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