

Mike Horvath <mik### [at] gmailcom> wrote:
> How do I split such a function into parts? For instance
It's already in parts.
What you can do is limit the range of U and V over which those functions are
evaluated.
If you limit U or V to the range from 0 to 0, then you will get a circle.
Swap U and V, and you get "the other" circle
Limit the ranges of both to a fraction of (2*pi), and you get a curved patch
that only covers part of the torus' surface.
Keep one full 2*pi range, and limit the other, and you get a "hoop sweep" around
the torus, one way or the other.
The set of macros takes this concept and just does it 4 times (0, +TStep,
+PStep, +both) to generate 4 corners, and fills that in with 2 smooth triangles.
See clipka's favorite link:
https://nylander.wordpress.com/2008/08/25/crosssectionofthequinticcalabiyaumanifold/
There is no magic.
Look back through the forums, and you can see that we've been exploring this
basic concept  for years.
#version version;
#include "colors.inc"
light_source {
<5, 10, 20>
color White
fade_distance 20
fade_power 2
}
camera {
location <0, 2, 35>
look_at <0, 0, 10>
right x*image_width/image_height
up y
}
#declare U1 = 0; // small radius
#declare U2 = 2*pi; //
#declare V1 = 0; // large radius
#declare V2 = 2*pi; //
#declare r0 = 10;
#declare r1 = 4;
#declare SphereRadius = 0.1;
// Create a set of points on the surface of a Torus
#declare X = function (T, P, R, r) {cos(T) * ( R + r * cos(P) )}
#declare Y = function (T, P, R, r) {sin(T) * ( R + r * cos(P) )}
#declare Z = function (T, P, r, n) {r * sin(P)}
#declare TStep = 0.02;
#declare PStep = 0.05;
#for (Theta, U1, U2, TStep)
#for (Phi, V1, V2, PStep)
#local XYandZResultOfParametricFunctionsEvaluatedForThisImmediateUandV =
<X (Theta, Phi, r0, r1), Y (Theta, Phi, r0, r1), Z (Theta, Phi, r0, r1)>;
sphere {XYandZResultOfParametricFunctionsEvaluatedForThisImmediateUandV
SphereRadius pigment {White}}
#end
#end
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