How can I draw (using a computer) spaces that I can't parametrize easily?

Question

I am studying algebraic topology and I came around the following problem:

I have to describe the space obtained when I identify the circles marked with different letters in the following figure:

I came to the conclusion that the resulting space is something which looks like a torus (due to the b circles) with something attached that is constructed similar to the Klein bottle (due to the a circles).

I am wondering now how can I draw an space like this with a computer. Is there any specific program that I can use or the best way is to use some pen tablet (like Wacom's tablets)?

Answer 1

What I do is draw several steps. I consider identifiying the b rings as trivial, so start with this:

(This was actually going well until I added the torus)

A series of drawings, easily verified at each step is the way forward.

Note: it is not a tube going through the torus (then it'd just be a 2-torus anyway). Can you see why? Try moving the right extrusion down to the opposite side of the left and then going "through" the torus.

---

Drawing with a computer

You cannot use a single pare of parameters to parameterise this as it has 2 holes. Well... I'll investigate this (they're sort of holes, right?) For example the 2-torus has to be 2 partial toruses joined together along a boundary.

It's so complicated to do these things that I'd urge you not to try. This is why a lot of topology books actually have hand-drawn illustrations.

You learn tricks like:

• To draw a torus draw a big ) then a slightly smaller ( touching the big one at two points, and draw an oval around it.

• Bold around an edge between the surface and the background help add depth.

• Use a pencil first, then go over it in pen. (I struggle with this one)