Intuitively, the length of a Curve or an arc (portion) of curve is the length of string which would have to be unrolled to traverse it completely. This length can be obtained if one knows run time and speed.

To give a general standard length of an arc, it is necessary to start by formalizing the concept of distance, in general within the framework of a Euclidean Espace. One can then measure the length of simple curves: the polygonal lines.

The Old ones without having an explicit process of calculation, were satisfied to approach the lengths of curve, by considering polygonal lines uniting of the points of the curve. It is the method of exhaustion, which had been initiated by Eudoxe de Cnide and Archimedes for calculations of surfaces.

These approximate calculations length can be used as base with a general standard making it possible to exceed the intuitive vision length. The length of the arc will be the upper limit, if it exists, lengths of such polygonal lines.

When the curve is parameterized in a sufficiently regular way, one obtains an explicit formula for the length, resulting from the differential Calculus. One can then use the curvilinear concept of Abscisse which is a kind length algebraic, taking account of the orientation, and which allows reparamétrer the curve in order to be freed from the considerations on the speed of course.

Modern presentation: rectifiable curve and length

Approaches by the concept speed

Here a first way of introducing the length, starting from a little fuzzy concept “length of a vector infinitesimal displacement”. As historically the Infinitesimal calculus preceded the precise definition by the concepts of Limite and of upper limit, this first definition length raises of a tradition different from following and can seem more speaking.

One places for this calculation in the Euclidean plan, brought back to an orthonormal reference mark. One considers a parameterized arc of class $\backslash \; mathcal\; C^1$ given by a function $\backslash \; overrightarrow\; \{F\}\; (T)\; =\; (X\; (T),\; there\; (T))$ for T variable in a segment ''. One will obtain a formula for the length by handling freely the notations Différentielle S, which could be made perfectly rigorous.

One can speak about the vector infinitesimal displacement

$\backslash \; overrightarrow\; \{df\}$

\ overrightarrow {F} (t+dt) - \ overrightarrow {F} (T) \ overrightarrow {\ frac {df} {dt}} dt~

Let us note its standard ds : it is the infinitesimal length traversed during the time interval dt . Then the length of the arc is obtained by summoning these elementary lengths

$L=\; \backslash \; int\; ds\; =\; \backslash \; int\_a^b\; \backslash \; frac\; \{ds\}\; \{dt\}\; dt\; =\; \backslash \; int\_a^b\; \backslash \; left\; \backslash |\backslash \; frac\; \{D\; \backslash \; overrightarrow\; \{F\}\}\; \{dt\}\; \backslash \; right\; \backslash |\; dt$

\ int_a^b \ sqrt {x' (T) ^2+y' (T) ^2} dt~

One will be able to summarize this formula by expressing the value infinitesimal length in the form

$ds^2=dx^2+dy^2$

Other formulas can be established in the same way (for curves of Euclidean space with 2,3 Dimension S), with, according to the selected Frame of reference Cartesian

• $ds^2=dx^2+dy^2+dz^2$ coordinated in space
• $ds^2=dr^2+r^2d\; \backslash \; theta^2$ coordinated polar in the plan
• $ds^2=dr^2+r^2d\; \backslash \; theta^2+dz^2$ coordinated cylindrical in space
• $ds^2=dr^2+r^2d\; \backslash \; theta^2+r^2\; \backslash \; sin^2\; \backslash \; theta\; D\; \backslash \; phi^2$ coordinated spherical in space
• $ds^2=dx\_1^2+\dots \; +dx\_n^2$ coordinated Cartesian in Euclidean space with N dimensions

To give to these Formula Ones a rigorous direction, it would be necessary to introduce the general concepts of quadratic Forme and metric Tenseur. To obtain the usual formulas, it is however enough to handle interpretation in terms of infinitesimal elements length.

The S which points its nose in these formulas is however a quantity interesting for itself: the curvilinear X-coordinate, algebrized version length.

General concepts of rectifiable curve and length of arc

The preceding paragraph masking a certain number of difficulties and being valid only for derivable arcs (for which one can speak about Flight Path Vector), one proceeds to a more general and more geometrical definition.

Definitions

A Courbe is rectifiable if the polygonal lines registered on this one are uniformly limited length.

If $t\; \backslash \; mapsto\; F\; (T)$ follows the curve ( T in ''), then a polygonal Ligne $P$ registered is given by its tops $M\_i=f\; (t\_i)$, for any

The curve is known as rectifiable if the length $L\; (P)$ is raised by a constant $C$ independent of the choice of the $t\_i$.

The length of the curve is then by definition the $sup\; L\; (P)$ taken on all the possible polygonal lines.

Note:

• this definition has a direction since space is provided with a Norme

• the concept of rectifiable curve, length are independent of the choice of parameter;

• the curve given by the graph of $x\; \backslash \; mapsto\; X\; cos\; (1/x^2)$ is not rectifiable.

• the graph of a numerical function ( y=f (X) ) continuously derivable definite on a segment '' is rectifiable with the formula

$L\; (F)\; =\; \backslash \; int\_a^b\; \backslash \; sqrt\; \{1+\; (f\text{'}\; (T))^2\}\; dt$

• an arc parameterized by T in '' and continuously derivable is rectifiable with the formula

$L\; (F)\; =\; \backslash \; int\_a^b\; \backslash \; left\; \backslash |\backslash \; frac\; \{D\; \backslash \; overrightarrow\; \{F\}\}\; \{dt\}\; \backslash \; right\; \backslash |\; dt$

If the arc were lipschitzien, it would be still rectifiable.

Traditional calculations lengths

The formula of computation length utilizing an integral of square root, it is frequent that the length cannot be calculated using usual functions.

Thus a problem seemingly as simple as to calculate the circumference of the ellipse according to the semi-axes leads to integrals as one cannot clarify front: besides one speaks about integral elliptic (of second species in fact).

Curved for which calculation is possible using the usual functions

• the segment of right
• the arc of Cercle
• the arc of Parabole
• the arc of Chaînette
• the arc of Cycloïde, of Hypocycloïde, epicycloid
• the arc of Spirale logarithmic curve
• …

Continuity of the function length

Let us put the question in vague terms: do two “close” curves have close lengths?

Here a negative example. One takes the graph of the constant function equalizes to 0 on. This one is length 1. One easily manufactures a succession of continuous functions on, rectifiable, which converges uniformly towards F and of which the length does not converge towards 1.

For example: f₁ is a function Triangle with slopes 1 on and -1 on. Then f₂ is a function formed of two triangles, with slopes 1 on, -1 on, 1 on, -1 on, and so on (4,8,16 triangles,…). Each function f_n is a graph length $\backslash \; sqrt\; \{2\}$, and in addition there is uniform convergence well towards F .

To obtain results of continuity for the application “length”, one thus should not work with the standard of uniform convergence. One would rather need a standard of the type of those of the spaces of Sobolev.

History of the calculation lengths of arc

For very a long period of the History of mathematics, the concept of length of arc appeared perfectly inaccessible to calculation. The possibility of defining such a length was even often questioned, as it was the case for the Limite S.

Method of exhaustion

The first calculations concerning the lengths of arc were thus the calculations approached, according to the Méthode of exhaustion. Various geometricians, with an increasingly large virtuosity, ingénièrent themselves to register on the remarkable curves of the polygonal lines, with an increasingly fine cutting. They obtained an increasingly precise value thus approached for the length. The same method was used to carry out the calculations approached for the surface S.

The XVIIe century, the method of exhaustion allowed correction, by geometrical processes, of several curved transcendent: the Spiral logarithmic curve by Torricelli in 1645 (allotted by some to John Wallis in the years 1650), the Cycloid by Christopher Wren in 1658, and the Catenoid by Gottfried Leibniz in 1691.

In 1659 took place the correction of the first algebraic Courbe noncommonplace, the semicubical Parabole (or parabola of Neile, of the name of its discoverer).

First steps in infinitesimal calculus

Before even the full advent of the infinitesimal calculus, the first foundations to obtain the integral formula giving the length of arc were thrown independently by Hendrik van Heuraet and Pierre de Fermat.

In 1659 van Heuraet published a construction by which the length of arc could be interpreted like the surface under a curve - thus indeed an integral - and applied that to the case of the Parabole. In 1660, Fermat published a more general theory, including this result, in Of linearum curvarum cum lineis rectis comparatione dissertatio geometrica .

Problems of minimum

Shorter way between two points

It is well-known that in Euclidean Géométrie, the straight line is the shortest way between two points.

• if one takes the general standard of rectifiable arc, the property is immediate

Indeed the length of the arc is higher than that of the straight line uniting origin and end of the arc (which is a particular polygonal line). All the other polygonal lines are besides bigger length by the triangular inequality.

• if one uses as expression length the integral of the standard of the derived vector:

One calls has and B the ends of the arc and one compares the length of the arc with that of the arc obtained by orthogonal Projection on (has, B) . As orthogonal projection decreases the standards, our arc is longer than an arc traced on a line and connecting has with B , him even longer than the arc ''.

Length and energy

In many problems of minimization, one can try to use the energy of the arc

$E=\; \backslash \; int\_a^b\; \backslash \; left\; \backslash |\backslash \; frac\; \{D\; \backslash \; overrightarrow\; \{F\}\}\; \{dt\}\; \backslash \; right\; \backslash |^2\; dt~$

who with the advantage of not utilizing square root.

Moreover energy and length are not without bond: when the arc admits a normal parameter setting (at uniform speed 1), length and energy are equal. The curve of minimal energy between two points is still the line, traversed at uniform speed.

This energy represents a “elastic energy of deformation”. One utilizes it for example in the isoperimetric inequality or Géodésique S. seeks it.

Other problems of minimization

• isoperimetric Inequality: the circle is the closed Courbe smaller length enclosing a field of surface given.

• the Spline S minimize energy between two points with initial Flight Path Vectors given.

• the search for shorter ways in a curved space, or Geodetic S, utilizes a framework much more general: the Geometry riemannienne. One gives below the expression length within this framework.

• the Brachistochrone minimizes run time between two points, for particles subjected to a constant acceleration (gravity). This makes it possible to make a parallel between presence of this field of acceleration and search for shorter way in a curved space (cf General relativity).

The last two problems require to call upon the techniques of the Calcul of the variations.

Generalization: length of an arc in a riemannienne variety

Let us suppose that M is a Variété riemannienne and that γ: '' B '' → M is a Courbe continuement derivable on M , then one can define the length and the energy of this curve by:

$L\; (\backslash \; gamma)\; =\; \backslash \; int\_a^b\; \backslash |\backslash \; gamma\text{'}\; (T)\; \backslash |\backslash ;\; dt\; \backslash \; qquad\; E\; (\backslash \; gamma)\; =\; \backslash \; int\_a^b\; \backslash |\backslash \; gamma\text{'}\; (T)\; \backslash |^2\; \backslash ;\; dt$ where $\backslash |\backslash \; gamma\text{'}\; (T)\; \backslash |$ is the standard of γ' (T) induced by the scalar product taken on tangent space in γ (T).

The search for shorter ways, than it is necessary to consider under two aspects: room and total, concerns then the calculation of Géodésique S.

Definition

The length of the chart of a function could be connected with the Périmètre of a geometrical object such as a Cercle, a Triangle. Here one further pushes the concept while generalizing with all the functions.

The length of the chart of $\backslash \; displaystyle\; F\; (X)$ on $\backslash \; displaystyle\; has;\; B$ ($\backslash \; displaystyle\; a$ and $has,\; B\; \backslash \; in\; \backslash \; R$) is:

$\backslash \; int\_\; \{has\}\; ^\; \{B\}\; \backslash \; sqrt\; \{1+f\text{'}\; (X)\; ^\; \{2\}\}\; \backslash ,\; dx$

External bonds

• Demonstration

See too

• Theorem isoperimetric

• Geodetic Parameter setting

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