Re: Question, Geomview

[slevy]

Well... you can use a "4OFF" object, and display the (triangular) 2-faces of
the (tetrahedral) 3-faces of the simplicial complex.  (There's no provision
for graphic display of 3-D solid objects, only surfaces, lines, and points.)

Basically you could list all the vertices of each simplex, then define a face
for each combination of three vertices of that simplex.

An example with a single 4-simplex:

4OFF

5  10  0	# 5 vertices, C(5,3) = 10 faces, never mind edges

0 0 0 1		# position of vertex 0
1 0 0 1		# vertex 1
0 1 0 1		# vertex 2
0 0 1 1		# vertex 3
0 0 0 2		# vertex 4

	# Now follow the 10 faces

3	0 1 2
3	0 1 3
3	0 1 4
3	0 2 3
3	0 2 4
3	0 3 4
3	1 2 3
3	1 2 4
3	1 3 4
3	2 3 4

You could build a simplicial complex by just combining all the simplices,
though this would be wasteful: there'd be lots of duplicated faces, so
it may be worth explicitly eliminating them.


As for viewing 4-D objects, the simplest approach would be to load the
following geomview script:

(dimension 4)
(new-camera xyz)
(new-camera yzw)
(ND-axes xyz cluster1 0 1 2)
(ND-axes yzw cluster1 1 2 3)

which creates two windows, one looking at the objects' projection in the
X-Y-Z 3-D subspace and the other at the Y-Z-W subspace.  Moving the mouse in
either view causes rotation, translation, etc. in that subspace.

Using geomview's built-in 4-D capacity would also let you periodically
change the object being viewed, under control of a program.  For example
if your program ran as an external module to geomview, it could periodically
emit ``(geometry somename { 4OFF ... })'' commands to replace the object
being displayed.

It may help to display just the edges of a 4-complex (i.e. turn faces off
and edges on when viewing it).  In any case, beware: even simple 4-dimensional
objects can become awfully hard to understand.

I'm copying this to Olaf Holt, the designer of the N-D viewing features,
in case he'd like to add more.

    Stuart Levy

> Date: Sun, 26 Dec 1993 13:46:45 -0600
> From: Nimish Shah <shah at hal.cs.uiuc.edu>
> To: software@geom
> Subject: Question, Geomview
> 
> Hi,
> 
> I would like to display a 4-dimensional simplicial complex.
> The complex is generated by a triangulation program (and
> may change as the program executes). The manual explains
> how polyhedra can be displayed by their faces. How do
> I actually input a 4-simplex?
> 
> Is there something special to be done for external modules
> for 4-D? 
> 
> Happy hols!
> 
> Thanks,
> Nimish