Cartesian parametrization: .

The 3D Lissajous curves are the trajectories of a point in space the rectangular components of which have a sinusoidal motion.
The projections on the 3 coordinate planes are the classic 2D Lissajous curves.

For n = 1 or n = m, we get a cylindrical sine wave.
We get a closed curve if and only if n and m are rational.

When the curve does not have double points, nor a cusp, it forms a knot in space, called Lissajous knot, equivalent to a cubic billiard knot.
 

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© Robert FERRÉOL  2018