Hypocycloid:

Whenever a circle of diameter D rolls on the inside of another circle of radius R, the path of the point A is like on the rolling circle is called a hypocycloid as displayed in diagram.

To sketch the hypocycloid curve the construction is rather similar to such of the epicycloids.

Describe a rolling circle of diameter D and a guiding circle of radius R tangent to it at point A. Divide the angle α such that is equal to 180 D/R and the rolling and guiding circles into 12 equal parts. From O₀ and via the points of division on the guiding circle, sketch radial lines that will intersect the auxiliary circle of radius R - D/2 at point O₁ via O₁₂.

 Along with O₀ as a center, describe auxiliary arcs passing through the points of division D/2 of the rolling circle. From O₁ via O₁₂ like centers and along with a radius equal to sketch arcs to intersect the corresponding auxiliary arcs at the points A₁ via A₁₂ of the desired hypocycloid. A smooth curve is then displayed through these points to get the desired hypocycloid.