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 '.