Length (math)

In mathematics, length is a property that can be assigned to routes , paths and curves . The length of a curve is also known as the arc length or rectification line.

Lengths of routes

If and are two points in the (two-dimensional) drawing plane ( ) with the respective Cartesian coordinates and , then the length of the line is the same according to the Pythagorean theorem ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle A(a_{1}|a_{2})}$ ${\displaystyle B(b_{1}|b_{2})}$ ${\displaystyle AB}$

${\displaystyle {\overline {AB}}={\sqrt {(b_{1}-a_{1})^{2}+(b_{2}-a_{2})^{2}}}.}$

In the three-dimensional visual space ( ) with the respective coordinates and applies ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle A(a_{1}|a_{2}|a_{3})}$ ${\displaystyle B(b_{1}|b_{2}|b_{3})}$

${\displaystyle {\overline {AB}}={\sqrt {(b_{1}-a_{1})^{2}+(b_{2}-a_{2})^{2}+(b_{3}-a_{3})^{2}}}.}$

There are essentially two ways in which such formulas can be generalized:

• The length of the line is interpreted as the length of the vector and length measures are defined for vectors. The corresponding generalized concept of length for vectors is called norm . ${\displaystyle AB}$ ${\displaystyle {\overrightarrow {AB}}}$
• The approach to consider the distance between the end points instead of the length of the route is even more general. General concepts of distance are called metrics .

Lengths of paths

A path is a continuous mapping from an interval into a topological space . In order to be able to assign a length to Wegen, however, this space must have an additional structure. In the simplest case, the plane or the visual space with the usual concept of length for lines; Generalizations are possible for Riemannian manifolds or any metric spaces . The length of the path is then referred to as . ${\displaystyle \gamma \colon [a,b]\to X}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \gamma \,}$ ${\displaystyle L(\gamma )\,}$

Paths in the plane and in space

A path in a plane or in space is given by two or three coordinate functions:

${\displaystyle t\mapsto (x(t),y(t))}$or for . ${\displaystyle t\mapsto (x(t),y(t),z(t))}$ ${\displaystyle a\leq t\leq b}$

For paths that are continuously differentiable piece by piece , the length of the path is given by the integral over the length of the derivative vector:

${\displaystyle L=\int \limits _{a}^{b}{\sqrt {{\dot {x}}(t)^{2}+{\dot {y}}(t)^{2}}}\,\mathrm {d} t}$ or. ${\displaystyle \int \limits _{a}^{b}{\sqrt {{\dot {x}}(t)^{2}+{\dot {y}}(t)^{2}+{\dot {z}}(t)^{2}}}\,\mathrm {d} t.}$

motivation

The flat path is first approximated by small straight lines , which are divided into two components and parallel to the coordinate axes. After the Pythagorean theorem applies: . The total length of the path is approximated by the sum of all straight lines: ${\displaystyle {\begin{matrix}f(t)=(x(t),y(t))\end{matrix}}}$ ${\displaystyle \Delta s}$ ${\displaystyle \Delta x}$ ${\displaystyle \Delta y}$ ${\displaystyle (\Delta s)^{2}=(\Delta x)^{2}+(\Delta y)^{2}}$

${\displaystyle L=\sum \Delta s=\sum {\sqrt {(\Delta x)^{2}+(\Delta y)^{2}}}=\sum {\sqrt {\left({\frac {\Delta x}{\Delta t}}\right)^{2}+\left({\frac {\Delta y}{\Delta t}}\right)^{2}}}\Delta t}$

If one assumes the convergence of the facts and gives the result without an exact limit value calculation , then the length is the sum of all infinitesimally small straight lines, i.e.: ${\displaystyle L}$

${\displaystyle L=\int \mathrm {d} s=\int {\sqrt {{\dot {x}}^{2}+{\dot {y}}^{2}}}\,\mathrm {d} t}$.

Physically (kinematically) the integrand can also be understood as the amount of the instantaneous velocity and the integration variable as the time. This is what motivates the definition of the length of a path best.

Examples

• The circle with a radius ${\displaystyle r}$

${\displaystyle t\mapsto (r\cdot \cos t,\ r\cdot \sin t)}$ For ${\displaystyle 0\leq t\leq 2\pi }$

has the length

${\displaystyle \int \limits _{0}^{2\pi }{\sqrt {r^{2}\sin ^{2}t+r^{2}\cos ^{2}t}}\ \mathrm {d} t=\int \limits _{0}^{2\pi }r\,\mathrm {d} t=2\pi r.}$

• A piece of a helix with radius and pitch ${\displaystyle r}$ ${\displaystyle h}$

${\displaystyle t\mapsto \left(r\cdot \cos t,\ r\cdot \sin t,\ {\tfrac {h}{2\pi }}\cdot t\right)\quad {\text{für}}\;0\leq t\leq 2\pi }$

has the length

${\displaystyle {\begin{aligned}\int \limits _{0}^{2\pi }{\sqrt {r^{2}\sin ^{2}t+r^{2}\cos ^{2}t+\left({\frac {h}{2\pi }}\right)^{2}}}\ \mathrm {d} t&=\int \limits _{0}^{2\pi }{\sqrt {r^{2}+\left({\frac {h}{2\pi }}\right)^{2}}}\ \mathrm {d} t\\&={\sqrt {(2\pi r)^{2}+h^{2}}}\end{aligned}}}$

Special cases

Length of a function graph

Let the function a continuously differentiable on , then the length is calculated as the function graph between the points and as follows: ${\displaystyle f\colon [a,b]\to \mathbb {R} }$ ${\displaystyle [a,b]\subset \mathbb {R} }$ ${\displaystyle L}$ ${\displaystyle (a|f(a))}$ ${\displaystyle (b|f(b))}$

${\displaystyle L(a,b)=\int \limits _{a}^{b}{\sqrt {1+(f'(x))^{2}}}\;\mathrm {d} x\qquad (*)}$

Example : The circumference of a circle can be calculated using. A circle with radius satisfies the equation and the derivative is: . ${\displaystyle {\begin{matrix}(*)\end{matrix}}}$ ${\displaystyle r}$ ${\displaystyle x^{2}+y^{2}=r^{2}}$ ${\displaystyle f(x)={\sqrt {r^{2}-x^{2}}}.}$ ${\displaystyle f'(x)={\frac {-x}{\sqrt {r^{2}-x^{2}}}}}$

Applying the formula , it follows: ${\displaystyle {\begin{matrix}(*)\end{matrix}}}$

${\displaystyle L=2\int \limits _{-r}^{r}{\sqrt {1+{\frac {x^{2}}{r^{2}-x^{2}}}}}\,\mathrm {d} x=2r\int \limits _{-r}^{r}{\frac {\mathrm {d} x}{\sqrt {r^{2}-x^{2}}}}\,=2r\arcsin(1)-2r\arcsin(-1)=2\pi r}$

Polar coordinates

If there is a flat path in polar coordinate representation , that is ${\displaystyle r(\varphi )}$

${\displaystyle \varphi \mapsto (r(\varphi )\cos \varphi ,r(\varphi )\sin \varphi )}$for , ${\displaystyle \varphi _{0}\leq \varphi \leq \varphi _{1}}$

so one gets from the product rule

${\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} \varphi }}=r^{\prime }(\varphi )\cos \varphi -r(\varphi )\sin \varphi }$

and

${\displaystyle {\frac {\mathrm {d} y}{\mathrm {d} \varphi }}=r^{\prime }(\varphi )\sin \varphi +r(\varphi )\cos \varphi }$,

thus so

${\displaystyle \left({\frac {\mathrm {d} x}{\mathrm {d} \varphi }}\right)^{2}+\left({\frac {\mathrm {d} y}{\mathrm {d} \varphi }}\right)^{2}=\left(r^{\prime }(\varphi )\right)^{2}+r^{2}(\varphi )}$.

The length of the path in polar coordinate representation is therefore

${\displaystyle L=\int \limits _{\varphi _{0}}^{\varphi _{1}}{\sqrt {\left(r^{\prime }(\varphi )\right)^{2}+r^{2}(\varphi )}}\,\mathrm {d} \varphi }$.

Paths in Riemannian manifolds

If, in general, there is a piecewise differentiable path in a Riemannian manifold , then the length of can be defined as ${\displaystyle \gamma \colon [a,b]\to M}$ ${\displaystyle \gamma }$

${\displaystyle L(\gamma )=\int \limits _{a}^{b}\|{\dot {\gamma }}(t)\|\,\mathrm {d} t.}$

Rectifiable paths in any metric space

Let it be a metric space and a way in . Then is called rectifiable or extensible , if the supremum ${\displaystyle (X,d)}$ ${\displaystyle \gamma \colon [0,1]\to X}$ ${\displaystyle X}$ ${\displaystyle \gamma }$

${\displaystyle L(\gamma )=\sup \left\{\left.\sum _{i=0}^{k-1}d(\gamma (t_{i}),\gamma (t_{i+1}))\right|k\in \mathbb {N} ,0=t_{0}<t_{1}<\ldots <t_{k-1}<t_{k}=1\right\}}$

is finite. In this case one calls the length of the way . ${\displaystyle L(\gamma )}$ ${\displaystyle \gamma }$

The length of a rectifiable path is therefore the supremum of the lengths of all approximations of the path by means of segments . For the differentiable paths considered above, the two definitions of length agree.

There are steady paths that cannot be rectified, for example the Koch curve or other fractals , space-filling curves , and almost certainly the paths of a Wiener process .

The word rectify or rectification means to make straight , that is, to take the curve (the thread) at the ends and pull it apart, stretch it out so that you get a line whose length you can measure directly. Nowadays this word mainly appears in the term rectifiable . The term often used instead of rectifiable in the older mathematical literature is stretchable .

