Epicycloid

A planar curve given by the trajectory of a point on a circle rolling on the exterior side of another circle. The parametric equations are:

where is the radius of the rolling and that of the fixed circle, and is the angle between the radius vector of the point of contact of the circles (see Fig. a, Fig. b) and the -axis.

Figure: e035860a

Figure: e035860b

Depending on the value of the modulus , the resulting epicycloid has different forms. For it is a cardioid, and if is an integer, the curve consists of distinct branches. The cusps have the polar coordinates , , . The vertices of the curve have the coordinates , . When is a rational fraction, the branches intersect each other in the interior; when is irrational there are infinitely many branches and the curve does not return to a point describing a position obtained previously; for rational the epicycloid is a closed algebraic curve. The arc length from the point is:

and from it is

The area of a sector bounded by two radius vectors of the curve and its arc is

The radius of curvature is

When the point is not situated on the rolling circle, but lies in its exterior (or interior) region, then the curve is called an elongated (respectively, shortened) epicycloid or epitrochoid (see Trochoid). Epicycloids belong to the so-called cycloidal curves (cf. Cycloidal curve).

References

[1]	A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)

Comments

Epicycloids (and hypocycloids, cf. Hypocycloid) have many equivalent definitions. See, e.g., [a3], pp. 273-277. Epicycloids and, more generally, trochoids are important for kinematical constructions, cf. [a1].

References

[a1]	H.-R. Müller, "Kinematik" , de Gruyter (1963)
[a2]	K. Strubecker, "Differential geometry" , I , de Gruyter (1964)
[a3]	M. Berger, "Geometry" , I , Springer (1977)
[a4]	M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)