Hopf Torus (3/3): the sinusoidal case

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

In this first and last part of our articles about Hopf tori, we will take a sinusoidal curve on \(S^2\).

In order to draw a sinusoidal curve on the sphere, I used this equation, that I found on mathcurve.com:

\[ \begin{equation} x = \frac{\cos u}{\sqrt{1+k^2\cos^2(nu)}} \\ y = \frac{\sin u}{\sqrt{1+k^2\cos^2(nu)}} \\ z = \frac{k \cos(nu)}{\sqrt{1+k^2\cos^2(nu)}} \end{equation} \]

This time, we obtain a Hopf torus with three lobes (because we took \(n=3\) in the formula above).

Below is an interactive rendering with three.js. Go to this post if you want to know how I’ve drawn the circles.