A curtate epicycloid is a fascinating type of curve generated by a point fixed at a distance less than the radius of a small circle as it rolls without slipping along the outside of a larger fixed circle. Unlike the standard epicycloid—where the tracing point lies exactly on the circumference—the curtate version creates loops that do not reach out as far, forming rounded, petal-like lobes that remain closer to the larger circle. This curve appears in various applications, including gear design a