Arc length

 ( Integral Calculus ) 

Arc length is the distance between two points along a curve. It can be formalized mathematically for using vector calculus and differential geometry, or for curves that might not necessarily be smooth as a limit of lengths of . The curves for which this limit exists are called rectifiable curves, and the process of determining their arc length in this way is called curve rectification.

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Definitions

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As an integral In the most basic formulation of arc length for a parametric curve (thought of as the trajectory of a particle, moving in the plane with position $(x(t),y(t))$ at time $t$) the arc length is obtained by integrating speed (the magnitude of the velocity vector) over the curve with respect to time. Thus the length of a continuously differentiable curve in the Euclidean plane, parameterized as $(x(t),y(t))$, for $a\backslash le\; t\backslash le\; b$, is given as the integral

(2025). 9780131469686, Pearson Prentice Hall. . ISBN 9780131469686

$$L\; =\; \backslash int\_a^b\; \backslash sqrt\{x\text{'}(t)^2+y\text{'}(t)^2\}\backslash ,dt.$$ Here the integrand (the square root inside the integral) is the particle's speed. This defining integral of arc length does not always have a closed-form expression, and numerical integration may be used instead to obtain numerical values of arc length.

More generally, for curves that are not necessarily in the plane, let $f\backslash colona,b\backslash to\backslash R^n$ be continuously differentiable (i.e., the derivative is a continuous function) function. The length of the curve is given by the formula $$L(f)\; =\; \backslash int\_a^b\; |f\text{'}(t)|\backslash ,dt$$ where $|f\text{'}(t)|$ is the Euclidean norm of the tangent vector $f\text{'}(t)$ to the curve.

A curve can be parameterized in infinitely many ways. The arc length of the curve is the same regardless of the parameterization used to define the curve.

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As a limit A curve in the Euclidean space can be approximated by connecting a number of points on the curve, in consecutive order, using (straight) to create a polygonal chain. One may calculate the length of each linear segment using the Pythagorean theorem, and sum these lengths to obtain the total length of the chain; that approximation is known as the (cumulative) chordal distance.

(1967). 9780080955452, Academic Press. . ISBN 9780080955452

If additional points are placed along the curve, the chordal distance will not decrease. It can increase to a finite bound, in the limit, as the length of the longest segment in the chain decreases to zero, or it can increase without bound. In the case where the length of the refined polygonal chain increases to a finite bound, this bound is the length of the curve. For a smooth curve of finite length, this limiting length always equals the length obtained from integration, but this limit-based definition of length also applies to certain non-smooth curves. A curve with the property that every arc between two points of the curve has finite length, when measured in this way, is called a rectifiable curve, even if the whole curve has infinite length.

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Sign A signed arc length can be defined to convey a sense of orientation or "direction" with respect to a reference point taken as origin in the curve (see also: curve orientation and signed distance).

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Finding arc lengths by integration If a plane curve in $\backslash R^2$ is defined by the equation $y\; =\; f(x),$ where $f$ is continuously differentiable, then it is simply a special case of a parametric equation where $x\; =\; t$ and $y\; =\; f(t).$ The Euclidean distance of each infinitesimal segment of the arc can be given by: $$\backslash sqrt\{dx^2\; +\; dy^2\}\; =\; \backslash sqrt\{1\; +\; \backslash left(\backslash frac\{dy\}\{dx\}\backslash right)^2\; \backslash ,\}dx.$$

The arc length is then given by:

$$s=\backslash int\_a^b\; \backslash sqrt\{1\; +\; \backslash left(\backslash frac\{dy\}\{dx\}\backslash right)^2\; \backslash ,\}dx.$$

Curves with closed-form solutions for arc length include the catenary, circle, cycloid, logarithmic spiral, parabola, semicubical parabola and straight line. The lack of a closed form solution for the arc length of an elliptic and Hyperbola arc led to the development of the elliptic integrals.

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Numerical integration In most cases, including even simple curves, there are no closed-form solutions for arc length and numerical integration is necessary. Numerical integration of the arc length integral is usually very efficient. For example, consider the problem of finding the length of a quarter of the unit circle by numerically integrating the arc length integral. The upper half of the unit circle can be parameterized as $y\; =\; \backslash sqrt\{1\; -\; x^2\}.$ The interval $x\; \backslash in\; \backslash left-\backslash sqrt\{2\}/2,$ corresponds to a quarter of the circle. Since $dy/dx\; =\; -x\; \backslash big/\; \backslash sqrt\{1\; -\; x^2\}$ and $1\; +\; (dy/dx)^2\; =\; 1\backslash big/\backslash left(1\; -\; x^2\backslash right),$ the length of a quarter of the unit circle is

$$\backslash int\_\{-\backslash sqrt\{2\}/2\}^\{\backslash sqrt\{2\}/2\}\; \backslash frac\{dx\}\{\backslash sqrt\{1\; -\; x^2\}\}\backslash ,\; .$$

The 15-point Gauss–Kronrod rule estimate for this integral of differs from the true length of

$$\backslash arcsin\; x\backslash bigg|^\{\backslash sqrt\{2\}/2\}\_\{-\backslash sqrt\{2\}/2\}\; =\; \backslash frac\{\backslash pi\}\{2\}$$

by and the 16-point Gaussian quadrature rule estimate of differs from the true length by only . This means it is possible to evaluate this integral to almost Machine epsilon with only 16 integrand evaluations.

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Curve on a surface Let $\backslash mathbf\{x\}(u,v)$ be a surface mapping and let $\backslash mathbf\{C\}(t)\; =\; (u(t),\; v(t))$ be a curve on this surface. The integrand of the arc length integral is $\backslash left|\backslash left(\backslash mathbf\{x\}\backslash circ\backslash mathbf\{C\}\backslash right)\text{'}(t)\backslash right|.$ Evaluating the derivative requires the chain rule for vector fields:

$$D(\backslash mathbf\{x\}\; \backslash circ\; \backslash mathbf\{C\})\; =\; (\backslash mathbf\{x\}\_u\; \backslash \; \backslash mathbf\{x\}\_v)\backslash binom\{u\text{'}\}\{v\text{'}\}\; =\; \backslash mathbf\{x\}\_u\; u\text{'}\; +\; \backslash mathbf\{x\}\_v\; v\text{'}.$$

The squared norm of this vector is

$$\backslash left(\backslash mathbf\{x\}\_u\; u\text{'}\; +\; \backslash mathbf\{x\}\_v\; v\text{'}\backslash right)\; \backslash cdot\; (\backslash mathbf\{x\}\_u\; u\text{'}\; +\; \backslash mathbf\{x\}\_v\; v\text{'})\; =\; g\_\{11\}\backslash left(u\text{'}\backslash right)^2\; +\; 2g\_\{12\}u\text{'}v\text{'}\; +\; g\_\{22\}\backslash left(v\text{'}\backslash right)^2$$

(where $g\_\{ij\}$ is the first fundamental form coefficient), so the integrand of the arc length integral can be written as $\backslash sqrt\{g\_\{ab\}\; \backslash left(u^a\backslash right)\text{'}\; \backslash left(u^b\backslash right)\text{'}\; \backslash ,\}$ (where $u^1\; =\; u$ and $u^2\; =\; v$).

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Other coordinate systems Let $\backslash mathbf\{C\}(t)\; =\; (r(t),\; \backslash theta(t))$ be a curve expressed in polar coordinates. The mapping that transforms from polar coordinates to rectangular coordinates is

$$\backslash mathbf\{x\}(r,\; \backslash theta)\; =\; (r\backslash cos\backslash theta,\; r\backslash sin\backslash theta).$$

The integrand of the arc length integral is $\backslash left|\backslash left(\backslash mathbf\{x\}\backslash circ\backslash mathbf\{C\}\backslash right)\text{'}(t)\backslash right|.$ The chain rule for vector fields shows that $D(\backslash mathbf\{x\}\; \backslash circ\; \backslash mathbf\{C\})\; =\; \backslash mathbf\{x\}\_r\; r\text{'}\; +\; \backslash mathbf\{x\}\_\{\backslash theta\}\; \backslash theta\text{'}.$ So the squared integrand of the arc length integral is

$$\backslash left(\backslash mathbf\{x\_r\}\backslash cdot\backslash mathbf\{x\_r\}\backslash right)\backslash left(r\text{'}\backslash right)^2\; +\; 2\backslash left(\backslash mathbf\{x\}\_r\backslash cdot\backslash mathbf\{x\}\_\{\backslash theta\}\backslash right)r\text{'}\backslash theta\text{'}\; +\; \backslash left(\backslash mathbf\{x\}\_\{\backslash theta\}\backslash cdot\backslash mathbf\{x\}\_\{\backslash theta\}\backslash right)\backslash left(\backslash theta\text{'}\backslash right)^2\; =\; \backslash left(r\text{'}\backslash right)^2\; +\; r^2\backslash left(\backslash theta\text{'}\backslash right)^2.$$

So for a curve expressed in polar coordinates, the arc length is: $$$$

 \int_{t_1}^{t_2} \sqrt{\left(\frac{dr}{dt}\right)^2 + r^2\left(\frac{d\theta}{dt}\right)^2 \,} dt =
 \int_{\theta(t_1)}^{\theta(t_2)} \sqrt{\left(\frac{dr}{d\theta}\right)^2 + r^2 \,} d\theta.
     

The second expression is for a polar graph $r\; =\; r(\backslash theta)$ parameterized by $t=\backslash theta$.

Now let $\backslash mathbf\{C\}(t)\; =\; (r(t),\; \backslash theta(t),\; \backslash phi(t))$ be a curve expressed in spherical coordinates where $\backslash theta$ is the polar angle measured from the positive $z$-axis and $\backslash phi$ is the azimuthal angle. The mapping that transforms from spherical coordinates to rectangular coordinates is $$\backslash mathbf\{x\}(r,\; \backslash theta,\; \backslash phi)\; =\; (r\; \backslash sin\backslash theta\; \backslash cos\backslash phi,\; r\backslash sin\backslash theta\; \backslash sin\backslash phi,\; r\backslash cos\backslash theta).$$

Using the chain rule again shows that $D(\backslash mathbf\{x\}\backslash circ\backslash mathbf\{C\})\; =\; \backslash mathbf\{x\}\_r\; r\text{'}\; +\; \backslash mathbf\{x\}\_\{\backslash theta\}\backslash theta\text{'}\; +\; \backslash mathbf\{x\}\_\{\backslash phi\}\backslash phi\text{'}.$ All $\backslash mathbf\{x\}\_i\; \backslash cdot\; \backslash mathbf\{x\}\_j$ where $i$ and $j$ differ are zero, so the squared norm of this vector is $$\backslash left(\backslash mathbf\{x\}\_r\; \backslash cdot\; \backslash mathbf\{x\}\_r\backslash right)\backslash left(r\text{'}^2\backslash right)\; +\; \backslash left(\backslash mathbf\{x\}\_\{\backslash theta\}\; \backslash cdot\; \backslash mathbf\{x\}\_\{\backslash theta\}\backslash right)\backslash left(\backslash theta\text{'}\backslash right)^2\; +\; \backslash left(\backslash mathbf\{x\}\_\{\backslash phi\}\; \backslash cdot\; \backslash mathbf\{x\}\_\{\backslash phi\}\backslash right)\backslash left(\backslash phi\text{'}\backslash right)^2\; =\; \backslash left(r\text{'}\backslash right)^2\; +\; r^2\backslash left(\backslash theta\text{'}\backslash right)^2\; +\; r^2\; \backslash sin^2\backslash theta\; \backslash left(\backslash phi\text{'}\backslash right)^2.$$

So for a curve expressed in spherical coordinates, the arc length is $$\backslash int\_\{t\_1\}^\{t\_2\}\; \backslash sqrt\{\backslash left(\backslash frac\{dr\}\{dt\}\backslash right)^2\; +\; r^2\backslash left(\backslash frac\{d\backslash theta\}\{dt\}\backslash right)^2\; +\; r^2\backslash sin^2\backslash theta\; \backslash left(\backslash frac\{d\backslash phi\}\{dt\}\backslash right)^2\; \backslash ,\}\; dt.$$

