Pilz Surface

pilz morph53 001
Mosrse transitions: two tori touch at 4 singular points then deform into a genus 5 surface. later two holes are closed leaving a genus 3 surface.

People have played with explicitly parametrized surfaces to produce shapes like the Snailshell or many others. One does not get closed surfaces of higher genus that way. Implicitly defined closed surfaces are easily obtained with high genus. This animation is one of the simplest examples: multiply two functions which each have an ellipse as minimum set and vary the distance of the two ellipses. — Join of 2 Tori and Bretzel5 Implicit Surface are similar examples.

The Pilz surface is a algebraic surface given by

  f(x,y,z) :=  ((x^2 + y^2 - 1)^2 + (z -0.5)^2)^2) *
                   ( (y^2/aa^2 + (z + hh)^2 -1)^2 +x^2) -
                    ff * (1 + bb*(0*x^2 + 0*y^2 + (z-0.5)^2))
           default:  ff = 0.28  hh = 0.03
pilz morph level 001
Fix the position of the two minimum-set-ellipses and vary the level value of the function.
pilz anaglyph geodesic
A closed geodesic is computed on one of the level surfaces. (anaglyph rendering)
pilz anaglyph ray 001
pilz morph310 001
Morse transition: two holes of a genus 3 surface are closed to deform into a genus 1 surface (torus). the torus is rotated and develops two conical singularities, the deformation ends with an outer and an inner sphere - see the following anaglyph picture.
pilz sphere inside ana
Passing through two singularities the torus deforms into two spheres, one inside the other.
pilz st

ImplicitCompact.pdf