I am trying to draw surfaces with complete hyperbolic structures and surfaces which are topologically tori. The hyperbolic surfaces I need to draw are torus with one or two holes on it, or torus with punctures on it, or torus with a cusp attached to it . They could also be torus torus with one or more than one handles attached to it.
For example, see the diagrams on : http://www.maths.bris.ac.uk/~mazag/hyperbolic/index.html
Or see the diagrams on : http://lamington.wordpress.com/2010/04/18/hyperbolic-geometry-notes-4-fenchel-nielsen-coordinates/
to get ideas about what surfaces I am talking about. They are not given by any easy equations.
Is there a software I can use to draw them ? People who study Riemann Surfaces or Hyperbolic Geometry or Teichmmuller Theory would definitely know exactly what surfaces I am talking about.Please let me know if you use such a software. Thanks a lot in advance !!
Best Answer
The solutions that people have mentioned are good if you are happy with a two-dimensional line drawing. It would be better to have three-dimensional equations that could be plotted using Maple, or something like that, but that seems to be surprisingly hard. The equation $$ 3x_3^2x_4-2(x_1^2+x_2^2)x_4-2x_4^3+2(x_1^2-x_2^2)x_3 = 0 $$ defines a highly symmetric surface of genus 2 embedded in $S^3$, and one can project stereographically into $\mathbb{R}^3$ to get a nice picture like this:
(There's a lot to be said about this example; I will have an undergraduate working on it over the summer.) However, I do not know similarly nice equations for surfaces of higher genus, or with the two tori in the same plane rather than at right angles, or with cusps.