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

§19.30 Lengths of Plane Curves

Keywords:: elliptic integrals, plane curves
Permalink:: http://dlmf.nist.gov/19.30

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

§19.30(i) Ellipse

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

The arclength of the ellipse

19.30.1

$x=a\mathop{\sin\/}\nolimits\phi,$

$y=b\mathop{\cos\/}\nolimits\phi$ , $0\leq\phi\leq 2\pi$ ,

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function and $\phi$ : real or complex argument
Permalink:: http://dlmf.nist.gov/19.30.E1
Encodings:: TeX, TeX, pMML, pMML, png, png

with , is given by

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

Symbols:: : differential of , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : 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

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

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

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables, $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : 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

where

19.30.4

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

$c={\mathop{\csc\/}\nolimits^{{2}}}\phi.$

Symbols:: $\mathop{\csc\/}\nolimits z$ : 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

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 $L(a,b)=4a\mathop{E\/}\nolimits\!\left(k\right)=8a\mathop{R_{G}\/}\nolimits\!% \left(0,b^{2}/a^{2},1\right)=8\!\mathop{R_{G}\/}\nolimits\!\left(0,a^{2},b^{2}% \right)=8ab\mathop{R_{G}\/}\nolimits\!\left(0,a^{{-2}},b^{{-2}}\right),$

Symbols:: $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind, $\mathop{E\/}\nolimits\!\left(k\right)$ : 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

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

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{\csc\/}\nolimits z$ : cosecant function, $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , $\phi$ : 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

§19.30(ii) Hyperbola

Keywords:: elliptic integrals, hyperbola, lengths of plane curves
Permalink:: http://dlmf.nist.gov/19.30.ii

The arclength of the hyperbola

19.30.7

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

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

Permalink:: http://dlmf.nist.gov/19.30.E7
Encodings:: TeX, TeX, pMML, pMML, png, png

is given by

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

Symbols:: : differential of , $\int$ : integral and : arclength
Permalink:: http://dlmf.nist.gov/19.30.E8
Encodings:: TeX, pMML, png

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

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

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : 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

For in terms of $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ , $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ , 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 $\mathop{R_{{-a}}\/}\nolimits$ that differ only in the sign of $b^{2}$ .

§19.30(iii) Bernoulli’s Lemniscate

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

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

19.30.10 $r^{2}=2a^{2}\mathop{\cos\/}\nolimits(2\theta),$ $0\leq\theta\leq 2\pi$ ,

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function and : length
Permalink:: http://dlmf.nist.gov/19.30.E10
Encodings:: TeX, pMML, png

is given by

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

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, : differential of , $\int$ : integral, $\mathop{\sec\/}\nolimits z$ : secant function, : length, and : arclength
Permalink:: http://dlmf.nist.gov/19.30.E11
Encodings:: TeX, pMML, png

or equivalently,

19.30.12 $s=a\mathop{F\/}\nolimits\!\left(\phi,1/\sqrt{2}\right),$ $\phi=\mathop{\mathrm{arcsin}\/}\nolimits\sqrt{2/(q+1)}=\mathop{\mathrm{arccos}% \/}\nolimits\!\left(\mathop{\tan\/}\nolimits\theta\right)$ .

Symbols:: $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\mathop{\mathrm{arccos}\/}\nolimits z$ : arccosine function, $\mathop{\mathrm{arcsin}\/}\nolimits z$ : arcsine function, $\mathop{\tan\/}\nolimits z$ : tangent function, $\phi$ : real or complex argument, and : arclength
Permalink:: http://dlmf.nist.gov/19.30.E12
Encodings:: TeX, pMML, png

The perimeter length of the lemniscate is given by

19.30.13 $P=4\sqrt{2a^{2}}\mathop{R_{F}\/}\nolimits\!\left(0,1,2\right)=\sqrt{2a^{2}}% \times 5.24411\;51\ldots=4a\mathop{K\/}\nolimits\!\left(1/\sqrt{2}\right)=a% \times 7.41629\;87\dots.$

Notes:: For more digits see OEIS Sequence A064853; see also Sloane (2003).
Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind and : length
Permalink:: http://dlmf.nist.gov/19.30.E13
Encodings:: TeX, pMML, png

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

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 19.29 Reduction of General Elliptic Integrals 19.31 Probability Distributions

§19.30 Lengths of Plane Curves

Contents

§19.30(i) Ellipse

§19.30(ii) Hyperbola

§19.30(iii) Bernoulli’s Lemniscate