The curve has a closed form when the ratio of the radius of the rolling circle and the
radius of the other circle
(a) is equal to a rational number. When giving this ratio its simplest form, the numerator
is the number of revolutions inside the resting circle, before the curve closes. The
denominator is the number of rotations of the rolling circle before this happens.
We let the rolling circle be smaller than the rolled circle 3), so that a < 1. And because the curve for a* = 1 - a is the same as for a, we can state that 0 < a <= 1/2.
A hypocycloid with parameters a and b is the same as a epicycloid with parameters a-1 and 1/b.
The resulting curve is a kind of star with a number of cusps. This amount is equal to the numerator of the constant a.
Some special hypocycloids are the following:
The curve is a cycloidal curve.
Now the point being followed is not lying on the rolling circle. When the point lays outside the circle (b>1), the curve is called a prolate hypocycloid. When the point lays inside the rolling circle (<1), the curve is called a curtate hypocycloid.
When the rolling circle is large than the rolled circle, the latter is on the
inside of the rolling circle. In this situation the curve has also the name of a
For a=1/3, b=1/2, we see an easy way to drill a hole with three edges.
The Wankel rotary engine uses the the hypotrochoid principle. The rotary
engine has a triangular rotor surrounded by a peripheral housing. This
combination completes the same four cycles of intake, compression, combustion
and exhaust and replaces the pistons, crankshafts cams and valves of the four
1) Hypo (Gr.) = under
2) Let a circle with radius r roll on the inside of a circle with radius R. Take as
origin the center of the rolled circle. Now let the starting point be on a distance b r
from the center of the rolling circle. Then the coordinates of the hypocycloid as a
function of the rolled angle t are:
3) If not, the rolling circle and the rolled circle can be interchanged.
4) Trochus (Lat.) = hoop.