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19 Elliptic IntegralsApplications

§19.30 Lengths of Plane Curves

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§19.30(i) Ellipse

The arclength s of the ellipse

19.30.1 x =asinϕ,
y =bcosϕ,
0ϕ2π,

with a>b, is given by

19.30.2 s=a0ϕ1-k2sin2θdθ.

When 0ϕ12π,

19.30.3 s/a=E(ϕ,k)=RF(c-1,c-k2,c)-13k2RD(c-1,c-k2,c),

where

19.30.4 k2 =1-(b2/a2),
c =csc2ϕ.

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(k)=8aRG(0,b2/a2,1)=8RG(0,a2,b2)=8abRG(0,a-2,b-2),

showing the symmetry in a and b. Approximations and inequalities for L(a,b) are given in §19.9(i).

Let a2 and b2 be replaced respectively by a2+λ and b2+λ, where λ(-b2,), to produce a family of confocal ellipses. As λ increases, the eccentricity k decreases and the rate of change of arclength for a fixed value of ϕ is given by

19.30.6 s(1/k)=a2-b2F(ϕ,k)=a2-b2RF(c-1,c-k2,c),
k2=(a2-b2)/(a2+λ), c=csc2ϕ.

§19.30(ii) Hyperbola

The arclength s of the hyperbola

19.30.7 x =at+1,
y =bt,
0t<,

is given by

19.30.8 s=120y2/b2(a2+b2)t+b2t(t+1)dt.

From (19.29.7), with aδ=1 and bδ=0,

19.30.9 s=12I(e1)=-13a2b2RD(r,r+b2+a2,r+b2)+yr+b2+a2r+b2,
r=b4/y2.

For s in terms of E(ϕ,k), F(ϕ,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 b2.

§19.30(iii) Bernoulli’s Lemniscate

For 0θ14π, the arclength s of Bernoulli’s lemniscate

19.30.10 r2=2a2cos(2θ),
0θ2π,

is given by

19.30.11 s=2a20rdt4a4-t4=2a2RF(q-1,q,q+1),
q=2a2/r2=sec(2θ),

or equivalently,

19.30.12 s=aF(ϕ,1/2),
ϕ=arcsin2/(q+1)=arccos(tanθ).

The perimeter length P of the lemniscate is given by

19.30.13 P=42a2RF(0,1,2)=2a2×5.24411 51=4aK(1/2)=a×7.41629 87.

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