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Invariance of the Length and the Area of Cycloids

Abstract To make a marked point on the boundary of a circular wheel of radius 1 trace the curve known as a cycloid, let this wheel roll along an arbitrary fixed curve called the base curve. There are two cycloids, each of which is drawn on one side of the base curve and share cusps with each other. Consider the two arcs joining the same pair of adjacent cusps, one arc from each cycloid. T. M. Apostol and M. A. Mnatsakanian showed that the sum of the lengths of these arcs is independent of the shape of the base curve. They also showed that the sum of the areas bounded by two cycloids is independent of the shape of the base curve. Their remarkable proof does not use integral calculus. We generalize these results to the general trochoid case. Our proof is based on simple undergraduate calculus.