An involute is the locus of a point on a string, as the string unwinds itself from a line or polygon, or a circle, keeping always the string taut.

Involute Of A Circle And Square

CYCLOIDAL:

Cycloid curves are formed by a point on the circumstance of a circle, rolling upon a line or an another circle. The rolling circle is called the generating circle. The line on which the generating circle rolls is called base line. The circle on which the generating circle rolls is called directing or base circle. A cycloid is a curve traced by a point on the circumference of a circle which rolls without slipping along a line.

HYPOCYCLOID:

A hypocycloid is a curve traced by a point on the circumference of a circle when it rolls without slipping on the inside of another circle