Hopf Torus (1/3): the equatorial case

Posted on May 1, 2018 by Stéphane Laurent
Tags: R, graphics, rgl

In this series of three articles, we will show how to construct Hopf tori.

Hopf tori take their origins in the Hopf map \(S^3 \to S^2\) defined by \[ H(p) = \begin{pmatrix} 2(p_2 p_4 - p_1 p_3) \\ 2(p_1 p_4 + p_2 p_3) \\ p_1^2 + p_2^2 - p_3^2 - p_4^2 \\ \end{pmatrix}. \]

Obviously, this map is not bijective. Its main property is that it sends each great circle of \(S^3\) to a same point of \(S^2\). Thus, it has, say a “pseudo-inverse” \(H^{-1} \colon S^2 \times S^1 \to S^3\) defined by \[ H^{-1}(q,t) = \frac{1}{\sqrt{2(1+q_3)}} \begin{pmatrix} q_1 \cos t + q_2 \sin t \\ \sin t (1+q_3) \\ \cos t (1+q_3) \\ q_1 \sin t - q_2 \cos t \\ \end{pmatrix}. \]

Now let’s introduce the stereographic projection: \[ Stereo \colon S^3 -> \mathbb{R}^3 \\ Stereo \, x = \frac{1}{1-x_4}(x_1, x_2, x_3). \]

Then, we can draw a circle in \(\mathbb{R}^3\) as follows:

  • take a point \(q\) on \(S^2\);

  • calculate the great circle on \(S^3\) by applying \(H^{-1}(q, S^1)\);

  • maps this great circle to \(\mathbb{R}^3\) with the stereographic projection.

In this article, we will show what happens when we take for \(q\) a point on the equator on \(S^2\). Below are the necessary R functions.

This is the sphere with the equator:

Each point of the equator is mapped to a circle, and the circles form a torus:

Indeed, we get a torus, made of circles: