dennis Roseman <roseman at dimension4.math.uiowa.edu>
To: techstaff at geom
Subject: higher dimensional geoms

To those that have some interest in the topic of higher dimensional
extensions of OOGL objects:

First of all I want to thank all on the staff who so patiently helped
me out in my recent visit to the Center.

Here are some thoughts on one particular topic generated from my
recent visit.


I would like once again to focus on what I see is a need to extend
oogl formats and geomview to higher dimensions than is currently
implemented.  I know it has been mentioned as a "future direction", so
it is not so much a question of whether it is a reasonable thing to
do, but more a matter of what is its priority relative to the many
other wonderful/important things that must/should get done.

Thus what I would like to do is to present my views on why I think it
would be useful and why I think others might find it useful also.


As some of you know, my interest is higher dimensions is due, in large
part, to my interest in knotted surfaces in four-dimensional space.
That still is my main motivation for wanting higher dimensional geoms.
But what I will speak about, here, is the application of a higher
dimensional point of view for use in THREE-DIMENSIONAL ANIMATIONS and
of possibilities of creating such animations in an INTERACTIVE WAY.
This topic is of interest to many at the Center.


I will focus on two possibilities: three-dimensional geoms in
four-dimensional space, and three-dimensional geoms in five
dimensional space.  But once I have finished I think it is clear that
there are other useful possibilities.


(Technically, perhaps I shouldn't really want to call three
dimensional object in four-dimensional space, etc, a "geom", but I
think you know what I mean.  The technicality is that if such a thing
were a geom object then--according to the object-oriented point of
view, it would have to have a "Draw" method defined for it, as
mentioned in section 3 of A Guide to Using geomview and OOGL.  I must
say that a *reasonable* general method *is* a problem.  However, I am
not really interested in this problem, for my interest is in drawing
only PARTS of such a geom, and these parts already have Draw methods.)

To give a feeling for the technique I have in mind, you might want to
look at an animation of hole* in my directory ~/oogl4d.  (Note: there
were some bad frames that I eliminated, so there hole.29, hole.30 &
hole.31 are "missing") The idea is to illustrate that "a torus with a
puncture looks like a disk with two bands attached".  This is to be
part of a larger animation I am planning.  The idea is to show an
expanding hole so that at the end, the two "bands" become evident.
This is a standard topic in introductions to topology.

The problem in doing this animation is that the "hole", as it expands,
has an irregular shape and is not particularly compatible with the
original polygons that make up the unpunctured torus.  It would be an
unpleasant task to do the calculations for each of the geoms in this
sequence.


The way I produced these was as follows.  I embedding the torus into
4-space by using a natural "bump function" to give the 4-th coordinate
of the place where the hole was to be made.  I then used the 4d-slicer
on this and cut the surface, in effect cutting off part of the bump.
The projection of the remaining surface into 3-space made by the
4d-viewer gave the geoms which were then saved to make the frames
hole*.


How did I know where to do the slices?  I just used the 4d-viewer and
interactively sliced off "about the same amount each time".  This was
easy and quick--if I didn't like the slice, I just wouldn't save it.

Six points I would like to make:

1) many thanks to Daeron for developing these wonderful tools
2) higher dimensions were used even thought the original problem did
   not call for this
3) a lot of calculational time/aggrivation was eliminated
4) there was an important interactive component.  The "bump function"
   I used was "quick & dirty"--it would have been a real calculational
   problem to CALCULATE which slices to take.  Interactively, I just did
   what I thought would look right.
5) This annimation, is still not exactly the way I want it to look--I
   think taking a different family of cutting (perhaps not even all
   parallel) hyperplanes would give a better animation.  Using
   interactive methods, it would not take too long to try it out.
6) The whole animation sequence was derived from ONE geom. (A
   different "bump" would be easy to calculate also, I might try this to
   get improvement.  One new bump would give a whole new set of frames.)


This is what can be done with existing tools--much more exciting
things could be done with extensions of these tools--extensions which
do not, I think take a lot of development effort.  These extensions
are a) defining higher dimensional geoms and b) higher dimensional
slicing methods.

I think it would be very useful to have higher dimensional geoms.  I
don't think one has to solve the problem of how to "draw" a
3-dimensional simplex , let alone a four dimensional one.  These
objects would be useful if we could (only) use them to calculate
two-dimensional geoms in 3-space.

I'm thinking here of say a homotopy of a surface in 3-space. The trace
of such a homotopy gives rise to a three-dimensional geom in
four-space.  Conversely, we can obtain a homotopy of a surface in
3-space by slicing an appropriate geom in 4-space.

A more advanced idea involves use of 4-dimensional geoms in 5-space to
give homotopies of surfaces in 3-space.  One could construct a
1-parameter family of homotopies which would then give such an object.
We would then slice this with a 3-dimensional subspace giving an image
of a surface in 3-space.

For example, suppose one wanted to describe an isotopy of a knotted
tube by giving an isotopy of the "center-circle" of the tube.  The
problem here is that in order to do this we might have to "shrink the
tube" as we follow the given isotopy of the center-circle in order to
avoid self intersections of the tube.  Later we might wish to expand
this tube.  The one-parameter family of isotopies would be determined
by the radius of the tube.  How to do the slices could be quickly done
in an interactive way.  I admit this is not a *very* interesting
application, but I'm just trying to illustrate the nature of the
approach.  Also, the problem of how to have the user input a
3-dimensional subspace of 5-space as well as how to generally display
the output is a non-trivial design problem.

I have some other ideas along these lines, but this note is getting
long enough.

Dennis Roseman, Department of Mathematics, 

University of Iowa, Iowa City, Iowa, USA (52242)
E-mail: roseman at dimension4.math.uiowa.edu, or 

        roseman at oak.math.uiowa.edu
Tel: Office ((555) 555-5555  Dept. ((555) 555-5555
Fax: ((555) 555-5555