From: dret@dgp.toronto.edu (George Drettakis)
Subject: Properties of a function expressing illumination
Originator: dan@symcom.math.uiuc.edu
X-Submissions-To: sci-math-research@uiuc.edu
Organization: CSRI, University of Toronto
X-Administrivia-To: sci-math-research-request@uiuc.edu
Date: Wed, 16 Dec 1992 20:03:09 GMT

The following integral gives the illumination from an area light
source to a receiving point on a plane. Assume a rectangular source
defined as (-xa<x1<xa, -ya<y1<ya),
embedded in the plane z=d, and a receiver plane z=0.

Then, if I(x,y) is the value of illumination at (x,y,0), we have:

                       ya1    xa1
                         /      /
                        |      |                 1
         I(x, y) :=     |      |   ---------------------------- dx1 dy1
                        |      |            2           2    2 2
                       /      /    ((x - x1)  + (y - y1)  + d )
                     - ya1  - xa1
                                          1

This integral has an analytic solution that looks like the following:


              (x - xa1)               (x + xa1)             (y - ya1) 
I(x,y)=- ------------------ w1 + ------------------ w3  - ------------------ w4
          2          2  1/2        2           2 1/2        2           2 1/2
        (d + (x - xa1) )         (d + (x + xa1) )         (d + (y - ya1) )

                   (y + ya1) 
              + ------------------  w2
                  2           2 1/2
                (d + (y + ya1) )

where, wi is the angle (in radians) formed by the two consecutive vertices 
of the source and the receiving point. 
(In the figure w1 is the angle < v1 P v2 )

 source:      v1
     ---------- +
    /        /   +
   /   +    /     +
  /        /       + <-angle w1
 ----------+        +
-ya1     v2   +      +
        +ya1     +    +
                __+___+________________________________
               /    + + P=(x,y,0)  receiver plane     /
              /                                      /
             /                                      /
            /______________________________________/


Each w i is usually expressed as an arccos or an arctan.
As an example of the difficulty of these expressions, for xa1=ya1=1,
w1 is:

                                   2   2          2
                                  d + y  - 2 y + x
w1=arccos(-----------------------------------------------------------------)
            2   2              2       1/2   2   2              2       1/2
          (d + y  - 2 y + 2 + x  - 2 x)    (d + y  - 2 y + 2 + x  + 2 x)


What I want to show is that the function I(x,y) has only one maximum, and is 
(radially) decreasing everywhere else. 
Ideally I want to show this for more general sources, but just the rectangle 
is a good start.

I have some ideas, but before I spend more time on this, I would like to
know whether this class of functions is well known, and whether there are
easy ways to prove its properties, or whether I am missing the obvious. 
Derivative analysis seems to be inconclusive, due to the complexity of 
the expressions. (The partials result in very high-degree equations).

Pointers to books, papers etc. are particularily welcome. 
I apologize beforehand if this is a well known problem with well known
solutions. I dont read this group often.

Thanks,

George Drettakis                          Dynamic Graphics Project
Grad Student                              Dept. of Computer Science
Phone +1 (416) 978-5473                   University of Toronto 
FAX   +1 (416) 978-0458                   Toronto, Ontario CANADA M5S 1A4 
e-mail: dret@dgp.toronto.edu  or dret@dgp.utoronto.ca 

Newsgroups: sci.math.research