The arclength of the ellipse
19.30.1 | ||||
, | ||||
, | ||||
with , is given by
19.30.2 | |||
When ,
19.30.3 | |||
where
19.30.4 | ||||
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 | |||
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
19.30.6 | |||
, . | |||
For , the arclength of Bernoulli’s lemniscate
19.30.10 | |||
, | |||
is given by
19.30.11 | |||
, | |||
or equivalently,
19.30.12 | |||
. | |||
The perimeter length of the lemniscate is given by
19.30.13 | |||