Harmonic Motion in Two Dimensions -the Locus of the Foci

The theory supporting this applet is presented in the paper ﻿Cassini Ovals in Harmonic Motion Orbits, ﻿Journal of Geometry and Symmetry in Physics, JGSP 47 (2018) 1–9.The initial position is chosen on the horizontal axis and the initial velocity is a vector , cutting angle with the -axis. For fixed and we change the angle from to in order to generate a family of ellipses. The enveloping curve for this family is also an ellipse (ellipse of safety) with center and foci at and Its axes are and . Very interesting is the locus of the foci of the changing ellipses (when changes): this locus is a Cassini oval. When this oval is the Lemniscate of Bernoulli. See an applet for Rotating the Lemniscate of Bernoulli