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Final Answers
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Related topics on this website include:
- Hypergeometric functions.
- Surface of an ellipse.
- Volume of an ellipsoid.
- Ellipses and Hyperbolae.
- Elliptic arc: Length of the arc of an ellipse between two points.
- Perimeter of an ellipse.
Exact formulas and simple ones.
- Circumference of an ellipse: Unabridged discussion.
- Surface area of an ellipsoid of revolution (oblate or prolate).
- Volume of a hypersphere in any number of dimensions. Hyper-surface area too!
Related Links (Outside this Site)Ellipse by Dr. James B. Calvert, University of Denver (Colorado).Circumference of an Ellipse by Robert L. Ward in "MathForum@Drexel". Perimeter of an Ellipse by Stanislav Sýkora (2005-05-30). On the Perimeter of an Ellipse (pdf) by Paul Abbott (Avignon, June 2006). A few articles posted by David W. Cantrell :
2001-05-08 :
New Approximation for [the] Perimeter of an Ellipse
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(Jaleigh. B. of Minonk, IL.
(the maximum relative error is for the
circumference of an ellipse whose eccentricity is about 0.979811,
which means an ellipse [ pictured at right ]
with a minor axis very slightly less than one fifth of its major axis).
For small values of e, the relative error of the
YNOT formula is about
corresponding to an aspect ratio b/a around 7.167 %, as pictured at left).
In thrifty implementations, a Gauss-Kummer term is roughly twice as fast to compute
(3 additions, 3 multiplications, 1 division)
as a term in Cayley's series
(6 additions, 5 multiplications, 2 divisions).
This makes it most efficient to switch from one method
to the other at an eccentricity of about 96%,
which may be considered an arbitrary boundary
between round and elongated ellipses
(corresponding to an aspect ratio b/a
of exactly 28 %, as pictured above).
This is a far cry from the "extremely elongated" ellipse described in many popular accounts
about the Comet (whose authors may have been impressed by a number "so close" to unity).
Unlike the aspect ratio (b/a = 25% in the case of Halley's comet),
the eccentricity e is a misunderstood measure of the elongation of an ellipse.
This is why we felt compelled to sprinkle this page with small graphics indicating what the
ellipses of various eccentricities actually look like...
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The next formula along the same road features a ratio of two cubic polynomials.
It's certainly not an easy-to-remember expression,
but it's much more accurate than any of the above for round ellipses,
as it boasts a relative error of about
Such formulas are probably best obtained using a device which is to analytic functions
what truncated continued fractions are to real numbers.
The above ratio is equal to the following expression,
truncated "at order m=6",
where the coefficients to use are:
Truncating at m=3 gives Hudson's formula,
while Jacobsen's formula corresponds to m=4.
Truncation at m=2 yields a
The proper sequence of coefficients [1, 4, -4, 4/3, -4, 12/11...]
is not too difficult to obtain:
Consider a function f(h) which is to be approximated in this way
[if we only want the above coefficients up to am,
we may replace an analytic function
In general, a relation of the type
In the above, an odd order of truncation (m) gives a rational approximation
where the degree of the numerator is one unit higher than the degree of the denominator.
If we apply the method to 1/f(h) instead
[and take the reciprocal of the result]
we obtain a numerator of lower degree than the denominator.
(At even orders of truncation, the results of the two approaches coincide.)
For this "reciprocal" approach, the sequence of coefficients is:
1,
David W. Cantrell (2004-05-24) Improving Ramanujan's second formula [over the entire range of h]. Like most historical approximations to the perimeter of an ellipse, the second formula of Ramanujan reaches its worst relative error for a degenerate ellipse: Its O(h5 ) relative error reaches a maximum of (7p/22-1) for a flat ellipse (h = 1).
