Winding number

The number of turns (also called winding number or index ) is a topological invariant, which plays a crucial role in the theory of functions.

Preview

The number of turns of a curve with respect to a point, the number of orbits is the counterclockwise direction in order, by following the shape of the curve. A flight in a clockwise direction results in a negative number of turns.

Definition

Is a closed curve, and is also not lie on a point in, the number of turns is defined with respect to a

The number of turns ( after the English index) is in the literature often referred to with or. The winding number of a closed curve is independent of the reference point ( which may not lie on the curve ) is always an integer.

Calculation

Intuitively, can the number of turns by

Calculate. The calculation of the definition is often not possible without further notice. Start one can, by looking at the curve on the edge of the unit circle

Considered. After the intuitive rule for any and all latter follows immediately with the integral theorem of Cauchy and definition. now let

It is

Is by interchanging differentiation and integration

And because a primitive function of the integrand is, is because is continuous, ie for all

Application in the theory of functions

The number of turns is mainly used in the calculation of curve integrals in the complex plane. Be

A meromorphic function with singularities then you can go to the residue theorem by the integral of a ( passing through any of the singularities ) curve

Calculate.

Algorithm

In computational geometry, the winding number is used to determine whether a point is inside or outside of a non- simple polygon ( polygon whose edges overlap ) is. For simple polygons, the algorithm simplifies to the even-odd rule.

For polygons (closed edge coatings ) are used to calculate the number of turns of the following algorithm:

• Cut the half-line a polygon edge that is oriented " right to left " (dot is on the left side of the edge ), increase by 1
• Cut the half-line a polygon edge that is oriented "left to right" ( point lies on the right side of the edge ) and scale by 1

The number of turns is 0, then the point is outside of the polygon, or within.

In the example at left is the half-line, starts with, the vertical arrow. He cuts three edges of the polygon. Regarding the red edge of the point is right Regarding the next edge is the point also right and with respect to the last edge is the point on the left, the point is inside the polygon. The polygon area is grayed out.

An analogous algorithm gives course for non- rectilinear (closed ) curves, the number of turns around a point, but is there checking the intersections not so easy to implement.

Generalization for n- dimensional manifolds

A generalization for - dimensional manifolds Nikolai Nikolaevich Bogolyubov is from: Using the general Stokes' theorem for one can

. Write is the unit sphere in the looked -dimensional closed manifold, on which is to be integrated.