Lissajous (in the works)

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


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:


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]}.


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


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*Pi/b,2*Pi].


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