I came across this question as I have been helping someone decorate styrofoam eggs by gluing wrapping paper onto it. The question is quite simple but I found myself stumped.
Given an egg what is the most efficient way to produce the prettiest egg?
So prettiest here would mean with the least amount of creases when the wrapping paper has been glued onto the egg. After a bit of thought one realises that a crease occurs because the chart maps onto the egg are not isometries. We can produce a prettier egg by using more charts with smaller domain. However, we don't want to be here all day cutting out tiny shapes in wrapping paper and gluing them onto eggs so we only want to use 2 charts. So the question can be rephrased as
Given an egg what 2 shapes should be cut out in the wrapping paper so that the charts are as close to an isometry as possible?
The types of eggs I am thinking of are as follows: Glue two halves of two different ellipses together and then rotate about the axis of symmetry. So in parametric equations consider $x \colon [0, 2\pi) \to \mathbb{R}$ and $y \colon [0,2 \pi) \to \mathbb{R} $ where $$x(\theta) = \begin{cases} a \mathrm{cos} (\theta) \quad \text{for } \theta \in [0,\pi/2) \cup [3\pi/2,2\pi) \\ b\mathrm{cos}(\theta) \quad \text{for } \theta \in[\pi/2,3\pi/2) \end{cases}$$ and $$y(\theta) =\mathrm{sin}(\theta) $$ for $a,b \geq 1$. To get the egg rotate the image of $(x(\theta),y(\theta))$ about the x-axis.