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Families of Figure-Eight Curves

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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.

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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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Contributed by: Claude Fabre (September 2009)
Open content licensed under CC BY-NC-SA

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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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