Families of Figure-Eight Curves
Take a pair of equal short and a pair of equal longer rods. Fix a rod with two nails (here the black points). Let both rods freely rotate around the nails. Hub both free ends together using the remaining fourth rod. Insert a pencil somewhere in this fourth rod (at the cyan point) that will draw the curve.[more]
As the lengths of the rods and the location of the pencil vary, you get not only the whole Cassini family of curves (including Bernoulli's lemniscate) but also closed asymmetric curves.[less]
The linkage, an isosceles trapezoid (called an isosceles trapezium by the British), shows a very simple mechanism hidden behind curves with complicated Cartesian coordinates. A circular motion gives rise to a large family of curves when throttling the length of the rods.