The Double Locus Surface of the Minimax Sphere Eversion.
Abstract
Consider the set of double points of a closed surface immersed in 3-space.
In general, this double-locus consists of a finite set of closed curves,
which self-intersect at the triple points of the surface. A regular homotopy
of the surface produces a homotopy of the double-locus. Generically, the
curves in the double locus move across themselves and each other, or recombine
like the level set of a saddle. Collectively, the stages of the double-locus
homotopy may be regarded as 3-dimensional crosssections of a surface located in
4-space. This project visualizes the double-locus surface of the minimax
sphere eversion.
Authors
George Francis and Chris Hartman, with Alexei Bourd, Glenn Chappell,
John Sullivan, Ken Brakke, Rob Kusner and Francois Apery.
Disclaimer
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Introduction
During June and July of 1996, Chris Hartman created a set of real-time
interactive computer animation (rtica) tools for the visualization of the
double-locus surface (dls) of the minimax sphere eversion (ivert). For this
he modified the following rticas. These were previously developed and used for
other purposes. All files are listed by the proper name of the project, followed
by the nickname(s). All source code (e.g. rtica.c) is in C/gl. The
console binary is then written rtica.x to distinguish if from the CAVE
executable, rtica.cave or rtica.mute, for example.
- illiVert: vn4.c =becomes=> vn.c in the present project.
- illiVert: rc35.savepipe a.k.a. movie35 and movie.
- illiLevel: ilevel.c [no changes to this rtica]
- illiSlice: slice.c [no changes to this rtica]
These are the new rtica, filters and display lists. When possible, they are
listed in the context of their usage.
- iris%vn.x -d -f movie calculates the file
- movie.dl, which is automatically read in when
- iris% vn.x -f movie is entered.
- half2whole.x converts movie.dl to longmovie.dl.
- dl2oe.x converts movie.dl to movie.oe.
- dl2oe2x converts movie.dl to movie.oe2.
- dl2oebis.xalso converts movie.dl to movie.oe2.
- iris% dl.x -f movie.oe2 ``doublelocus''
reads ``orderededge'' files.
- dl2sps.x converts movie.dl to to object.sps files.
- iris% ilevel.x surface.sps splines patch lists in .sps format.
- sps24off.x converts an object.sps into an object.4off.
- iris%slice.x saddle.4off to view objects.4off.
History of the Problem
The Minimax Sphere Eversion
The Double Locus
The standard illiVert viewer, still called vn.x, now also computes
(with the -d option) an unordered list of facets which
intersect other facets for each tope
in any sequence of topes, for example movie35. It also appends to the
movie.dl file the unordered list of edges in the double locus.
It can do that to a stream
of topes coming over the net from a remote translator. (Recall that the
translator can either transmit topes coming from the evolver or read them
from a stored file named rc35.savepipe, for example.) Or, with the -f
option, vn.x reads both the movie and the list movie.dl of facets neighboring
the double locus. Use the (L)key to cycle through the whole eversion, the
eversion of a regular neighborhood of the double locus, or its complement.
In order to isolate the curves making up the double locus, the filter
dl2oe.x orders the edges of the double locus, dl2oe2.x recomputes the
100 vertices on each cycle proportional to arclengh. The third filter,
dl2oe2bis.x does a little better, by making sure each cycle in the
double locus has the same number of vertices (except a the saddle levels.)
The rtical dl.x is designed to read and explore the .oe lists. It is
depth cued and may need to be adjusted to the particular hardware it is
running on. Only the movie.oe2 file created by dl2oe2bis.x applied to
the particular double locus of the movie35 eversion is fully successful
in computing proper adjacency of vertices in successive levels.
At any event, the fourth filter, dl2sps.x, a modification of dl2oe2bis.x,
produces the correct format for the patch splining rtica, ilevel.x
Thus, the other filters are not necessary if all you want is an object
defined by a stack of cycles of vertices along 3-D cross sections of
a surface in 4-D.
The splining tool, ilevel.x, knits together the curves into a surface
suitable for viewing with the illiSlice rtica, slice.x.
The Saddle Ribbons
Last edited 21jul96 by gkf.