§19.30 Lengths of Plane Curves

The arclength $s$ of the ellipse

with $a>b$, is given by

When $0\le \varphi \le \frac{1}{2}\pi$,

where

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

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 $\lambda \in (-b^2,\mathrm{\infty })$, 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 $\varphi$ is given by

The arclength $s$ of the hyperbola

is given by

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

For $s$ in terms of $E(\varphi ,k)$, $F(\varphi ,k)$, 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 $b^2$.

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

is given by

or equivalently,

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

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