This site (Ovals and Egg Curves) displays some interesting equations for egg curves. One such equation modifies the equation of an ellipse slightly by multiplying the “yÂ²” term by a suitable function t(x), so that y becomes larger on the left side of the y-axis and smaller on the right side.
For example, starting with the equation of the ellipse xÂ²/9+yÂ²/4=1, you can multiply yÂ² by t(x) to get: xÂ²/9+yÂ²/4*t(x)=1. The three examples below show three such functions for t(x) giving egg shaped curves:
Image Source: http://www.mathematische-basteleien.de/eggcurves.htm
For more egg curves check out the following two links: