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

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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 $s$ of the ellipse

19.30.1		$\displaystyle x$	$\displaystyle=a\sin\phi,$
		$\displaystyle y$	$\displaystyle=b\cos\phi$ ,
	$0\leq\phi\leq 2\pi$ ,
ⓘ 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 $a>b$ , is given by

19.30.2		$s=a\int_{0}^{\phi}\sqrt{1-k^{2}{\sin}^{2}\theta}\,\mathrm{d}\theta.$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus and $s$ : 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		$s/a=E\left(\phi,k\right)={R_{F}\left(c-1,c-k^{2},c\right)-\tfrac{1}{3}k^{2}R_{% D}\left(c-1,c-k^{2},c\right)},$
ⓘ 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, $k$ : real or complex modulus and $s$ : 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	$\displaystyle k^{2}$	$\displaystyle=1-(b^{2}/a^{2}),$
19.30.4	$\displaystyle c$	$\displaystyle={\csc}^{2}\phi.$
ⓘ Symbols: $\csc\NVar{z}$ : cosecant function, $\phi$ : real or complex argument and $k$ : 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		$L(a,b)=4aE\left(k\right)=8aR_{G}\left(0,b^{2}/a^{2},1\right)=8R_{G}\left(0,a^{% 2},b^{2}\right)=8abR_{G}\left(0,a^{-2},b^{-2}\right),$
ⓘ 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, $k$ : real or complex modulus and $L(a,b)$ : 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 $a$ and $b$ . Approximations and inequalities for $L(a,b)$ 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 $k$ decreases and the rate of change of arclength for a fixed value of $\phi$ is given by

19.30.6		$\frac{\partial s}{\partial(1/k)}=\sqrt{a^{2}-b^{2}}F\left(\phi,k\right)=\sqrt{% a^{2}-b^{2}}R_{F}\left(c-1,c-k^{2},c\right),$
$k^{2}=(a^{2}-b^{2})/(a^{2}+\lambda)$ , $c={\csc}^{2}\phi$ .
ⓘ 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 $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\phi$ : real or complex argument, $k$ : real or complex modulus and $s$ : 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 $s$ of the hyperbola

19.30.7		$\displaystyle x$	$\displaystyle=a\sqrt{t+1},$
		$\displaystyle y$	$\displaystyle=b\sqrt{t}$ ,
	$0\leq t<\infty$ ,
ⓘ 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		$s=\frac{1}{2}\int_{0}^{y^{2}/b^{2}}\sqrt{\frac{(a^{2}+b^{2})t+b^{2}}{t(t+1)}}% \,\mathrm{d}t.$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $s$ : 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		$s=\tfrac{1}{2}I(\mathbf{e}_{1})=-\tfrac{1}{3}a^{2}b^{2}R_{D}\left(r,r+b^{2}+a^% {2},r+b^{2}\right)+y\sqrt{\frac{r+b^{2}+a^{2}}{r+b^{2}}},$
$r=b^{4}/y^{2}$ .
ⓘ Symbols: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $I(\mathbf{m})$ : integral and $s$ : 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 $s$ 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 $s$ of Bernoulli’s lemniscate

19.30.10		$r^{2}=2a^{2}\cos\left(2\theta\right),$
$0\leq\theta\leq 2\pi$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function and $r$ : 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		$s=2a^{2}\int_{0}^{r}\frac{\,\mathrm{d}t}{\sqrt{4a^{4}-t^{4}}}=\sqrt{2a^{2}}R_{% F}\left(q-1,q,q+1\right),$
$q=2a^{2}/r^{2}=\sec\left(2\theta\right)$ ,
ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sec\NVar{z}$ : secant function, $r$ : length, $q$ and $s$ : arclength Permalink: http://dlmf.nist.gov/19.30.E11 Encodings: TeX, pMML, png See also: Annotations for §19.30(iii), §19.30 and Ch.19

or equivalently,

19.30.12		$s=aF\left(\phi,1/\sqrt{2}\right),$
$\phi=\operatorname{arcsin}\sqrt{2/(q+1)}=\operatorname{arccos}\left(\tan\theta\right)$ .
ⓘ Symbols: $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\operatorname{arccos}\NVar{z}$ : arccosine function, $\operatorname{arcsin}\NVar{z}$ : arcsine function, $\tan\NVar{z}$ : tangent function, $\phi$ : real or complex argument, $q$ and $s$ : arclength Permalink: http://dlmf.nist.gov/19.30.E12 Encodings: TeX, pMML, png See also: Annotations for §19.30(iii), §19.30 and Ch.19

The perimeter length $P$ of the lemniscate is given by

19.30.13		$P=4\sqrt{2a^{2}}R_{F}\left(0,1,2\right)=\sqrt{2a^{2}}\times 5.24411\;51\ldots=% 4aK\left(1/\sqrt{2}\right)=a\times 7.41629\;87\dots.$
ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind and $P$ : length Notes: For more digits see OEIS Sequence A064853; see also Sloane (2003). Permalink: http://dlmf.nist.gov/19.30.E13 Encodings: TeX, pMML, png See also: Annotations for §19.30(iii), §19.30 and Ch.19

For other plane curves with arclength representable by an elliptic integral see Greenhill (1892, p. 190) and Bowman (1953, pp. 32–33).

19.29 Reduction of General Elliptic Integrals 19.31 Probability Distributions

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§19.30 Lengths of Plane Curves

Contents

§19.30(i) Ellipse

§19.30(ii) Hyperbola

§19.30(iii) Bernoulli’s Lemniscate

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