Cycloid curves
A circle rolls along a line or around another circle, and a point on (or near) the rim traces a curve. Cycloids, epicycloids, hypocycloids — cardioids, nephroids, astroids — all from one mechanism: rolling without slipping.
x(t) = (R ± r)cos(t) − d·cos((R ± r)t/r) y(t) = (R ± r)sin(t) − d·sin((R ± r)t/r)
The family of cycloid curves
When a circle rolls along a straight line without slipping, a point on its rim traces a cycloid. Move that point inside the rim and you get a curtate cycloid; move it outside and you get a prolate cycloid. These three curves — basic, curtate, and prolate — form the cycloid family for straight-line rolling.
Epicycloids and hypocycloids
When the rolling circle moves around the outside of a fixed circle, the traced curve is an epicycloid. When it rolls around the inside, the curve is a hypocycloid. The ratio of the fixed radius R to the rolling radius r determines the number of cusps. Integer ratios produce closed curves; irrational ratios produce space-filling patterns that never close.
Famous special cases
A cardioid is an epicycloid with R/r = 1 (one cusp). A nephroid has R/r = 2 (two cusps). A deltoid is a hypocycloid with R/r = 3 (three cusps), and an astroid has R/r = 4 (four cusps). The astroid is the envelope of a line segment of fixed length whose endpoints slide along perpendicular axes — a trammel of Archimedes.
The brachistochrone connection
The cycloid is the solution to two classical problems: the brachistochrone (curve of fastest descent under gravity) and the tautochrone (curve where the period of oscillation is independent of amplitude). Johann Bernoulli posed the brachistochrone in 1696; Newton, Leibniz, l’Hôpital, and the elder Bernoulli all solved it. The answer was the same curve traced by a rolling wheel — a beautiful convergence of mechanics and geometry.