Adam Coffman wrote:
> 
> I am working on a web site describing "Steiner surfaces," which are
> geometric shapes
> defined by specific types of polynomial equations.  They are of some
> interest in the theory of
> computer graphics, since Steiner surface patches contain many conic
> curves (ellipses, etc.) and
> have nice ray-tracing properties.
<snipped>
> Adam Coffman
> Assistant Professor of Mathematics
> Indiana Purdue Univ. Fort Wayne

... have nice ray-tracing properties.  ???

  I would be interested to hear an elaboration on that particular statement.

  This message reminds me of something I have been meaning to ask for a few
months now but always seem to forget. Below are three Parametric equations
that describe shapes that are either interesting, useful, or both. Alas my
math skills are prehistoric compared to what it takes to plug them into the
quartic matrix that Pov uses to display these types of advanced shapes.

Anyone out there up to the challenge ?

   
  The Triaxial tritorus is defined parametrically as:
  
  x = sin(u) (1+cos(v)) 
  y = sin(u+2 PI/3) (1+cos(v+2 PI/3)) 
  z = sin(u+4 PI/3) (1+cos(v+4 PI/3) 
  
  Where -PI <= u <= PI and -PI <= v <= PI 
 
  -------------------------------------------------------

  A Clien Cycloid Shape - defined parametrically as:

  x= cos(u/c)*cos(u/b)*(a+cos(v))+sin(u/b)*sin(v)*cos(v); 
  y= sin(u/c)*cos(u/b)*(a+cos(v))+sin(u/b)*sin(v)*cos(v); 
  z=-sin(u/b)*(a+cos(v))+cos(u/b)*sin(v)*cos(v); 

  Where: 0<=u<=2 b c PI and 0<=v<=4 PI 

  a=10;
  b=3;
  c=2;
  X= Cos[u/c]*Cos[u/b]*(a+Cos[v])+Sin[u/b]*Sin[v]*Cos[v];
  Y= Sin[u/c]*Cos[u/b]*(a+Cos[v])+Sin[u/b]*Sin[v]*Cos[v];
  Z=-Sin[u/b]*(a+Cos[v])+Cos[u/b]*Sin[v]*Cos[v];

  ParametricPlot3D [{X, Y, Z}, {u, 0, 2*b*c*Pi}, {v, 0, 4 Pi};
        PlotPoints -> {120,10};
              Axes -> False;
             Boxed -> False;
         ViewPoint -> {5.265, -6.828, 2.580}]

  ---------------------------------------


   This describes the parametric equations which approximatly model a
   drop of water, for example, a tear drop.

   Note:  I don't know the name for the symbol and my keybord refuses
          to reproduce it for me. Where you see Note:1 it is in reference
          to a symbol that resembles a circle with a line through it
          sometimes drawn at a slant.
  
   The equations as functions of longitute phi and lattitude theta are:
  
   x = 0.5 *(1-cos(8)) sin(8) cos(Note:1)
   y = 0.5 *(1-cos(8)) sin(8) sin(Note:1)
   z = cos(8)
  
    where 0 <= Note:1 <= 2pi
      and 0 <= 8 <= pi

   When theta is 0 there is a discontinuity at the apex where 
   x = 0  y = 0  z = 1



Regards,

-- 
Ken Tyler

mailto://tylereng@pacbell.net

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