# §19.30 Lengths of Plane Curves

## §19.30(i) Ellipse

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 and . Approximations and inequalities for are given in §19.9(i).

Let and be replaced respectively by and , where , to produce a family of confocal ellipses. As increases, the eccentricity decreases and the rate of change of arclength for a fixed value of is given by

## §19.30(ii) Hyperbola

The arclength of the hyperbola

19.30.7
,,

is given by

19.30.8

From (19.29.7), with and ,

For in terms of , , 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 that differ only in the sign of .

## §19.30(iii) Bernoulli’s Lemniscate

For , the arclength of Bernoulli’s lemniscate

19.30.10,

is given by

or equivalently,

The perimeter length of the lemniscate is given by

19.30.13

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