9474

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.
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.

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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.
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