Cycloidal curve

The plane curve described by a point that is connected to a circle rolling along another circle.

If the generating point lies on the circle, then the cycloidal curve is called an epicycloid or a hypocycloid, depending on whether the rolling circle is situated outside or inside the fixed circle. If the point is situated outside or inside the rolling circle then the cycloidal curve is called a trochoid.

The form of a cycloidal curve depends on the ratio between the radii of the circles. E.g., if the ratio of the radii is rational, then the cycloidal curve is a closed algebraic curve, but if the ratio is irrational, then the cycloidal curve is not closed. Among the epicycloids the best known is the cardioid, among the hypocycloids — the astroid and the Steiner curve.

The parametric equation of a cycloidal curve can be written in complex form:

$$z=l_1e^{\omega_1ti}+l_2e^{\omega_2ti},\quad z=x+iy.$$

This is a special case of an equation

$$z=l_0+l_1e^{\omega_1t_i}+\dotsb+l_ne^{\omega_nti},$$

which describes cycloids of higher order.

References

[1]	A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)
[a1]	J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972)
[a2]	K. Fladt, "Analytische Geometrie spezieller ebener Kurven" , Akad. Verlagsgesell. (1962)