Length of an arc

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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.

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 length s of the arc bounded by a and b is found by the formula

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

if function is defined parametrically where x=f(t)\, and y=g(t)\,

s=\int _{{a}}^{{b}}{\sqrt  {[g'(t)]^{2}+[f'(t)]^{2}}}dt.

if function is defined via polar coordinates where r=f(\theta ) then

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

In most cases, including even simple curves, there is no closed form solution of arc length available and numerical integration is necessary. For instance, applying this formula to the circumference of an ellipse leads to elliptic integrals of the second kind.

Curves with closed form solutions for arc length include the catenary, circle, cycloid, logarithmic spiral, parabola, semicubical parabola and (mathematically, a curve) straight line.

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.

1600s

In the 1600s, the method of exhaustion led to the rectification by geometrical methods of several transcendental curves: the logarithmic spiral by 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.

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.

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.

File:Arc length, Fermat.png
Fermat's Method of determining arc length

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

y=x^{{3 \over 2}}

whose tangent at x=a had a slope of

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

so the tangent line would have the equation

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

Next, he increased a by a small amount to a+e, 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:

AC^{2}\, =AB^{2}+BC^{2}\,
={3 \over 2}a^{{1 \over 2}}+e^{2}
=e^{2}\left(1+{9 \over 4}a\right)

which, when solved, yields

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

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

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

nl:Booglengte