Hypocycloid

A hypocycloid is the curve produced by a fixed point on the circumference of a small circle of radius rolling around the inside of a large circle of radius . A hypocycloid is therefore a hypotrochoid with .

To derive the parametric equations of the hypocycloid, call the angle by which a point on the small circle rotates about its center , and the angle from the center of the large circle to that of the small circle . Then

so

Call . If , then the first point is at minimum radius, and the Cartesian parametric equations of the hypocycloid are

If instead so the first point is at maximum radius (on the circle), then the parametric equations of the hypocycloid are

For the parameterization with , the signed curvature, arc length measured from the initial cusp, and oriented tangential angle on the first smooth arch are given by

For an integer , the hypocycloid has cusps. With , its parametric equations become

The arc length of the parameter traversal and the enclosed area are

When , this traversal covers the line segment twice, so its geometric length is .

A 2-cusped hypocycloid is a line segment (Steinhaus 1999, p. 145; Kanas 2003), as can be seen by setting in equations (4) and (6) and noting that the equations simplify to

This result was noted by the Persian astronomer and mathematician Nasir Al-Din al-Tusi (1201-1274), and is sometimes known as a "Tusi couple" in his honor (Sotiroudis and Paschos 1999, p. 60; Kanas 2003). Al-Tusi's astronomical works describe the construction as a way to produce linear motion from two uniform circular motions (Ragep 1993).

The following table summarizes the names given to this and other hypocycloids with special integer values of .

If for relatively prime positive integers , the curve has parameter period and has distinct cusps. Hypocycloids for several rational values of are illustrated above.

If is irrational, then the curve never closes on itself. Hypocycloids for a number of irrational values of are illustrated above.

-cusped hypocycloids can also be constructed by beginning with the diameter of a circle, offsetting one end by a series of steps while at the same time offsetting the other end by steps times as large in the opposite direction and extending beyond the edge of the circle. After traveling around the circle once, an -cusped hypocycloid is produced, as illustrated above (Madachy 1979).

On the first arch of the parameterization with , let be the radius of curvature and the arc length from the initial cusp. For , these satisfy the Cesàro equation

where and .

For the parameterization with , let be the radial distance from the center of the fixed circle, and let be the oriented angle from the radius vector to the tangent. Away from the cusps and for , a hypocycloid satisfies

The corresponding differential relation is

where is the polar angle.

The parametric equations also give a form useful for calculus of variations problems with radial symmetry. The radial distance satisfies

But , so , which gives

Now let

so

then

The polar angle satisfies

But

so

The trigonometric addition formulas give

For and , the principal value of the inverse tangent gives

Substituting for on the same branch gives

An expression retaining the quadrant for any , including , is

where denotes the complex argument and . The polar angle is undefined at the origin.

This form is useful in the solution of the sphere with tunnel problem, which is the generalization of the brachistochrone problem, to find the shape of a tunnel drilled through a sphere (with gravity varying according to Gauss's law) in a gravitational field such that the travel time between two points on the surface of the sphere under the force of gravity is minimized.