  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.
