§19.30 Lengths of Plane Curves

§19.30(i) Ellipse

The arclength $s$ of the ellipse

 19.30.1 $\displaystyle x$ $\displaystyle=a\mathop{\sin\/}\nolimits\phi,$ $\displaystyle y$ $\displaystyle=b\mathop{\cos\/}\nolimits\phi$, $0\leq\phi\leq 2\pi$,

with $a>b$, is given by

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

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

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

where

 19.30.4 $\displaystyle k^{2}$ $\displaystyle=1-(b^{2}/a^{2}),$ $\displaystyle c$ $\displaystyle={\mathop{\csc\/}\nolimits^{2}}\phi.$ Symbols: $\mathop{\csc\/}\nolimits\NVar{z}$: 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)

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

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

§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)

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)

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

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

For $s$ 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

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

 19.30.10 $r^{2}=2a^{2}\mathop{\cos\/}\nolimits(2\theta),$ $0\leq\theta\leq 2\pi$, Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function and $r$: length Permalink: http://dlmf.nist.gov/19.30.E10 Encodings: TeX, pMML, png See also: Annotations for 19.30(iii)

is given by

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

or equivalently,

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

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

 19.30.13 $P=4\sqrt{2a^{2}}\mathop{R_{F}\/}\nolimits\!\left(0,1,2\right)=\sqrt{2a^{2}}% \times 5.24411\;51\ldots=4a\mathop{K\/}\nolimits\!\left(1/\sqrt{2}\right)=a% \times 7.41629\;87\dots.$ Symbols: $\mathop{R_{F}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind, $\mathop{K\/}\nolimits\!\left(\NVar{k}\right)$: 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)

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