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 !!