Lissajous knot
Knot defined by parametric equations defining Lissajous curves / From Wikipedia, the free encyclopedia
Dear Wikiwand AI, let's keep it short by simply answering these key questions:
Can you list the top facts and stats about Lissajous knot?
Summarize this article for a 10 year old
In knot theory, a Lissajous knot is a knot defined by parametric equations of the form
where , , and are integers and the phase shifts , , and may be any real numbers.[1]
The projection of a Lissajous knot onto any of the three coordinate planes is a Lissajous curve, and many of the properties of these knots are closely related to properties of Lissajous curves.
Replacing the cosine function in the parametrization by a triangle wave transforms every Lissajous knot isotopically into a billiard curve inside a cube, the simplest case of so-called billiard knots. Billiard knots can also be studied in other domains, for instance in a cylinder[2] or in a (flat) solid torus (Lissajous-toric knot).