The osculating circle

Drag the gold point on the picture and it settles on the nearest place on the curve. The gold circle is the osculating circle there: it runs in the curve's direction and bends exactly as much, and its radius R is one over the curvature κ. Where the curve bends harder the circle shrinks; where it straightens the circle grows. On the peanut, drag the point across an inflection: the circle swells without limit and comes back on the other side of the curve, and κ changes sign. The ice-blue line is the evolute, the path of all the centres of curvature; its cusps face the gold points on the curve, where the curvature reaches a maximum or a minimum, the vertices. Try the other curves and count: a closed curve that never crosses itself has at least four vertices; the figure eight and the limaçon cross themselves, so the rule does not bind them, and the limaçon has only two.