Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

For questions about winding numbers. The winding number of a continuous curve counts how many times it "loops" around a given point.

Consider a curve in the plane parameterized in polar coordinates by $r = r(t)$ and $\theta = \theta(t)$, with $0 \le t \le 1$. Assuming that $r$ and $\theta$ are continuous, and the curve does not pass through the origin, we can define the winding number to be

$$\text{winding number} = \frac{\theta(1) - \theta(0)}{2\pi}$$

This counts the change in angle as a point moves along the curve containing the origin: Adding $1$ every counterclockwise loop, and subtracting $1$ for every counterclockwise loop.

Alternatively, in the complex plane, the winding number of a curve $\gamma$ not passing through a point $a$ can be defined as

$$\text{winding number} = \oint_{\gamma} \frac{dz}{z - a}$$

This can be generalized in geometry and algebraic topology, and the winding number of a map can also be called its degree.

history | excerpt history