Justin Koepke 
3/21/03 
Math198 
Proposal 
        	illiLight: A Demonstration Polytope Lighting

A question that was posed by Professor Joseph O'Rourke is as follows: 
        "Let P be a polygon and imagine that all of its edges are perfect
	mirrors.  Is there always at least one interior point from which P is
	completely illuminable by a point light bulb?  Is P always illuminable
	from each of its points?  Assume that the light bulb sends out rays in all
	directions and that Snell's Law applies.  Further assume that a light ray
	is absorbed if it hits a vertex.  Surprisingly these problems are unsolved
	for polygons.  However, Klee showed the answers to be 'no' if curved
	(difffentiable) arcs are permitted." 
	I plan to create a program that will compute whether a region of
the polygon is dark, which is to say that any of the rays from the point
light source do not reflect such that the region is bounded by two rays
from the light source.  I also plan to implement a version for
3-dimensional objects.  I will utilize the existing illiSkel code to place
the objects in a universe.  Then, the user will be able to fly through the
polytope to see for him/herself whether all of the regions are
illuminated.  The program will also allow the user to change the location
of the light source inside the polytope.  I will use DPgraph to design the
two and three-dimensional objects that the user can choose from at the
start of the program.  I plan to show that if any curved arcs are allowed,
then the answer to the question is 'no'.  For the CAVE version of the
program, I will change the skel to be controlled gesturally from the wand. 
        In order to make the program run, I can only calculate a finite
number of rays from the point light source using a fixed delta theta, the
angle that the ray is directed away from the axis, which will be adjusted
if greater accuracy is desired.  For the graphics, if a region enclosed by
two rays (whether reflected or the original ones) and a wall of the
polytope, then that region will appear lit, while areas not enclosed will
appear dark inside the polytope. 
        If sufficient time if available, I will try to implement the
program for a polytope in 4-space, if that is even possible, as my
research hasn't resulted in any answers for this. 

Sources: 

"Art Gallery Theorems and Algorithms," Joseph O'Rourke p. 265-6.