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Encyclopedia > Arc length

Determining the length of an irregular arc segment—also called rectification of a curve—was historically difficult. Although many methods were used for specific curves, the advent of calculus led to a general formula that provides closed-form solutions in some cases. Rectification has the following technical meanings. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... Calculus (from Latin, pebble or little stone) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ... In mathematics, an equation or system of equations is said to have a closed-form solution if, and only if, at least one solution can be expressed analytically in terms of a bounded number of well-known operations. ...

Contents

Precise definition

Choose a finite number of points along a curve and connect each point to the next with a straight line. The sum of the lengths of such line segments is the length of a "polygonal path". Image File history File links This is a lossless scalable vector image. ... A polygonal chain, polygonal curve, polygonal path, piecewise linear curve, a connected series of line segments. ...


Definition: The length of the curve is the smallest number that such lengths of polygonal paths can never exceed, no matter how close together the discretely placed endpoints of line segments are.


In the language of mathematicians, the arc length is the supremum of all lengths of such polygonal paths. In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ...


This definition does not require the curve to be "smooth"; it need not be either the graph or the image of a differentiable function.


Modern methods

Consider a function f(x) such that f(x) and f′(x) (its derivative with respect to x) are continuous on [ab] . The length s of the part of the graph of f between x = a and x = b is found by the formula Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...

s = int_{a}^{b} sqrt { 1 + [f'(x)]^2 }, dx.

which is derived from the distance formula approximating the arc length with many small lines. As the number of line segments increases (to infinity by use of the integral) this approximation becomes an exact value. For distance between people, see proxemics. ...


If a curve is defined parametrically by x = X(t) and y = Y(t), then its arc length between t = a and t = b is

s = int_{a}^{b} sqrt { [X'(t)]^2 + [Y'(t)]^2 }, dt.

This is more clearly a consequence of the distance formula where instead of a Δx and Δy , we take the limit. A useful mnemonic is

s = lim sum_a^b sqrt { Delta x^2 + Delta y^2 } = int_{a}^{b} sqrt { dx^2 + dy^2 } = int_{a}^{b} sqrt { left(frac{dx}{dt}right)^2 + left(frac{dy}{dt}right)^2 },dt.

If a function is defined in polar coordinates by r = f(θ) then the arc length is given by A polar grid with several angles labeled in degrees In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by an angle and a distance. ...

s = int_a^b sqrt{r^2+left(frac{dr}{dtheta}right)^2} , dtheta.

In most cases, including even simple curves, there are no closed-form solutions of arc length and numerical integration is necessary. In mathematics, an equation or system of equations is said to have a closed-form solution just in case a solution can be expressed analytically in terms of a bounded number of well-known operations. ... Numerical Integration with the Monte Carlo method: Nodes are random equally distributed. ...


Curves with closed-form solution for arc length include the catenary, circle, cycloid, logarithmic spiral, parabola, semicubical parabola and (mathematically, a curve) straight line. The lack of closed form solution for the arc length of an elliptic arc led to the development of the elliptic integrals. For its use in ring theory, see Catenary ring. ... Circle illustration This article is about the shape and mathematical concept of circle. ... Cycloid (red) generated by a rolling circle A cycloid is the curve defined by a fixed point on a wheel as it rolls, or, more precisely, the locus of a point on the rim of a circle rolling along a straight line. ... A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: παραβολή) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane... Semicubical parabolas for different values of a. ... A representation of one line Three lines — the red and blue lines have same slope, while the red and green ones have same y-intercept. ... For other uses, see Ellipse (disambiguation). ... In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler. ...


Derivation

A representative linear element of the function begin{cases} y = t^5  x = t^3 end{cases}

In order to approximate the arc length of the curve, it is split into many linear segments. To make the value exact, and not an approximation, infinitely many linear elements are needed. This means that each element is infinitely small. This fact manifests itself later on when an integral is used. Image File history File links No higher resolution available. ... The word linear comes from the Latin word linearis, which means created by lines. ... It has been suggested that this article or section be merged with estimation. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... In calculus, the integral of a function is an extension of the concept of a sum. ...


Begin by looking at a representative linear segment (see image) and observe that its length (element of the arc length) will be the differential ds. We will call the horizontal element of this distance dx, and the vertical element dy. In mathematics, differential has several meanings: Differential (infinitesimal), an infinitesimal change in the value of a function In differential topology: Differential form, a generalization that accommodates multiplication and differentiation of differentials In addition, differentials and differential forms on manifolds come with the following notions of differentiation: Exterior derivative, a notion...


The distance formula tells us that For distance between people, see proxemics. ...

ds = sqrt{dx^2 + dy^2}.,

Since the function is defined in time, segments (ds) are added up across infintesimally small intervals of time (dt) yielding the integral

int_a^b sqrt{bigg(frac{dx}{dt}bigg)^2+bigg(frac{dy}{dt}bigg)^2},dt,

If convenient values for t were chosen, i.e. t = x, it would yield:

int_a^b sqrt{1+bigg(frac{dy}{dx}bigg)^2},dx,

which is the arc length from t = a to t = b of the parametric function f(t).


For example, the curve in this figure is defined by

begin{cases} y = t^5,  x = t^3 end{cases}.

Subsequently, the arc length integral for values of t from −1 to 1 is

int_{-1}^1 sqrt{(3t^2)^2 + (5t^4)^2},dt = int_{-1}^1 sqrt{9t^4 + 25t^8},dt.

Using computational approximations, we can obtain a very accurate (but still approximate) arc length of 2.905. An expression in terms of the hypergeometric function can be obtained: it is 2,{}_2F_1left(-frac{1}{2},frac{3}{4};frac{7}{4};-frac{25}{9}right) In mathematics, a hypergeometric series is a power series in which the ratios of successive coefficients k is a rational function of k. ...


Historical methods

Ancient

For much of the history of mathematics, even the greatest thinkers considered it impossible to compute the length of an irregular arc. Although Archimedes had pioneered a rectangular approximation for finding the area beneath a curve with his method of exhaustion, few believed it was even possible for curves to have definite lengths, as do straight lines. The first ground was broken in this field, as it often has been in calculus, by approximation. People began to inscribe polygons within the curves and compute the length of the sides for a somewhat accurate measurement of the length. By using more segments, and by decreasing the length of each segment, they were able to obtain a more and more accurate approximation. For a list of biographies of mathematicians, see list of mathematicians. ... Archimedes of Syracuse (Greek: c. ... The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. ... Calculus (from Latin, pebble or little stone) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ... It has been suggested that this article or section be merged with estimation. ... Look up polygon in Wiktionary, the free dictionary. ...


1600s

In the 1600s, the method of exhaustion led to the rectification by geometrical methods of several transcendental curves: the logarithmic spiral by Evangelista Torricelli in 1645 (some sources say John Wallis in the 1650s), the cycloid by Christopher Wren in 1658, and the catenary by Gottfried Leibniz in 1691. Many inventions and institutions are created, including Hans Lippershey with the telescope (1608, used by Galileo the next year), the newspaper Avisa Relation oder Zeitung in Augsburg, and Cornelius Drebbel with the thermostat (1609). ... In mathematics, a transcendental curve is a curve that is not an algebraic curve. ... A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. ... Evangelista Torricelli portrayed on the frontpage of Lezioni dEvangelista Torricelli. ... // Events January 10 - Archbishop Laud executed on Tower Hill, London. ... John Wallis John Wallis (November 22, 1616 - October 28, 1703) was an English mathematician who is given partial credit for the development of modern calculus. ... Significant Events and Trends World Leaders King Frederick III of Denmark (1648 - 1670). ... Cycloid (red) generated by a rolling circle A cycloid is the curve defined by a fixed point on a wheel as it rolls, or, more precisely, the locus of a point on the rim of a circle rolling along a straight line. ... Sir Christopher Wren, (20 October 1632–25 February 1723) was a 17th century English designer, astronomer, geometrician, and the greatest English architect of his time. ... Events January 13 - Edward Sexby, who had plotted against Oliver Cromwell, dies in Tower of London February 6 - Swedish troops of Charles X Gustav of Sweden cross The Great Belt (Storebælt) in Denmark over frozen sea May 1 - Publication of Hydriotaphia, Urn Burial and The Garden of Cyrus by... For its use in ring theory, see Catenary ring. ... It has been suggested that this article be split into multiple articles. ... Events March 5 - French troops under Marshal Louis-Francois de Boufflers besiege the Spanish-held town of Mons March 20 - Leislers Rebellion - New governor arrives in New York - Jacob Leisler surrenders after standoff of several hours March 29 - Siege of Mons ends to the city’s surrender May 6...