A very similar calculation shows that the arc length of a curve expressed in cylindrical coordinates is $$\backslash int\_\{t\_1\}^\{t\_2\}\; \backslash sqrt\{\backslash left(\backslash frac\{dr\}\{dt\}\backslash right)^2\; +\; r^2\backslash left(\backslash frac\{d\backslash theta\}\{dt\}\backslash right)^2\; +\; \backslash left(\backslash frac\{dz\}\{dt\}\backslash right)^2\; \backslash ,\}\; dt.$$

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Simple cases

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Arcs of circles Arc lengths are denoted by s, since the Latin word for length (or size) is spatium.

In the following lines, $r$ represents the radius of a circle, $d$ is its diameter, $C$ is its circumference, $s$ is the length of an arc of the circle, and $\backslash theta$ is the angle which the arc subtends at the centre of the circle. The distances $r,\; d,\; C,$ and $s$ are expressed in the same units.

• $C\; =\; 2\backslash pi\; r,$ which is the same as $C\; =\; \backslash pi\; d.$ This equation is a definition of $\backslash pi.$
• If the arc is a semicircle, then $s\; =\; \backslash pi\; r.$
• For an arbitrary circular arc:
• If $\backslash theta$ is in then $s\; =\; r\backslash theta.$ This is a definition of the radian.
• If $\backslash theta$ is in degrees, then $s\; =\; \backslash frac\{\backslash pi\; r\; \backslash theta\}\{180^\backslash circ\},$ which is the same as $s\; =\; \backslash frac\{C\; \backslash theta\}\{360^\backslash circ\}.$
• If $\backslash theta$ is in grads (100 grads, or grades, or gradians are one right-angle), then $s\; =\; \backslash frac\{\backslash pi\; r\; \backslash theta\}\{200\backslash text\{\; grad\}\},$ which is the same as $s\; =\; \backslash frac\{C\backslash theta\}\{400\backslash text\{\; grad\}\}.$
• If $\backslash theta$ is in turns (one turn is a complete rotation, or 360°, or 400 grads, or $2\backslash pi$ radians), then $s\; =\; C\backslash theta/1\backslash text\{\; turn\}$.

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Great circles on Earth Two units of length, the nautical mile and the metre (or kilometre), were originally defined so the lengths of arcs of on the Earth's surface would be simply numerically related to the angles they subtend at its centre. The simple equation $s\; =\; \backslash theta$ applies in the following circumstances:

• if $s$ is in nautical miles, and $\backslash theta$ is in ( degree), or
• if $s$ is in kilometres, and $\backslash theta$ is in .

The lengths of the distance units were chosen to make the circumference of the Earth equal kilometres, or nautical miles. Those are the numbers of the corresponding angle units in one complete turn.

Those definitions of the metre and the nautical mile have been superseded by more precise ones, but the original definitions are still accurate enough for conceptual purposes and some calculations. For example, they imply that one kilometre is exactly 0.54 nautical miles. Using official modern definitions, one nautical mile is exactly 1.852 kilometres, which implies that 1 kilometre is about nautical miles.CRC Handbook of Chemistry and Physics, p. F-254 This modern ratio differs from the one calculated from the original definitions by less than one part in 10,000.

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Other simple cases

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Historical methods

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Antiquity 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 way of 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 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. In particular, by inscribing a polygon of many sides in a circle, they were able to find approximate values of π.

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17th century In the 17th century, 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 1659, Wallis credited William Neile's discovery of the first rectification of a nontrivial algebraic curve, the semicubical parabola. The accompanying figures appear on page 145. On page 91, William Neile is mentioned as Gulielmus Nelius.

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Integral form Before the full formal development of calculus, the basis for the modern integral form for arc length was independently discovered by Hendrik van Heuraet and Pierre de Fermat.

