📂Geometry

Winding Number of a Closed Curve

Buildup

Before discussing how much the tangent of a plane curve rotates, let’s first think about something like an appropriate angle function. In the plane, let’s denote the size of the angle created by the tangent $t$ from point $p$ to the horizontal line (x-axis) as $\overline{\theta} (p)$. The issue is that since the value is $0 \le \overline{\theta} \le 2\pi$, it’s not continuous from $0$ to $\overline{\theta}$.

To overcome this, the angle $\theta$ we will consider is defined as being continuous in the direction of progress in the four semiplanes created by the connection of quadrants, as described above. If a plane curve is a regular curve, there’s no worry about the tangent suddenly jumping to an adjacent, unrelated quadrant, so the continuity of $\theta$ is assured. More simply, without setting restrictions like $0 \le \theta \le 2\pi$, as long as it continues to rotate multiple times, the angle can keep increasing in the direction it’s going.

Definition ¹

For a closed curve $\alpha$ with unit speed and length $L$, the following integer is called the rotation number of $\alpha$. $$i_{\alpha} = {{ \theta (L) - \theta (0) } \over { 2\pi }} = {{ \theta (L) } \over { 2\pi }}$$