# Way (math)

In topology and analysis , a **path** or a **parameterized curve is** a continuous mapping of a real interval into a topological space. The image of a path is called a *curve* , *beam* , *track* or *curve* .

## definition

Let be a topological space , a real interval . If a continuous function , then a *path is called* in . The set of images is called the *curve* in .

The points and are called the *start point* and *end point of* the curve.

A path is called a *closed path* if it is. A closed path provides a continuous mapping of the unit circle (1- sphere ) . A closed path is also called a *loop* .

A path is called a *simple path* (or also without a colon) if is injective . In particular, is therefore permitted. A simple way is also called the *Jordan Way* .

This definition includes what we intuitively think of as a “curve”: a coherent geometric figure that is “like a line” (one-dimensional). But there are also curves that you would not call intuitively as such.

You have to distinguish between a path and a curve (the image of a path). Two different ways can have the same picture. Often, however, we are only interested in the image and then call the path a *parameter representation* or *parameterization of* the curve.

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

## Examples

The graph of a continuous function is a Jordan curve in . One parameterization is the Jordan way with . Here, on the product topology used.

The unit circle is a closed Jordan curve.

## Rectifiable ways

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

- .

A rectifiable path is a path of finite length.

If further , then the following applies:

Every piecewise continuously differentiable path is rectifiable, and its length is the integral over the amount of the derivative :

- .

A curve is the image set of a path , the path is then a parametric representation of the curve . For a given curve , the path integral and thus the path length - if finite - is independent of the choice of the parametric representation . Therefore it can be defined:

A piecewise smooth curve is called rectifiable if there is a parametric representation for it that is a rectifiable path. The length of a curve is the path length of its parametric representation .

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

## Other ways

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

## See also

## literature

- Klaus Fritzsche: Basic Course Analysis 1 . Differentiation and integration in a mutable. 2nd Edition. Spektrum Akademischer Verlag (Springer-Verlag), Heidelberg 2008, ISBN 978-3-8274-1878-4 , p. 257 ff .
- Stefan Hildebrandt: Analysis 2 . Springer-Verlag, Berlin / Heidelberg 2003, ISBN 978-3-540-43970-7 , pp. 110 ff . ( limited preview in Google Book search).