New approx. for perimeter of an ellipse
David W. Cantrell
perimeter of an ellipse. I believe the approximation to be new, and of
interest due to its combination of accuracy and simplicity: For an ellipse
having semiaxes of lengths a and b, the perimeter is approximately
4(a+b) - 2(4-Pi)ab/H_p(a,b)
where H_p(a,b) = ((a^p+b^p)/2)^(1/p) is a Holder mean
and p is a positive constant, to be considered below.
Note that the formula gives the perimeter exactly both when b=a and when
b=0. I have called such approximations "extreme-perfect" since they give
the perimeter exactly at both extremes of eccentricity. A few other such
approximations can be found in my article at
<http://forum.swarthmore.edu/epigone/sci.math/sumbrabo>.
At least to my prejudiced eye, the new approximation has a very attractive
form. Consider the rectangle of dimensions 2a and 2b which circumscribes
the ellipse. Then the approximation has the form
(perimeter of the circumscribing rectangle) - 2(area of region between
that rectangle and the ellipse)/(a mean radius).
What is the geometrical motivation behind an approximation in this form?
Think of the special case b=a, when the rectangle is then a square and
the ellipse a circle. Note that 2(area)/radius = 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.
To obtain the mean radius, a Holder mean, H_p(a,b), of the lengths of the
semiaxes is used. Many well known approximations for the perimeter of
an ellipse -- such as those of Kepler, Euler, and Muir -- as well as the
more recent YNOT approximation, may be thought of as using Holder means.
What should we choose for p? If we wish an approximation of this type to
be as accurate as possible for small eccentricities, then a little calculus
gives p = (3Pi-8)/(8-2Pi), in which case the approximation has a maximum
relative error of about 2*10^(-4). Similarly, if we wish an approximation
of this type to be as accurate as possible for eccentricities very close to
1, then we should use p = 1/(1-lg(4-Pi)), where lg denotes the binary
logarithm, in which case the maximum of |relative error| is about
3*10^(-4). However, if we wish the maximum of |relative error| to be
minimized over _all_ eccentricities, then a value for p should be chosen
which is between the two previously mentioned values. Specifically, using a
value for p of approximately 0.82507 (obtained numerically), we can get
|relative error| < 8.3*10^(-5).
How good is the new type of approximation compared to well known
approximations? Consider the more accurate of the two well known
approximations due to Ramanujan (1914):
* 1 + 3m^2/(10 + Sqrt(4-3m^2)) where m = (a-b)/(a+b)
For its simplicity, * gives truly remarkable accuracy for small
eccentricities. But, when b=0, * has |relative error| = 7Pi/22-1 or about
4*10^(-4). Thus, the new approximation (using p = 0.82507) has a smaller
maximum of |relative error|. [Ramanujan also gave two extreme-perfect
approximations which, although of impressive accuracy, are not well known,
probably due to their complexities.]
Perhaps it should be noted that the new approximation can also be written
in the alternate form
4(a+b) - 4(2-Pi)H_r(a,b) where r = -p is a negative constant.
This looks somewhat simpler, I grant. However, in my opinion, the original
form has two advantages. First, it makes immediately clear that the second
term vanishes when b=0 [although of course, in the alternate form, it is
reasonable to expect H_r(a,0) to be taken as equal to 0 since r is
negative]. Second, the original form makes the geometrical motivation
behind the formula clearer that does the alternate form.
Your comments will be welcomed.
David W. Cantrell
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Osher Doctorow
I like the algebraic form of your approximation because it seems to be part
of a generalization of the important quasi-product or star product x * y = x
+ y - xy which is the basis of the Jacobson Radical of module and ring
theory. The generalization would be of the form x o y = c(x + y) + f(x, y)
for c real constant and f(x, y) a function of x and y with at least one
irreducible factor xy. Both products are actually representable as special
cases of my proximity funcion p1(x, y) = 1 + y - x, although your domain and
range of p1 may be generalized (mine usually has y < = x).
I haven't spent much time examining your geometric argument, although it
seems interesting. I am not that familiar with past approximations of the
perimeter of the ellipse.
Osher Doctorow Ph.D.
Ventura College, Doctorow Consultants, etc.