If the moving circle rolls internally on the fixed circle, a point on the circumference describes a "hypocycloid" (from inr6, under).
The hypocycloid derived from the same circles is shown as curve d, and is seen to consist of three cusps arranged internally to the fixed circle; the corresponding hypotrochoid consists of a three-foil and is shown in curve e.
The equations to the hypocycloid and its corresponding trochoidal curves are derived from the two preceding equations by changing the sign of b.
Leonhard Euler (Acta Petrop. 1784) showed that the same hypocycloid can be generated by circles having radii of; (a+b) rolling on a circle of radius a; and also that the hypocycloid formed when the radius of the rolling circle is greater than that of the fixed circle is the same as the epicycloid formed by the rolling of a circle whose radius is the difference of the original radii.
Therefore any epicycloid or hypocycloid may be represented by the equations p = A sin B+,' or p---A cos B,,G, s = A sin B11.
If the radius of the rolling circle be one-half of the fixed circle, the hypocycloid becomes a diameter of this circle; this may be confirmed from the equation to the hypocycloid.
If the ratio of the radii be as I to 4, we obtain the four-cusped hypocycloid, which has the simple cartesian equation x 2'3+ y 213 = a 21 '.