Introduction

Given an ellipse with semiaxes of lengths a and b, let AE denote its area,
let AR and PR denote, resp., the area and perimeter of the rectangle
circumscribing the ellipse, and let MR denote a mean radius (i.e., some
strict mean of a and b). The form to be used is then

#    PR - 2(AR - AE)/MR,

that is,  4(a + b) - 2(4 - pi) a b / MR.

There is good reason to consider approximating the perimeter of an ellipse
using form #. First, think of the extreme case b = a, when the rectangle is
then a square and the ellipse a circle. Note that 2(area)/radius gives the
perimeter for both circle and square [where the radius of a square is taken
to be half of its side length]. Thus, when b = a, the second term is
precisely what must be subtracted from the perimeter of the square to leave
that of the circle. Second, think of the other extreme case, b = 0.
Obviously the rectangle and ellipse are degenerate, both having area = 0,
and the perimeter of the rectangle is exactly that of the ellipse. Thus,
approximations in form # are always extreme-perfect (that is, they give the
perimeter of the ellipse perfectly at both extremes of eccentricity)
regardless of the mean chosen.

Of course there are many possible means of a and b which could be used for
MR. In my previous article (cited above), Hoelder means were used:

 MR = H_p(a,b) = ((a^p + b^p)/2)^(1/p).

The value of p may be chosen according to various criteria. For example, if
we choose to minimize the worst |relative error| over all eccentricities,
we find that p = 0.825056... and obtain |relative error| < 8.3*10^(-5). The
second approximation to be introduced below will not only reduce that error
bound substantially but also be simpler in that no transcendental functions
will be required. First, however, we consider

1.  The Simplest Approximation in Form #

Using the arithmetic mean of a and b to give the mean radius,
MR = (a + b)/2, we obtain the approximation

(1)    4( (a + b) - (4 - pi) a b / (a + b) ).

This approximation may also be written in other forms which might be of
interest. One such alternative form is

  8( A(a,b) - (1 - pi/4) H(a,b) )

where A(a,b) and H(a,b) denote, resp., the arithmetic and harmonic means
of the semiaxes lengths.

Between the extremes of eccentricity, this approximation overestimates the
perimeter of the ellipse, with the worst relative error being about 0.0063.
I'm surprised that I have not seen this simple approximation suggested
before. How does it compare with other approximations of similar
simplicity? The approximation pi(a + b)(1 + 1/4 ((a - b)/(a + b))^2),
obtained by using just the first two terms of the Gauss-Kummer series, is
both slightly more complicated and less accurate, its worst relative error
being about -0.018. (Using the first three terms of the Gauss-Kummer
series, we get Lindner's approximation, with worst relative error of about
-0.00598.)

2.  A More Accurate Approximation in Form #

To obtain an approximation which is more accurate than that presented in my
previous article (cited above), we use a mean radius MR in the form

    p (a+b) + Sqrt( q (a+b)^2 + r a b ).

[Note: This form -- written differently, but essentially the same -- is
discussed in "Modifying Ramanujan's first approximation for the perimeter
of an ellipse" at <http://mathforum.org/discuss/sci.math/t/604361>. There
the form was used, by itself, to approximate the perimeter; here, as
stated, we shall use this form to give the mean radius in form #.]

It is easy to verify that we must take r = (1-2p)^2 - 4q if this form is to
be a strict mean. That leaves us with the parameters p and q to be chosen
as desired. Here we shall choose to minimize worst |relative error| over
all eccentricities. Determining p and q numerically according to that
criterion, we find that

p =  0.410117...  and  q = 0.0004251...
and consequently r is about 0.030615.

The approximation then provides |relative error| < 4.2*10^(-6). Not only is
the worst |relative error| substantially reduced compared to that of the
approximation, using a Hoelder mean, given in the first link above, but the
new approximation requires no transcendental functions, only a single
square root.

This new approximation appears somewhat messy, having two numerically
determined parameters, and so it's natural to ask how it compares in
complexity to other approximations of similar accuracy. But there are no
well known approximations for the perimeter of an ellipse _per se_ having
similar accuracy. We can, nonetheless, look at an approximation for the
complete elliptic integral of the second kind, given by Abramowitz and
Stegun as item 17.3.35 on page 592:
<http://hcohl.shell42.com/as/page_592.htm>.
It requires four messy coefficients and computation of a logarithm, and yet
is still substantially less accurate than our new approximation.

It just so happens that the mean radius used above can be very nicely
approximated by a neater form:

  MR = p(a + b) + (1-2p)/75 Sqrt((a + 74b)(74a + b))

where the optimal value of p is, for all practical purposes, the same as
before.  (We may even, if desired, also approximate p using convergents of
its continued fraction, such as 73/178 or 381/929, without substantially
increasing worst |relative error|.)

We may then approximate the perimeter of an ellipse,
with |relative error| < 4.2*10^(-6), by

[please view using a fixed-width font]

                               150(4 - pi) a b
(2)   4(a + b) -  --------------------------------------------
                  75p(a + b) + (1-2p) Sqrt((a + 74b)(74a + b))

where p = 0.410117...

Thoughtful comments are welcomed.