Citizendia
Your Ad Here

Determining the length of an irregular arc segment—also called rectification of a curve—was historically difficult. In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object 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. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives

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". A polygonal chain, polygonal curve, polygonal path, or piecewise linear curve, is a connected series of Line segments More formally a polygonal

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.

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

Some curves have infinite length. A curve is rectifiable if its length is finite.

Modern methods

Consider a function f(x) such that f(x) and f′(x) (its derivative with respect to x) are continuous on [a, b] . The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function 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, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set The length s of the part of the graph of f between x = a and x = b is found by the formula

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. Distance is a numerical description of how far apart objects are As the number of line segments increases (to infinity by use of the integral) this approximation becomes an exact value.

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

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

In most cases, including even simple curves, there are no closed-form solutions of arc length and numerical integration is necessary. In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by In Numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite Integral, and by extension

Curves with closed-form solution for arc length include the catenary, circle, cycloid, logarithmic spiral, parabola, semicubical parabola and (mathematically, a curve) straight line. In Physics and Geometry, the catenary is the theoretical Shape of a hanging flexible Chain or Cable when supported at its ends and Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the A cycloid is the curve defined by the path of a point on the edge of circular wheel as the wheel rolls along a straight line Definition In Polar coordinates ( r, θ the curve can be written as r = ae^{b\theta}\ or \theta In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular In Mathematics, a semicubical parabola is a Curve defined Parametrically as x = t^2 \ y = at^3 The lack of closed form solution for the arc length of an elliptic arc led to the development of the elliptic integrals. In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a In Integral calculus, elliptic integrals originally arose in connection with the problem of giving the Arc length of an Ellipse.

Derivation

For a small piece of curve, Δs can be approximated with the Pythagorean theorem
For a small piece of curve, Δs can be approximated with the Pythagorean theorem
A representative linear element of the function
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. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry The word linear comes from the Latin word linearis, which means created by lines. To make the value exact, and not an approximation, infinitely many linear elements are needed. An approximation (represented by the symbol ≈ is an inexact representation of something that is still close enough to be useful Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness This means that each element is infinitely small. This fact manifests itself later on when an integral is used. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space

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. In mathematics and more specifically in Differential calculus, the term differential has several interrelated meanings We will call the horizontal element of this distance dx, and the vertical element dy.

The distance formula tells us that

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. Distance is a numerical description of how far apart objects are 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_1\left(-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{25}{9}\right)

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. In Mathematics, a hypergeometric series is a Power series in which the ratios of successive Coefficients k is a Rational function The area of study known as the history of mathematics is primarily an investigation into the origin of new discoveries in Mathematics and to a lesser extent an investigation 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. Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer 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 The first ground was broken in this field, as it often has been in calculus, by approximation. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives An approximation (represented by the symbol ≈ is an inexact representation of something that is still close enough to be useful People began to inscribe polygons within the curves and compute the length of the sides for a somewhat accurate measurement of the length. In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit By using more segments, and by decreasing the length of each segment, they were able to obtain a more and more accurate approximation.

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. In Mathematics, a transcendental curve is a Curve that is not an Algebraic curve. Definition In Polar coordinates ( r, θ the curve can be written as r = ae^{b\theta}\ or \theta Evangelista Torricelli ( ( October 15, 1608 &ndash October 25, 1647) was an Italian physicist and mathematician John Wallis ( November 23, 1616 - October 28, 1703) was an English mathematician who is given partial credit for the A cycloid is the curve defined by the path of a point on the edge of circular wheel as the wheel rolls along a straight line Sir Christopher Wren ( 20 October 1632 &ndash 25 February 1723) was a 17th century English Designer, Astronomer In Physics and Geometry, the catenary is the theoretical Shape of a hanging flexible Chain or Cable when supported at its ends and

In 1659, Wallis credited William Neile's discovery of the first rectification of a nontrivial algebraic curve, the semicubical parabola. In Algebraic geometry, an algebraic curve is an Algebraic variety of dimension one In Mathematics, a semicubical parabola is a Curve defined Parametrically as x = t^2 \ y = at^3

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 (1633 Haarlem - 1660? Leiden) was a Dutch mathematician Pierre de Fermat pjɛːʁ dəfɛʁ'ma ( 17 August 1601 or 1607/8 &ndash 12 January 1665) was a French Lawyer at the

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 Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular In 1660, Fermat published a more general theory containing the same result in his De linearum curvarum cum lineis rectis comparatione dissertatio geometrica.

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

y = x^{3/2} \,

whose tangent at x = a had a slope of

{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. For the tangent function see Trigonometric functions. For other uses see Tangent (disambiguation. Slope is used to describe the steepness incline gradient or grade of a straight line. To find the length of the segment AC, he used the Pythagorean theorem:

\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. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry

Curves with infinite length

The Koch curve.
The Koch curve.
The graph of .
The graph of x\,\sin(1/x).

As mentioned above, some curves are non-rectifiable, that is, they have infinite length. One such example is just a line, but the line is non-rectifiable for rather trivial reasons. There are continuous curves for which any arc on the curve (containing more than a single point) has infinite length. An example of such a curve is the Koch curve. The Koch snowflake (or Koch star) is a mathematical Curve and one of the earliest Fractal curves to have been described A less extreme example of a curve with infinite length is the graph of the function f(x)=x\,\sin (1/x) in the range 0\le x\le 1, where we take f(0) = 0. Sometimes, the Hausdorff dimension and Hausdorff measure are used to "measure" the size of infinite length curves. In Mathematics, the Hausdorff dimension (also known as the Hausdorff–Besicovitch dimension) is an extended non-negative Real number associated In Mathematics a Hausdorff measure is a type of Outer measure, named for Felix Hausdorff, that assigns a number in to each set in R

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 (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold.

The length of \gamma \, is by definition l(\gamma)=\int_{0}^{1} \sqrt{ \pm g(\dot\gamma(t),\dot\gamma(t)) }\,dt\ , where \dot\gamma(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. This page is about the scientific concept of relativity for philosophical or sociological theories about relativity see Relativism. 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

References

Farouki, Rida T. In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object In Geometry, an arc is a closed segment of a Differentiable Curve in the two-dimensional plane; for example a circular In Numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite Integral, and by extension In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces (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

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
 Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic