The magenta locus is a lissajous curve, the path of a point each of whose coordinates is under harmonic motion. Drag endpoints of the blue and green segments to alter the respective harmonics. The white segment (at the bottom of the JavaSketch) determines the length of the curve. If, as in physics, one considers the curve as the path of a particle, this segment's length represents time. The curve could thus be represented analytically as [sin(bt + k0), sin(gt + k1)], 0 ≤ t ≤ c, where k0 and k1 are constants, and b, g, and c the lengths of the blue, green, and white segments respectively.
Nick "Jackous," September 1997