---
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:

<iframe src="../frames/threejs_TennisBall.html" width="100%" height="500px" scrolling="no" frameborder="0"></iframe>

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.

<iframe src="../frames/threejs_HopfTorusParametric2.html" width="100%" height="500px" scrolling="no" frameborder="0"></iframe>
