© Mike Williams 2005

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Seashell surfaces

These are parametric isosurfaces using variations of functions like:
#declare W = function(u){u/(2*pi)}
#declare Fx = function(u,v){W(u)*cos(N*u)*(1+cos(v))}
#declare Fy = function(u,v){W(u)*sin(N*u)*(1+cos(v))}
#declare Fz = function(u,v){W(u)*sin(v)  + H*pow(W(u),2)}
  • H   controls the height of the shell
  • N   is the number of turns

u/(2*pi) simply gives a value that goes linearly from 0 to 1 as u goes from 0 to 2*pi. It turns up several times in the functions, so I've made it into a separate sub-function W(u)

I started with the parametric functions for a torus:

#declare Fx = function(u,v){cos(u)*(1+cos(v))}
#declare Fy = function(u,v){sin(u)*(1+cos(v))}
#declare Fz = function(u,v){sin(v)}
The first parts of Fx, Fy and Fz are multiplied by R(u) so that the radius of the tube increases linearly as u increases.

The N*u factors that occur in Fx and Fy cause the tube to go round the origin N times.

Then I added +H*pow(W(u),2) to Fz so that the turns of the spiral are offset in the z direction by a distance that varies from 0 to H. I found that I needed to square the u/(2*pi) term, otherwise the offset was too rapid at the start.

Download a zip file containing the POV source files for all the images that appear on this page. The zip file contains versions of each shape using real parametric isosurfaces and using the approximation macro.

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