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MathGroup Archive 2002

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Re: Ray Trace

  • To: mathgroup at smc.vnet.net
  • Subject: [mg33601] Re: Ray Trace
  • From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
  • Date: Wed, 3 Apr 2002 01:13:25 -0500 (EST)
  • Organization: Universitaet Leipzig
  • References: <a81juj$783$1@smc.vnet.net>
  • Reply-to: kuska at informatik.uni-leipzig.de
  • Sender: owner-wri-mathgroup at wolfram.com

Hi,

a) what is a *flat* polygon surface ? Do you mean a polygonal
approximation
   to a surface ?
b) a polygonal approximation to a surface uses *triangles* not
   quadrangles otherwise the surface between the points is not a polygon

c) say your points are {px[1],py[1],py[1]},{px[2],py[2],py[2]}, ..
   with these four points you can make a parametric patch
   { a[1]+b[1]*u+c[1]*v+d[1]*u*v,
     a[2]+b[2]*u+c[2]*v+d[2]*u*v,
     a[3]+b[3]*u+c[3]*v+d[3]*u*v}

  with u, v in [0,1]

  this can be solved by Mathematica and you have your parametric
  representation of the surface patch.
  For a[1],b[1],c[1],d[1] one gets
{a[1] ->  px[1], 
 b[1] -> -px[1] + px[2], 
 c[1] -> -px[1] + px[3], 
 d[1] ->  px[1] - px[2] - px[3] + px[4]}

 Such approximation is not smooth on the polygon edges. That's why
 spline surfaces are constructed, but a spline surface need more than
 four points

d) for a curved surface you need the two principal curvatures
   and the corresponding directions. 

Regards
  Jens

Regards
  Jens

Shawn O'Connor wrote:
> 
> flat polygon surfaces are relatively simple to define and use.  I would like
> specify a curved rectangular duct with a certain curvature.  Is there an
> easy way to define a curved surface given four points that pass through it?
> Then check weather a point lies on that plane bounded by the points.  I can
> do this already for flat surfaces.
> 
> Thank you


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