the evolute and involute
/ EV-oh-loot, IN-voh-loot /
Take a curve and, at each point, find the center of its osculating circle — the center of the kissing circle that best fits the curve there. As your point slides along the original curve, that center moves too, tracing out a brand-new curve. This new curve is the evolute. The reverse relationship has a vivid physical picture: unwind a taut string wrapped around a curve and the free end of the string sweeps out an involute. The two ideas are inverse partners — the original curve is an involute of its own evolute.
Precisely: the evolute of a plane curve is the locus of its centers of curvature. At a point P with principal normal N and curvature kappa, the center of curvature sits at P + (1/kappa) N, a distance equal to the radius of curvature along the inward normal; collecting all these centers as P runs along the curve gives the evolute. The involute is the dual construction by 'string unwinding': hold one end of a string fixed against the curve, keep it taut, and unwrap it; the path of the moving end is an involute. Because you can start unwinding with different initial string lengths, a curve has a whole family of parallel involutes, but only one evolute. A key geometric fact links them: the tangent line to the evolute at a center of curvature is the normal line to the original curve — the evolute is the envelope of the family of normals of the original curve.
Where these appear: the involute of a circle is the classic profile used to cut the teeth of gears, because it makes meshing teeth roll smoothly and transmit motion at a constant rate; it is also the shape of the curve a goat traces while a rope unwinds from a circular post. The evolute reveals the focusing structure of a curve — its cusps occur exactly where the curvature has a local extremum (the vertices of the curve, by the four-vertex theorem at least four for a simple closed convex curve). One honest caution: the evolute is typically more singular than the curve you started with, sprouting sharp cusps even when the original curve is perfectly smooth, precisely at those points where the radius of curvature is momentarily stationary.
The evolute of an ellipse is a four-cusped star-like curve (an astroid-shaped curve): its four cusps sit on the major and minor axes, at the four vertices of the ellipse where the curvature is greatest or least. Conversely, the involute of a circle of radius a is the spiral profile r(t) = (a(cos t + t sin t), a(sin t - t cos t)), the very shape machined onto gear teeth.
Evolute = locus of centers of curvature = envelope of the normals; involute = string unwound.
The evolute is usually MORE singular than the original curve: even a smooth curve has an evolute with sharp cusps, located exactly where the curvature reaches a local maximum or minimum (the curve's vertices).