From: israel@math.ubc.ca (Robert Israel) Subject: Re: A not-yet-solve fundamental problem on computational geometry Date: 4 Feb 2000 22:31:22 GMT Newsgroups: sci.math.symbolic,comp.graphics.algorithms,alt.algebra.help Summary: [missing] In article <389b2704.43439670@news.et.tudelft.nl>, derijken@hotmail.com (Dany) writes: > Given are: > a point source L=( - 4, 6, -1) > a receiver point R=(10,12,-2) > a metal sphere with radius = 3, centered at C=(4,1,1). > Question:: > Find the exact reflection point on the sphere.? I think the simplest way to do this is using Fermat's Principle: the path of the light ray is a stationary point (in this case a minimum) of the time of travel. So look for a stationary point of F = dist([-4,6,1],[x,y,z])+dist([10,12,-2],[x,y,z]) subject to the constraint (x-4)^2 + (y-1)^2 + (z-1)^2 = 3^2 (using Lagrange multipliers). Maple gives two solutions: 563367478 547534400 3 165862400 2 251590270 {x = --------- + --------- %4 - --------- %4 - --------- %4, 31553793 31553793 10517931 10517931 86526916 342209000 3 103664000 2 105530758 y = --------- - --------- %4 + --------- %4 + --------- %4, z = %4, t = 31553793 31553793 10517931 10517931 1040503819385 3 671200036015 2 3202182366455 - -------------- %5 %4 + -------------- %5 %4 + --------------- %5 %4 37645221184857 25096814123238 100387256492952 26934693466537 1040503819385 3 - ---------------- %5}, {z = %2, t = - -------------- %3 %2 1505808847394280 37645221184857 671200036015 2 3202182366455 26934693466537 + -------------- %3 %2 + --------------- %3 %2 - ---------------- %3, 25096814123238 100387256492952 1505808847394280 563367478 547534400 3 165862400 2 251590270 x = --------- + --------- %2 - --------- %2 - --------- %2, 31553793 31553793 10517931 10517931 86526916 342209000 3 103664000 2 105530758 y = --------- - --------- %2 + --------- %2 + --------- %2} 31553793 31553793 10517931 10517931 4 3 2 %1 := 9777400 _Z - 37519300 _Z + 49302825 _Z - 25516018 _Z + 4479973 %2 := RootOf(%1, 1.492548805) 2 3 2 %3 := RootOf(4377 _Z - 136883600 %2 - 126601300 + 124396800 %2 + 171261300 %2, 105.8038342) %4 := RootOf(%1, .5386167688) 2 3 2 %5 := RootOf(4377 _Z - 136883600 %4 - 126601300 + 124396800 %4 + 171261300 %4, 67.01577970) where the first is approximately (3.10699040, 3.826598554, .5386167688) and the second (4.71847729, -1.87074662, 1.492548805). Note that the first solution is the one you want (minimizing the time), while the second corresponds to reflection from the back of a transparent sphere (maximizing the time). Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 ============================================================================== From: anti@spamming.org (D. Mitrovich) Subject: Re: A not-yet-solve fundamental problem on computational geometry Date: Sat, 05 Feb 2000 12:06:29 +0500 Newsgroups: sci.math.symbolic,comp.graphics.algorithms,alt.algebra.help In article <87fjvq$qj3$1@nntp.itservices.ubc.ca>, israel@math.ubc.ca (Robert Israel) wrote: > In article <389b2704.43439670@news.et.tudelft.nl>, > derijken@hotmail.com (Dany) writes: >> Given are: >> a point source L=( - 4, 6, -1) >> a receiver point R=(10,12,-2) >> a metal sphere with radius = 3, centered at C=(4,1,1). >> Question:: >> Find the exact reflection point on the sphere.? > > I think the simplest way to do this is using Fermat's Principle: > the path of the light ray is a stationary point (in this case a minimum) > of the time of travel. So look for a stationary point of > F = dist([-4,6,1],[x,y,z])+dist([10,12,-2],[x,y,z]) > subject to the constraint (x-4)^2 + (y-1)^2 + (z-1)^2 = 3^2 > (using Lagrange multipliers). Maple gives two solutions: > > < big snip > Good grief! Why not just use the facts that the reflection point X has to lie on the plane defined by points L (source) and R (receiver) and sphere center C, and that the line from C thru X has to bisect the angle LXR? - Dushan Mitrovich