Trochoids — curves traced by rolling circles — were originally explored and defined by Archimedes, that lover of all things spiral, for sport. Later the curves and their precise definition caused so much strife between 17thcentury mathematicians that they came to be called the Helen of Geometers. But since then, the shape has actually been designed into many successful mechanisms.
Geometry distinguishes three major categories of trochoids, with three subcategories of each. Trochoids include a family of curves called cycloids — curves traced by a point located on the rolling circle's circumference. Mathematically, a cycloid through (0,0) created by a circle of radius r looks like
x = r(t  sin t)
y = r(1  cos t)
where t is the rolling circle's center.
Cyclone home
But what if our rolling circle rides not on a straight path, but around another circle? Then we have another set of curves.

When a circle rolls without slipping around the outside of a fixed circle, a point on the roller traces a loopy path called an epicycloid.

If our active circle rolls around the inside of a fixed circle, it traces a path called a hypocycloid.
Solid rolled
Most gear teeth are an involute shape. When they mesh, they slide together, roll at a pitch point to transmit power, and then slide apart again. But when the inefficiency of sliding is unacceptable, gear teeth can be made with profiles in the shape of trochoids. Trochoid drives are planetary sets that produce pure rolling action. More efficient than worm gearing, they often beat helical sets, too — especially when multiple stages are needed. (Cycloidal drives can reduce speed 200:1 in a single stage.) Harmonic and planocentric drives are two cycloidal subtypes.
Rolling lobes
Cycloids and trochoids are found in many designs. Galileo used cycloids in his telescopes to prove Copernicus's theories; cycloid compressors have been used since the 19th century. Other modern designs include Braytonthermalcycle engines for jetairplane propulsion, cyclodrives, and gerotor oil pumps like the one shown here. These use the lobes to collect input fluid and roll it to an output.
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