Path (topology)

In topology and the Analysis a way or a parameterized curve is a continuous mapping of a real interval into a topological space. The picture of a path is called curve, carrier, track, or arc.

Definition

Be a topological space, a real interval. Is a continuous function, then that means a way in. The image set is called curve.

The points and hot start point and end point of the curve.

A path is called closed path if it is. A closed path delivers to a continuous map from the unit circle (1- sphere ). A closed path is also called loop.

A path is called simple way (or even colon -free) if it is injective on. In particular, therefore is allowed. A simple way is also called Jordan - way.

This definition includes what we intuitively under a " curve " Imagine: a coherent geometric figure, " like a line " is ( one-dimensional). But there are also curves that one would intuitively not marked as such.

One must distinguish ( the image of a path ) between a stroke and a curve. Two different paths can have the same image. Often we are only interested in the image, and then call the way a parametric representation or parameterization of the curve.

If there is a curve, a parameterization that is a Jordan path then the curve is called a Jordan curve, as well as for closed curve.

Examples

The graph of a continuous function is a Jordan curve in. A parameterization is with the Jordan - way. This product is used in the topology.

The unit circle is a closed Jordan curve.

Rectifiable paths

If a metric space with metric, then we can define the length of a path in:

A rectifiable path is a path with finite length.

Continues, then:

Each piecewise smooth path is rectifiable and its length is the integral of the amount of the derivative:

The Koch curve and a trajectory of a Wiener process are examples of non- rectifiable paths.

Other ways

A fractal path is a path with a broken dimension. Since different definitions of fractional dimension exist, so there are also different definition of a fractal path. Typical examples are the Koch curve and the dragon curve.