In 1659, Wallis credited William Neile's discovery of the first rectification of a nontrivial algebraic curve, the semicubical parabola. // Events May 25 - Richard Cromwell resigns as Lord Protector of England following the restoration of the Long Parliament, beginning a second brief period of the republican government called the Commonwealth. ... In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ... Semicubical parabolas for different values of a. ...


Integral form

Before the full formal development of the calculus, the basis for the modern integral form for arc length was independently discovered by Hendrik van Heuraet and Pierre Fermat. Hendrik van Heuraet (1634-1660?) was a Dutch mathematician. ... Pierre de Fermat Pierre de Fermat (August 17, 1601 – January 12, 1665) was a French lawyer at the Parliament of Toulouse and a mathematician who is given credit for the development of modern calculus. ...


In 1659 van Heuraet published a construction showing that arc length could be interpreted as the area under a curve - this integral, in effect - and applied it to the parabola. In 1660, Fermat published a more general theory containing the same result in his De linearum curvarum cum lineis rectis comparatione dissertatio geometrica. // Events May 25 - Richard Cromwell resigns as Lord Protector of England following the restoration of the Long Parliament, beginning a second brief period of the republican government called the Commonwealth. ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: παραβολή) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane... // Events January 1 - Colonel George Monck with his regiment crosses from Scotland to England at the village of Coldstream and begins advance towards London in support of English Restoration. ...

Fermat's method of determining arc length
Fermat's method of determining arc length

Building on his previous work with tangents, Fermat used the curve Image File history File links Download high resolution version (1350x1350, 82 KB)This graph is meant to help accompany the article Length of an arc and descibe the method Fermat used. ... Image File history File links Download high resolution version (1350x1350, 82 KB)This graph is meant to help accompany the article Length of an arc and descibe the method Fermat used. ...

 y = x^{3/2} ,

whose tangent at x = a had a slope of In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonometry. ... Look up Slope in Wiktionary, the free dictionary. ...

 {3 over 2} a^{1/2}

so the tangent line would have the equation

 y = {3 over 2} {a^{1/2}}(x - a) + f(a).

Next, he increased a by a small amount to a + ε, making segment AC a relatively good approximation for the length of the curve from A to D. To find the length of the segment AC, he used the Pythagorean theorem: In mathematics, the Pythagorean theorem or Pythagoras theorem is a relation in Euclidean geometry among the three sides of a right triangle. ...

begin{align} AC^2 &{}= AB^2 + BC^2  &{} = varepsilon^2 + {9 over 4} a varepsilon^2  &{}=varepsilon^2 left (1 + {9 over 4} a right ) end{align}

which, when solved, yields

AC = varepsilon sqrt { 1 + {9 over 4} a }.

In order to approximate the length, Fermat would sum up a sequence of short segments.


Generalization to (pseudo-)Riemannian manifolds

Let be M , a (pseudo-)Riemannian manifold, gamma : [0,1] to M a curve in M , and g , the (pseudo-) metric tensor. In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ...


The length of gamma , is by definition l(gamma)=int_{0}^{1} sqrt{ pm g(dotgamma(t),dotgamma(t)) },dt , where dotgamma(t) in T_{gamma(t)}M , represents the tangent vector of gamma , at gamma (t) ,. The sign in the square root is chosen once for a given curve, to ensure that the square root is a real number. The positive sign is chosen for spacelike curves; in a pseudo-Riemannian manifold, the negative sign may be chosen for timelike curves.


In Relativity theory, arc-length of timelike curves (world lines) is the proper time elapsed along the world line. Albert Einsteins theory of relativity is a set of two theories in physics: special relativity and general relativity. ... In physics, the world line of an object is the unique path of that object as it travels through 4-dimensional spacetime. ... In relativity, proper time is time measured by a single clock between events that occur at the same place as the clock. ...


See also

In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... In Euclidean geometry, an arc is a closed segment of a differentiable curve in the two-dimensional plane; for example, a circular arc is a segment of a circle. ... Numerical Integration with the Monte Carlo method: Nodes are random equally distributed. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. In presence of a metric, geodesics are defined to be (locally) the shortest path between points on the space. ...

References

Farouki, Rida T. (1999). Curves from motion, motion from curves. In P-J. Laurent, P. Sablonniere, and L. L. Schumaker (Eds.), Curve and Surface Design: Saint-Malo 1999, pp.63-90, Vanderbilt Univ. Press. ISBN 0-8265-1356-5.


External links


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