# §19.30 Lengths of Plane Curves

## §19.30(i) Ellipse

The arclength $s$ of the ellipse

 19.30.1 $\displaystyle x$ $\displaystyle=a\sin\phi,$ $\displaystyle y$ $\displaystyle=b\cos\phi$, $0\leq\phi\leq 2\pi$,

with $a>b$, is given by

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

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

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

where

 19.30.4 $\displaystyle k^{2}$ $\displaystyle=1-(b^{2}/a^{2}),$ $\displaystyle c$ $\displaystyle={\csc^{2}}\phi.$ ⓘ 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 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 $L(a,b)=4aE\left(k\right)=8aR_{G}\left(0,b^{2}/a^{2},1\right)=8\!R_{G}\left(0,a% ^{2},b^{2}\right)=8abR_{G}\left(0,a^{-2},b^{-2}\right),$

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

## §19.30(ii) Hyperbola

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 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 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 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

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

 19.30.10 $r^{2}=2a^{2}\cos(2\theta),$ $0\leq\theta\leq 2\pi$, ⓘ 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 19

is given by

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

or equivalently,

 19.30.12 $s=aF\left(\phi,1/\sqrt{2}\right),$ $\phi=\operatorname{arcsin}\sqrt{2/(q+1)}=\operatorname{arccos}\left(\tan\theta\right)$.

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 19

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