In geometry, an epicycloid (also called hypercycloid) is a plane curve produced by tracing the path of a chosen point on the circumference of a circleâÂÂcalled an epicycleâÂÂwhich rolls without slipping around a fixed circle. It is a particular kind of roulette.
An epicycloid with a minor radius (R2) of 0 is a circle. This is a degenerate form.
If the rolling circle has radius , and the fixed circle has radius , then the parametric equations for the curve can be given by either:
or:
This can be written in a more concise form using complex numbers as
where
Assuming the initial point lies on the larger circle, when is a positive integer, the area and arc length of this epicycloid are
It means that the epicycloid is larger in area than the original stationary circle.
If is a positive integer, then the curve is closed, and has cusps (i.e., sharp corners).
If is a rational number, say expressed as irreducible fraction, then the curve has cusps.
Count the animation rotations to see and
If is an irrational number, then the curve never closes, and forms a dense subset of the space between the larger circle and a circle of radius .
The distance from the origin to the point on the small circle varies up and down as
where
The epicycloid is a special kind of epitrochoid.
An epicycle with one cusp is a cardioid, two cusps is a nephroid.
An epicycloid and its evolute are similar.
Assuming that the position of is what has to be solved, is the angle from the tangential point to the moving point , and is the angle from the starting point to the tangential point.
Since there is no sliding between the two cycles, then
By the definition of angle (which is the rate arc over radius), then
and
From these two conditions, the following identity is obtained
By calculating, the relation between and is obtained, which is
From the figure, the position of the point on the small circle is clearly visible.