Graphs of Eggs

I recently came across some nice material on John Cook's blog about equations that described eggs.

It turns out there are vast number of equations whose graphs are egg-shaped: that is, basically ellipse shape, but with one end "rounder" than the other.

You can see lots at Jürgen Köller's Mathematische Basteleien page. (Although this blog is mostly in German, there are enough English language pages for monoglots such as me). And plenty of egg equations can be found in the 2dcurves pages.

Another excellent source of eggy equations is TDCC Laboratory from Japan (the link here is to their English language page). For the purposes of experimenting we will use equations from this TDCC, adjusted as necessary. Many of their equations are given in parametric form, which means they can be easily graphed and explored using JSXGraph.

The first set of parametric equations, whose author is given to be Nobuo Yamamoto, is:

\begin{align*} x&=(a+b+b\cos\theta)\cos\theta\\ y&=(a+b\cos\theta)\sin\theta \end{align*}

If we divide these equations by \(a\), and use the parameter \(c\) for \(b/a\) we obtain slightly simpler equations:

\begin{align*} x&=(1+c+c\cos\theta)\cos\theta\\ y&=(1+c\cos\theta)\sin\theta \end{align*}

Here you can explore values of \(c\) between 0 and 1:

Another set of equations is said to be due to Tadao Ito (whose surname is sometimes transliterated as Itou):

\begin{align*} x&=\cos\theta\\ y&=c\cos\frac{\theta}{4}\sin\theta \end{align*}

Many more equations: parametric, implicit, can be found at the sites linked above.