On Sun, 18 Apr 1999 20:34:38 -0700, Ken <tyl### [at] pacbellnet> wrote:


>  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

You would need a series approximation if you really wanted to use a
quartic.  The accuracy would suck.  Why not just stick the actual
parametric equations directly into the parametric object type in the
Isosurface patch (and the Superpatch)?  Incidentally, as to the "8"
and "note 1" symbols, I'm afraid they're Greek to me! :)

Jerry Anning
clem "at" dhol "dot" com

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