Roulette (curve) application of this is that a square wheel could roll without bouncing on a road that is a matched series of catenary arcs. Roulette (curve) In the differential geometry of curves, a roulette is a kind of curve, generalizing cycloids, epicycloids, hypocycloids, trochoids, and involutes. Roughly speaking, a roulette is the curve described by a point (called the "generator" or "pole") attached to a given curve as that curve rolls without slipping, along a second given curve that is fixed. More precisely, given a curve attached to a plane which is moving so that the curve rolls, without slipping, along
Deltoid curve line segment can slide with each end on the deltoid and remain tangent to the deltoid. The point of tangency travels around the deltoid twice while each end travels around it once. The dual curve of the deltoid is which has a double point at the origin which can be made visible for plotting by an imaginary rotation y ↦ iy, giving the curve with a double point at the origin of the real plane. The area of the deltoid is formula_9 where again "a" is the radius of the rolling circle; thus the area of the deltoid is twice