In 1659 van Heuraet published a construction showing that the problem of determining arc length could be transformed into the problem of determining the area under a curve (i.e., an integral). As an example of his method, he determined the arc length of a semicubical parabola, which required finding the area under a parabola. In 1660, Fermat published a more general theory containing the same result in his De linearum curvarum cum lineis rectis comparatione dissertatio geometrica (Geometric dissertation on curved lines in comparison with straight lines).

Building on his previous work with tangents, Fermat used the curve

$y\; =\; x^\backslash frac\{3\}\{2\}\; \backslash ,$

whose tangent at x = a had a slope of

$\{3\; \backslash over\; 2\}\; a^\backslash frac\{1\}\{2\}$

so the tangent line would have the equation

$y\; =\; \{3\; \backslash over\; 2\}\; a^\backslash frac\{1\}\{2\}(x\; -\; a)\; +\; f(a).$

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:

$\backslash begin\{align\}$

 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

$AC\; =\; \backslash varepsilon\; \backslash sqrt\{1\; +\; \{9\; \backslash over\; 4\}\; a\; \backslash ,\}.$

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

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Curves with infinite length As mentioned above, some curves are non-rectifiable. That is, there is no upper bound on the lengths of polygonal approximations; the length can be made arbitrarily large. Informally, such curves are said to have infinite length. There are continuous curves on which every arc (other than a single-point arc) has infinite length. An example of such a curve is the Koch snowflake. Another example of a curve with infinite length is the graph of the function defined by f( x) =  x sin(1/ x) for any open set with 0 as one of its delimiters and f(0) = 0. Sometimes the Hausdorff dimension and Hausdorff measure are used to quantify the size of such curves.

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Generalization to (pseudo-)Riemannian manifolds Let $M$ be a (pseudo-)Riemannian manifold, $g$ the (pseudo-) metric tensor, $\backslash gamma:0,1\backslash rightarrow\; M$ a curve in $M$ defined by $n$ parametric equations

: $\backslash gamma(t)=\backslash gamma^1(t),,\backslash quad\; t\backslash in$

and

: $\backslash gamma(0)\; =\; \backslash mathbf\; x,\; \backslash ,\backslash ,\backslash gamma(1)\; =\; \backslash mathbf\; y$

The length of $\backslash gamma$, is defined to be

$\backslash ell(\backslash gamma)\; =\; \backslash int\backslash limits\_0^1\; ||\backslash gamma\text{'}(t)||\_\{\backslash gamma(t)\}\; dt$,

or, choosing local coordinates $x$,

$\backslash ell(\backslash gamma)\; =\; \backslash int\backslash limits\_0^1\; \backslash sqrt\{\backslash pm\; \backslash sum\_\{i,j=1\}^n\; g\_\{ij\}(x(\backslash gamma(t)))\backslash frac\{dx^i(\backslash gamma(t))\}\{dt\}\backslash frac\{dx^j(\backslash gamma(t))\}\{dt\}\}dt$,

where

$\backslash gamma\text{'}(t)\; \backslash in\; T\_\{\backslash gamma(t)\}\; M$

is the tangent vector of $\backslash gamma$ at $t.$ 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. Thus the length of a curve is a non-negative real number. Usually no curves are considered which are partly spacelike and partly timelike.

In theory of relativity, arc length of timelike curves () is the proper time elapsed along the world line, and arc length of a spacelike curve the proper distance along the curve.

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See also

• Arc (geometry)
• Circumference
• Crofton formula
• Elliptic integral
• Intrinsic equation
• Integral approximations
• Line integral
• Meridian arc
• Multivariable calculus
• Sinuosity

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Sources

• Rida T. Farouki (1999). 9780826513564, Vanderbilt Univ. Press. ISBN 9780826513564

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External links

• The History of Curvature
• Arc Length by Ed Pegg Jr., The Wolfram Demonstrations Project, 2007.
• Calculus Study Guide – Arc Length (Rectification)
• Famous Curves Index The MacTutor History of Mathematics archive
• Arc Length Approximation by Chad Pierson, Josh Fritz, and Angela Sharp, The Wolfram Demonstrations Project.
• Length of a Curve Experiment Illustrates numerical solution of finding length of a curve.

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