Harmonic motion and Cassini ovals

Abstract

We consider a two-dimensional free harmonic oscillator where the initial position is fixed and the initial velocity can change direction. All possible orbits are ellipses and their enveloping curve is an ellipse too. We show that the locus of the foci of all elliptical orbits is a Cassini oval. Depending on the magnitude of the initial velocity we observe all three kinds of Cassini ovals, one of which is the lemniscate of Bernoulli. These Cassini ovals have the same foci as the enveloping ellipse.