A circle that rolls leaves a law behind it.
The common cycloid is the path of a point on a rim of radius r. One arch runs 2Οr wide and 2r deep. Set the point outside the rim and the curve loops beneath the line; set it inside and the arch flattens. The same rolling rule, inside or outside another circle, draws the hypocycloid and epicycloid.
Beside them the spirals split by growth:
Archimedean r = aΞΈ keeps equal gaps, logarithmic r = ae^{kΞΈ} keeps equal angles, hyperbolic r = a/ΞΈ falls toward a straight asymptote.
Each curve is only a constraint, drawn until it becomes visible.