--- title: "The Hopf torus of the tennis ball curve" author: "Stéphane Laurent" date: "2018-08-30" output: md_document: variant: markdown preserve_yaml: true html_document: keep_md: no prettify: yes linenums: yes prettifycss: twitter-bootstrap tags: graphics, geometry, maths highlighter: kate --- {r setup, include=FALSE} knitr::opts_chunk$set(echo = TRUE, collapse= TRUE)  In [a previous post](https://laustep.github.io/stlahblog/posts/HopfTorusParametric.html) I explain the construction of the Hopf torus of a closed curve on the sphere, and I show the rendering for a certain spherical curve. On [Paul Bourke's site](http://paulbourke.net/geometry/spherical/index.html), I've found another spherical curve for which the corresponding Hopf torus is prettier. This spherical curve is defined by the three coordinates $$\begin{cases} p_1(t) & = & \sin\bigl(\pi/2 - (\pi/2 - A) \cos(t)\bigr) \cos\bigl(t/n + A \sin(2t)\bigr) \\ p_2(t) & = & \sin\bigl(\pi/2 - (\pi/2 - A) \cos(t)\bigr) \sin\bigl(t/n + A \sin(2t)\bigr) \\ p_3(t) & = & \cos\bigl(\pi/2 - (\pi/2 - A) \cos(t)\bigr) \end{cases}$$ for$t$varying from$0$to$2n\pi$. This is this curve: For$n=2$and$A \approx 0.44$it looks like the curve appearing on a tennis ball. For$n=3$and$A = 0.44$, here is the correspondiing Hopf torus (rendered with Asymptote): ![](figures/HopfTorusTennisBall.png) The parameter$n\$ defines the number of lobes of the torus. You can play with it below.