📂Geometry

Rotation Number of Plane Curves

Buildup

Before discussing how much the tangent of a plane curve rotates, let’s first consider something like the appropriate angle function. In the plane, let’s represent the size of the angle formed at point $p$ from the horizontal line (x-axis) to the tangent $t$ as $\overline{\theta} (p)$. The problem is that its value is $0 \le \overline{\theta} \le 2\pi$ so it is discontinuous from $0$ to $\overline{\theta}$.

To overcome this, the angle $\theta$ we consider is defined as continuous in the direction of progress in the four semi-planes formed by connecting quadrants as above. If the plane curve is regular, there is no worry that the tangent will suddenly jump to a non-adjacent quadrant, so the continuity of $\theta$ is guaranteed. More simply, do not impose limitations such as $0 \le \theta \le 2\pi$, and just keep increasing the angle in the direction it’s going as many times as it rotates.

Definition ¹

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

Examples

Counterclockwise Simple Curve

No matter where you start, if you move a length of $L$, that is, rotate one full circle, the angle changes by $2\pi$, so the rotation number is $\displaystyle i_{\alpha} = {{ 2 \pi } \over { 2 \pi }} = 1$.

Clockwise Simple Curve

It’s the same one round, but rotating in the opposite direction so the sign is opposite, making the rotation number $-1$.

Twice Wrapped Curve

If the definition of rotation number is correct, a twice wrapped curve should intuitively have a rotation number of $2$. Indeed, if you follow the line in any way, it is $\theta (L) = 4 \pi$. Of course, this curve is not a simple curve because it has tangled parts.

Figure-Eight Curve

If you start at one point on the right wing and follow the line directly, you enter the left wing and seem to end up with $2 \pi$ but actually finish with $\theta (L) = 0$. In a way, it seems like the rotations from the right $+1$ and from the left $-1$ cancel each other out.

Complex Curve

Let’s imagine a complex curve just by looking at it.

Of course, it’s good to grab the line directly and rotate it as in the previous examples, but from the examples seen so far, a mathematical intuition should come that the red rotation is $+1$ and the blue rotation is $-1$.

If you think separately about the overlapping parts of the twisted curve, the rotation number of this complex curve can also be calculated simply. Of course, each of these can not be called a simple curve due to the twisted part, but it is sufficient to consider the rotation number.