Lengths of curves

Definition of the length of a curve

The set of images belonging to a path is referred to as a curve (also called a track of the path ). The path is also referred to as the parameter display or parameterization of the curve . Two different paths can have the same picture, so the same curve can be parameterized through different paths. It is obvious to define the length of a curve as the length of an associated path; but this assumes that the length supplies the same value for each parameterization. This is graphically clear, and it can actually be shown for injective parameterizations. In particular: ${\displaystyle \gamma \colon [a,b]\to X}$ ${\displaystyle \Gamma =\gamma ([a,b])\,}$ ${\displaystyle \gamma \,}$ ${\displaystyle \gamma \,}$ ${\displaystyle \Gamma \,}$

Be and two injective parametrisations same curve , ie . Then: . ${\displaystyle \gamma _{1}\colon [a_{1},b_{1}]\to \mathbb {R} ^{n}}$ ${\displaystyle \gamma _{2}\colon [a_{2},b_{2}]\to \mathbb {R} ^{n}}$ ${\displaystyle \Gamma \,}$ ${\displaystyle \gamma _{1}([a_{1},b_{1}])=\gamma _{2}([a_{2},b_{2}])=\Gamma \,}$ ${\displaystyle L\left(\gamma _{1}\right)=L\left(\gamma _{2}\right)=L\left(\Gamma \right)}$

Parameterization of a curve according to the path length

As already said, there are different parameterizations for a curve. A special parameterization is the parameterization according to the path length (or arc length ).

Is a rectifiable curve with the parameterization ${\displaystyle \Gamma }$

${\displaystyle {\begin{matrix}\gamma :&[a,b]&\to &\mathbb {R} ^{n}\\&\tau &\mapsto &\gamma (\tau )\end{matrix}}}$

and for the partial curve with the parameterization , this is the name of the function ${\displaystyle \Gamma _{t}}$ ${\displaystyle t\in [a,b]}$ ${\displaystyle \gamma |[a,t]}$

${\displaystyle {\begin{matrix}s:&[a,b]&\to &\mathbb {R} \\&t&\mapsto &L\left(\Gamma _{t}\right)\end{matrix}}}$

as a path length function of . This path length function is steadily and monotonically increasing , for injective it is even strictly monotonically increasing and therefore also bijective . In this case there is an inverse function . The function ${\displaystyle \Gamma }$ ${\displaystyle s(t)}$ ${\displaystyle \gamma }$ ${\displaystyle t(s)}$

${\displaystyle {\begin{matrix}{\hat {\gamma }}:&[0,L(\gamma )]&\to &\mathbb {R} ^{n}\\&s&\mapsto &\gamma (t(s))\end{matrix}}}$

is referred to as the parameterization of with the arc length as a parameter. ${\displaystyle \gamma }$

If it is continuously differentiable and for all , the peculiarity of the parameterization according to the arc length is that it is also continuously differentiable and for all ${\displaystyle \gamma }$ ${\displaystyle {\dot {\gamma }}(\tau )\neq 0}$ ${\displaystyle \tau \in [a,b]}$ ${\displaystyle {\hat {\gamma }}}$ ${\displaystyle s\in [0,L(\Gamma )]}$

${\displaystyle \left\|{\frac {\mathrm {d} {\hat {\gamma }}(s)}{\mathrm {d} s}}\right\|=1}$

applies.

See also

• Length (physics)

literature

• Harro Heuser : Textbook of Analysis (=  Mathematical Guidelines . Part 2). 5th revised edition. Teubner Verlag, Wiesbaden 1990, ISBN 3-519-42222-0 .
• Konrad Knopp : Theory of functions I. Basics of the general theory of analytical functions (=  Göschen Collection . Volume 668 ). Walter de Gruyter Verlag, Berlin 1965.
• Hans von Mangoldt , Konrad Knopp : Introduction to higher mathematics . 13th edition. Volume 2: differential calculus, infinite series, elements of differential geometry and function theory. S. Hirzel Verlag, Stuttgart 1968.
• Hans von Mangoldt, Konrad Knopp: Introduction to higher mathematics . 13th edition. Volume 3: Integral calculus and its applications, function theory, differential equations. S. Hirzel Verlag, Stuttgart 1967.
• Wolfgang Ebeling : Lecture notes Analysis II. University of Hanover, Institute for Algebraic Geometry

References and comments

1. ↑ The non-rectifiable curves can thus be regarded as of infinite length ; see. Hans von Mangoldt, Konrad Knopp: Introduction to higher mathematics . 13th edition. S. Hirzel Verlag, Stuttgart, p. 227 .
2. ↑ The term stretchable is rarely used in modern mathematical literature. Likewise, when it comes to paths and curves, one usually speaks of rectifiability instead of extensibility . Konrad Knopp : Theory of functions I. Basics of the general theory of analytical functions (=  Göschen Collection . Volume 668 ). Walter de Gruyter Verlag, Berlin 1965, p. 22 . Harro Heuser : Textbook of Analysis (=  Mathematical Guidelines . Part 2). 5th revised edition. Teubner Verlag, Wiesbaden 1990, ISBN 3-519-42222-0 , pp.  349 .