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

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

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 .

See also

• Way (physics)

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).