Parametrization of torus-like object

Question

I'm trying to parameterize this surface so that I can have a geometric description of it and to find some closed geodesics on it.

How do you parameterize this surface?

The central cross section is defined as $\log^2 x +\log^2 y=1.$

I can parameterize this by $(e^{\cos(t)},e^{\sin(t)}).$ I'm having trouble parameterizing the surface though. I know that the surface is generated by taking the cross section and having another copy perpendicular to it, similar to the way the torus is generated with two circles being perpendicular and revolving them.

Answer 1

Writing in Wolfram Mathematica 12.2:

r[u_] = {Exp[Cos[u]], Exp[Sin[u]], 0}
    
t[u_] = Simplify[r'[u] / Sqrt[Total[r'[u]^2]], 0 < u < 2π]
n[u_] = Simplify[t'[u] / Sqrt[Total[t'[u]^2]], 0 < u < 2π]
b[u_] = Simplify[Cross[t[u], n[u]], 0 < u < 2π]

s[u_, v_] = r[u] + h Exp[Cos[v]] n[u] + k Exp[Sin[v]] b[u]

{h, k} = {1/6, 1/5};
ParametricPlot3D[s[u, v], {u, 0, 2π}, {v, 0, 2π}]

we get:

$$ \small \begin{aligned} & \mathbf{r}(u) = \left(e^{\cos u},\;e^{\sin u},\;0\right) \\ \\ & \hat{\mathbf{t}}(u) = \left(-\frac{\left(e^{\cos u}\,\sin u\right)}{\sqrt{\left(e^{\sin u}\,\cos u\right)^2 + \left(e^{\cos u}\,\sin u\right)^2}}, \; \frac{\left(e^{\sin u}\,\cos u\right)}{\sqrt{\left(e^{\sin u}\,\cos u\right)^2 + \left(e^{\cos u}\,\sin u\right)^2}}, \; 0\right) \\ \\ & \hat{\mathbf{n}}(u) = \left(-\frac{\left(e^{\sin u}\,\cos u\right)}{\sqrt{\left(e^{\sin u}\,\cos u\right)^2 + \left(e^{\cos u}\,\sin u\right)^2}}, \; -\frac{\left(e^{\cos u}\,\sin u\right)}{\sqrt{\left(e^{\sin u}\,\cos u\right)^2 + \left(e^{\cos u}\,\sin u\right)^2}}, \; 0\right) \\ \\ & \hat{\mathbf{b}}(u) = \left(0,\;0,\;1\right) \\ \\ & \mathbf{s}(u,\,v) = \left(e^{\cos u} - \frac{h\,e^{\cos v}\left(e^{\sin u}\,\cos u\right)}{\sqrt{\left(e^{\sin u}\,\cos u\right)^2 + \left(e^{\cos u}\,\sin u\right)^2}}, \; e^{\sin u} - \frac{h\,e^{\cos v}\left(e^{\cos u}\,\sin u\right)}{\sqrt{\left(e^{\sin u}\,\cos u\right)^2 + \left(e^{\cos u}\,\sin u\right)^2}}, \; k\,e^{\sin v}\right) \\ & \end{aligned} $$

with $0 \le u \le 2\pi$ and $0 \le v \le 2\pi$ (moreover, I chose $h = 1/6$ and $k = 1/5$):

which seems to me to be of the desired type.