the rotation index
Drive a closed loop of road and return to your starting point facing your starting direction. Along the way the direction you faced kept changing — sometimes you turned left, sometimes right — but by the time you got back, your heading must have made some whole number of complete turns. That whole number, counted with sign (counterclockwise positive, clockwise negative), is the rotation index of the loop. It records the total net turning of your direction, packaged as an integer count of full revolutions.
Precisely, for a smooth closed plane curve travelled in a fixed direction, watch the unit tangent vector T. As you go once around the loop, T sweeps around and, because the curve closes up, ends pointing exactly where it started — so the total angle it has swept is an integer multiple of 2 pi. The rotation index (also called the turning number) is that integer: total turning angle divided by 2 pi. You can also compute it as the integral of the signed curvature around the curve divided by 2 pi: rotation index = (1/(2 pi)) times the integral of kappa_s ds. A circle traced counterclockwise has rotation index +1; traced clockwise, -1; a figure-eight, where the two lobes turn in opposite senses and cancel, has rotation index 0.
The rotation index is a robust, discrete invariant: small wiggles of the curve cannot change an integer continuously, so it stays put under any continuous deformation that keeps the curve smooth and closed (it is, in fact, a homotopy invariant of the tangent direction). The deep companion result, the Hopf Umlaufsatz, pins it down for the simplest loops: every simple closed curve (one that does not cross itself) has rotation index exactly +1 or -1, the sign just recording orientation. Two honest cautions. First, do not confuse the rotation index with the winding number around a point: the rotation index tracks the turning of the tangent DIRECTION, whereas a winding number counts how many times the curve encircles a given point — different curves can share one and differ in the other. Second, the index is signed and orientation-dependent: reverse your direction of travel and its sign flips.
A counterclockwise circle has rotation index +1: the tangent turns through one full counterclockwise revolution. A limacon-style loop with one small inner loop traced the same overall sense can have rotation index +2 (the tangent makes two net revolutions). A figure-eight has rotation index 0, because its two crossings turn the tangent one way then the other and the net turning cancels.
Net turns of the tangent direction, counted with sign; an integer that survives smooth deformation.
The rotation index (turning of the tangent direction) is not the winding number (how many times a curve encircles a fixed point) — they answer different questions. The index is signed and reverses when you reverse the orientation of travel.