---
author: Stéphane Laurent
date: '2018-08-30'
highlighter: kate
linenums: True
output:
html_document:
keep_md: False
md_document:
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tags: 'graphics, geometry, maths'
title: The Hopf torus of the tennis ball curve
---
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):
The parameter $n$ defines the number of lobes of the torus. You can play
with it below.