Lissajous (in the works)

lissajous
Varieties of Lissajous {a*Sin[b*t*2*π+c],Sin[t*2*π]}, where a is the inverse of golder ratio 1/ϕ ≈ 0.618, and b, c are indicated in the corner. lissajous.nb.zip.

Description

Lissajous is a family of curves, given by {a*Sin[b*t+c], Sin[t]} with 3 parameters.

In studying nature there often arises the wave motion a*Sin[b*t+c]. For example, the motion of a pendulum. Lissajous is two such motions in perpendicular directions:

{a1*Sin[b1*t+c1],
a2*Sin[b2*t+c2]}

Changing the parameter by replacing t→1/b1*t then t→c2, then scale by 1/a2, we can simplify the parameters down to {a*Sin[b*t+c], Sin[t]}.

History

Studied by Nathaniel Bowditch in 1815 and Jules Antoine Lissajous (1822 – 1880).

Formula

Parametric: {a*Sin[b*t+c], Sin[t]}

The parameter a stretches the curve in one direction. Parameters b and c are more interesting. The period of the curve is the least common multiple of the two components, that is, LCM[2*π/b,2*π].

Properties

Related Web Sites

See: Websites on Plane Curves, Printed References On Plane Curves.

Lissajous figure.