2D Spirograph — Rolling Circle Hypotrochoid & Epitrochoid

This is the 2D companion to the 3D Spirograph simulator. A small circle of radius r rolls without slipping around the inside (hypotrochoid) or outside (epitrochoid) of a fixed ring of radius R, with a pen held a fixed distance d from the rolling circle's own centre. Unlike a purely traced curve, this view draws the actual rolling geometry — the ring, the rolling circle, its centre and the pen arm — live, frame by frame, on a plain flat canvas. Petal count and the number of trips needed to close the pattern are computed exactly from the greatest common divisor and least common multiple of R and r.