**The ballistic ellipse**

This is something I found when I was playing around with ballistic trajectories. I wondered what shape you would get if you connected all the apex points of all trajectories, if you only changed the angle and kept the same initial speed.

Surprisingly, you get an ellipse!

**EDIT: Also, here it is in 3D! Naturally, you get an ellipsoid.**

The equation for the ellipse is:

*x*^{2} / *a*^{2} + (*y* - *b*)^{2} / *b*^{2} = 1

Where *a* = *v*_{0}^{2} / (2*g*) and *b* = *v*_{0}^{2} / (4*g*). Naturally, v_{0} is the initial speed and *g* is the acceleration due to gravity.

In another curiosity, the eccentricity of this ellipse is constant for all values of *v*_{0} and *g*, and this value is *e* = √3 / 2.

Obviously, I wasn’t the first to find this. A quick search revealed a paper on arXiv from 2004 describing this. Still, it’s a nice little-known curiosity of a classical physics problem.

**Bonus points**: for which angles does the trajectory contain the foci of the ellipse?