19 Elliptic Integrals Applications 19.29 Reduction of General Elliptic Integrals 19.31 Probability Distributions

§19.30 Lengths of Plane Curves

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Keywords:: elliptic integrals, plane curves
Permalink:: http://dlmf.nist.gov/19.30
See also:: Annotations for Ch.19

§19.30(i) Ellipse
§19.30(ii) Hyperbola
§19.30(iii) Bernoulli’s Lemniscate

§19.30(i) Ellipse

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Keywords:: ellipse, elliptic integrals, lengths of plane curves
Notes:: See Carlson (1977b, §9.4). For (19.30.5) see (19.25.1). For (19.30.6) use (19.4.6).
Permalink:: http://dlmf.nist.gov/19.30.i
See also:: Annotations for §19.30 and Ch.19

The arclength of the ellipse

19.30.1

$x=a\sin\phi,$

$y=b\cos\phi$ , $0\leq\phi\leq 2\pi$ ,

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function and $\phi$ : real or complex argument
Permalink:: http://dlmf.nist.gov/19.30.E1
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §19.30(i), §19.30 and Ch.19

with , is given by

19.30.2 $s=a\int_{0}^{\phi}\sqrt{1-k^{2}{\sin}^{2}\theta}\,\mathrm{d}\theta.$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, : real or complex modulus and : arclength
Permalink:: http://dlmf.nist.gov/19.30.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §19.30(i), §19.30 and Ch.19

When $0\leq\phi\leq\tfrac{1}{2}\pi$ ,

19.30.3 $s/a=E\left(\phi,k\right)={R_{F}\left(c-1,c-k^{2},c\right)-\tfrac{1}{3}k^{2}R_{% D}\left(c-1,c-k^{2},c\right)},$

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\phi$ : real or complex argument, : real or complex modulus and : arclength
Referenced by:: §19.30(i)
Permalink:: http://dlmf.nist.gov/19.30.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §19.30(i), §19.30 and Ch.19

where

19.30.4

$k^{2}=1-(b^{2}/a^{2}),$

$c={\csc}^{2}\phi.$

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Symbols:: $\csc\NVar{z}$ : cosecant function, $\phi$ : real or complex argument and : real or complex modulus
Permalink:: http://dlmf.nist.gov/19.30.E4
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §19.30(i), §19.30 and Ch.19

Cancellation on the second right-hand side of (19.30.3) can be avoided by use of (19.25.10).

The length of the ellipse is

19.30.5 $L(a,b)=4aE\left(k\right)=8aR_{G}\left(0,b^{2}/a^{2},1\right)=8R_{G}\left(0,a^{% 2},b^{2}\right)=8abR_{G}\left(0,a^{-2},b^{-2}\right),$

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Symbols:: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, : real or complex modulus and : length
Referenced by:: §19.15, §19.24(i), §19.30(i), §19.33(i), §19.9(i)
Permalink:: http://dlmf.nist.gov/19.30.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §19.30(i), §19.30 and Ch.19

showing the symmetry in and . Approximations and inequalities for are given in §19.9(i).

Let $a^{2}$ and $b^{2}$ be replaced respectively by $a^{2}+\lambda$ and $b^{2}+\lambda$ , where $\lambda\in(-b^{2},\infty)$ , to produce a family of confocal ellipses. As $\lambda$ increases, the eccentricity decreases and the rate of change of arclength for a fixed value of $\phi$ is given by

19.30.6 $\frac{\partial s}{\partial(1/k)}=\sqrt{a^{2}-b^{2}}F\left(\phi,k\right)=\sqrt{% a^{2}-b^{2}}R_{F}\left(c-1,c-k^{2},c\right),$ $k^{2}=(a^{2}-b^{2})/(a^{2}+\lambda)$ , $c={\csc}^{2}\phi$ .

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\csc\NVar{z}$ : cosecant function, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of with respect to , $\,\partial\NVar{x}$ : partial differential of , $\phi$ : real or complex argument, : real or complex modulus and : arclength
Referenced by:: §19.30(i)
Permalink:: http://dlmf.nist.gov/19.30.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §19.30(i), §19.30 and Ch.19

§19.30(ii) Hyperbola

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Keywords:: elliptic integrals, hyperbola, lengths of plane curves
Permalink:: http://dlmf.nist.gov/19.30.ii
See also:: Annotations for §19.30 and Ch.19

The arclength of the hyperbola

19.30.7

$x=a\sqrt{t+1},$

$y=b\sqrt{t}$ , $0\leq t<\infty$ ,

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Permalink:: http://dlmf.nist.gov/19.30.E7
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §19.30(ii), §19.30 and Ch.19

is given by

19.30.8 $s=\frac{1}{2}\int_{0}^{y^{2}/b^{2}}\sqrt{\frac{(a^{2}+b^{2})t+b^{2}}{t(t+1)}}% \,\mathrm{d}t.$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral and : arclength
Permalink:: http://dlmf.nist.gov/19.30.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §19.30(ii), §19.30 and Ch.19

From (19.29.7), with $a_{\delta}=1$ and $b_{\delta}=0$ ,

19.30.9 $s=\tfrac{1}{2}I(\mathbf{e}_{1})=-\tfrac{1}{3}a^{2}b^{2}R_{D}\left(r,r+b^{2}+a^% {2},r+b^{2}\right)+y\sqrt{\frac{r+b^{2}+a^{2}}{r+b^{2}}},$ $r=b^{4}/y^{2}$ .

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Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $I(\mathbf{m})$ : integral and : arclength
Permalink:: http://dlmf.nist.gov/19.30.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §19.30(ii), §19.30 and Ch.19

For in terms of $E\left(\phi,k\right)$ , $F\left(\phi,k\right)$ , and an algebraic term, see Byrd and Friedman (1971, p. 3). See Carlson (1977b, Ex. 9.4-1 and (9.4-4)) for arclengths of hyperbolas and ellipses in terms of $R_{-a}$ that differ only in the sign of $b^{2}$ .

§19.30(iii) Bernoulli’s Lemniscate

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Keywords:: Bernoulli’s lemniscate, elliptic integrals, lengths of plane curves, plane curves
Notes:: See Carlson (1977b, Ex. 8.3-7, with solution on p. 312).
Permalink:: http://dlmf.nist.gov/19.30.iii
See also:: Annotations for §19.30 and Ch.19

For $0\leq\theta\leq\tfrac{1}{4}\pi$ , the arclength of Bernoulli’s lemniscate

19.30.10 $r^{2}=2a^{2}\cos\left(2\theta\right),$ $0\leq\theta\leq 2\pi$ ,

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function and : length
Permalink:: http://dlmf.nist.gov/19.30.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §19.30(iii), §19.30 and Ch.19

is given by

19.30.11 $s=2a^{2}\int_{0}^{r}\frac{\,\mathrm{d}t}{\sqrt{4a^{4}-t^{4}}}=\sqrt{2a^{2}}R_{% F}\left(q-1,q,q+1\right),$ $q=2a^{2}/r^{2}=\sec\left(2\theta\right)$ ,

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