# 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

A non-closed path with two colons

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 . ${\ displaystyle X}$${\ displaystyle I = [a, b]}$${\ displaystyle f \ colon I \ to X}$ ${\ displaystyle f}$${\ displaystyle X}$ ${\ displaystyle f (I)}$${\ displaystyle X}$

The points and are called the start point and end point of the curve. ${\ displaystyle f (a)}$${\ displaystyle f (b)}$

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 . ${\ displaystyle f}$${\ displaystyle f (a) = f (b)}$ ${\ displaystyle S ^ {1}}$${\ displaystyle X}$

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 . ${\ displaystyle f}$${\ displaystyle f}$${\ displaystyle [a, b)}$ ${\ displaystyle f (a) = f (b)}$

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. ${\ displaystyle h \ colon [a, b] \ to X}$${\ displaystyle \ mathbb {R} \ times X}$${\ displaystyle f \ colon [a, b] \ to \ mathbb {R} \ times X}$${\ displaystyle f (t) = (t, h (t))}$${\ displaystyle \ mathbb {R} \ times X}$

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 : ${\ displaystyle X}$${\ displaystyle d}$ ${\ displaystyle L}$${\ displaystyle f}$${\ displaystyle X}$

${\ displaystyle L (f) = \ sup \ left \ {\ sum \ limits _ {i = 1} ^ {n} d (f (t_ {i}), f (t_ {i-1})) \, {\ Bigg |} \, n \ in \ mathbb {N}, a \ leq t_ {0} .

A rectifiable path is a path of finite length.

If further , then the following applies: ${\ displaystyle X = \ mathbb {R} ^ {n}}$

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

${\ displaystyle L (f) = \ int \ limits _ {a} ^ {b} | f '(t) | \, \ mathrm {d} t}$.

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: ${\ displaystyle {\ mathcal {C}}}$${\ displaystyle f ([a, b])}$${\ displaystyle f}$${\ displaystyle f}$${\ displaystyle {\ mathcal {C}}}$${\ displaystyle {\ mathcal {C}}}$${\ displaystyle f}$

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 . ${\ displaystyle {\ mathcal {C}}}$${\ displaystyle f}$${\ displaystyle L ({\ mathcal {C}})}$${\ displaystyle {\ mathcal {C}}}$${\ displaystyle L (f)}$${\ displaystyle f}$

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 .