Let $T^2\cong S^1\times S^1$ be the one-holed torus surface (say, embedded in $\mathbb{R}^3$) and say I have a simple-ish map $f:T^2\to T^2$ which I'd like to visualize. How might I do that?
To make this concrete, let $f(z,w)=(zw,z^2)$ where $z,w\in S^1$. I'd like to see a decent representation of the image of this map on $T^2$.
Here's what I know:
From a previous answer here, I know I can visualize a torus in Mathematica (as well as plot a contour on it). For example, I can use something like
yourFunc = Function[{u, v}, Re[2 Exp[2 π I (u + 2 v)] + 3 Exp[2 π I (u - 2 v)]]];
ParametricPlot3D[{(2 + Cos[2 π v]) Sin[2 π u],
(2 + Cos[2 π v]) Cos[2 π u], Sin[2 π v]},
{u, 0, 1}, {v, 0, 1},
MeshFunctions -> Function[{x, y, z, u, v},
yourFunc[u, v]], Mesh -> {{0}},
MeshStyle -> Directive[Blue, Thick], PlotPoints -> 50]
to visualize (on a torus) the contour corresponding to the zero-set of a provided parametric function:
However, this sort of example hinges on a 3D parametric representation of a torus rather than a product of circles representation and my knowledge is currently insufficient to bridge the gap.
Edit: Per a comment by @Rahul below: I'm considering $S^1$ as a subset of $\mathbb{C}$. In particular, the map $f(z,w)$ can be converted to a map $[0,1]^2\mapsto[0,2]^2$ by converting:
$$z=e^{2\pi i\theta},w=e^{2\pi i \phi}\implies zw=e^{2\pi i(\theta+\phi)}\text{ and }z^2=e^{2\pi i(2\theta)}.$$
So, equivalently, we have a map $g:[0,1]^2\to[0,2]^2$ given by $g(\theta,\phi)=(\theta+\phi,2\theta)$. Using Mod
, we can visualize a parametric plot for the map $g$:
ParametricPlot[{Mod[u + v, 1], Mod[2 u, 1]}, {u, 0, 1}, {v, 0, 1}]
yields
. Does this help? Note that I tried substituting
yourFunc = Function[{u, v}, {u+v, 2u}];
into the original snippet of code above but that doesn't work.