Therefore,
the following formula is exact for a flat ellipse while retaining the same leading error
term for a roundish ellipse as Ramanujan's second formula,
provided f is
David W. Cantrell has proposed the O(h12 ) correction f = h6 , for a relative error never worse than ±15 ppm. This proposal was spurred by a more complex correction term presented to him by Edgar Erives. The exponent 6 is merely the integer closest to an optimal exponent which is only minutely better. Ricardo Bartolomeu (2004-06-06; e-mail) Beginner's Luck... On 2004-06-06, Ricardo Bartolomeu asked us to evaluate a [very messy] trigonometric approximation of the perimeter of an ellipse, which he had stumbled upon, with apparently good results. Miraculously, Bartolomeu's expression happens to be equivalent to a nice symmetric function of a and b: P » p (a-b) / arctg [ (a-b) / (a+b) ] = p (a+b) [ 1 + h/3 + O(h2 ) ] This simple formula is defined by continuity for a = b. It's exact for a circle and a flat ellipse and is about 5 times less accurate than the YNOT formula (both for low eccentricities and over the entire range, with a worst relative error exceeding 1.72 % for an ellipse of eccentricity near 0.983). This goes to show that even wild guesses can be fairly accurate if they turn out to be symmetrical. In 1932, E.H. Lockwood had proposed a similar approximation of the perimeter of an ellipse, which has a worst relative error of almost -0.9% (when e»0.9598). It's about four times less accurate than the YNOT formula for a round ellipse: P » (4b2/a) arctg (a/b) + (4a2/b) arctg (b/a) = p (a+b) [1 + (4-12/p)h + ...] Another attempt...On 2004-08-02, Ricardo Bartolomeu asked us again to evaluate yet another [slightly less messy] trigonometric approximation for the perimeter of an ellipse, which could be reduced to the following asymmetrical formula:
Again, this is defined by continuity for a = b (sin(x)/x tends to 1 as x tends to 0) and the formula is exact for a circle, a flat ellipse, or an ellipse of eccentricity: 0.88729428292031793745442629632208285906244622844440758655... Below that, the relative error is negative but never worse than -0.184 % (for eccentricities around 0.766764) above that, it's positive but never worse than +0.9822 % (almost reached for an ellipse of eccentricity around 0.991233). For the perimeter of an ellipse of low eccentricity e, the relative error is: -e4 (p2-6) / 384 + e6 (12-p2 ) / 768 + O(e8) That's more than 9.23 times worse than the commensurable YNOT formula... David F. Rivera (2004-02-24; e-mail) Rivera's Formula David Rivera (of the Naval Undersea Warfare Center, RI) has contacted us to share the approximative formula for the circumference of an ellipse which he developed around 1997 and has used in his antenna work:
This formula clearly gives the correct circumference for a circle (a = b) and a flat ellipse (b = 0). It is also exact for an ellipse of eccentricity e = 0.986118932960305275314772672749686388... For a rounder ellipse, Rivera's formula features a negative error which is never worse than -103.70 ppm (when e is around 0.9329538). For a flater ellipse, the error is positive, but never worse than +103.73 ppm (for e around 0.99811). All told, the relative error of Rivera's 1997 formula is always better than 104 ppm and is thus almost as good as Cantrell's 2001 formula. (104 ppm vs. 83 ppm) The value p = 89/146 is a good rational approximation to the value which minimizes the magnitude of the worst relative error(s) in a parametrized formula which, incidentally, could be rewritten as follows, for any value of p : P » p (a+b) [ 1 + (4/p-1) h - (p/4p) {1 - h - (1-h) 3/2 } ] For any p, the parametrized Rivera formula is exact in the case of a circle (h=0) or a flat ellipse (h=1). The optimal value of p for low eccentricities is 32-10p. David F. Rivera had submitted another approximation to the perimeter of an ellipse among errata to the 30th edition of the Standard Mathematical Tables and Formulae (CRC Press)... This formula is the only one of its kind given in the 31st edition. [ Thanks to David Cantrell for pointing this out. ] P » 2a [ 2 + (p - 2) (b/a) 1.456 ] This asymmetrical expression is exact for a circle or a flat ellipse. The relative error is also zero for an ellipse of eccentricity around 0.921271. Below that point, it's negative but never worse than -0.447 % (around 0.7129); above that point, it's positive but never worse than +0.439 % (around 0.9883). David W. Cantrell (2004-05-23) Cantrell's 2004 Ellipse Formula Cantrell considers approximations to the perimeter of the ellipse of the form: P » 4(a+b) - 2(4-p) ab / f In his original 2001 proposal, Cantrell had used f = [ ½ (ap+bp) ] 1/p, namely the Hölder mean of the principal radii ( p = 33/40 yields an 85 ppm accuracy). Seeking better accuracy and computational simplicity, Cantrell now introduces a two-parameter expression for f (the more parameters to optimize, the better) which makes f equal to a when a = b, for any choice (within limits) of p and k :
For an ellipse of low eccentricity, the formula yields an optimal O(h3) error when:
Numerically, that's approximatively k = 133 and p = 0.412. However, Cantrell's primary concern is to minimize the worst relative error. For this, he settles on k = 74 and uses an approximation of the corresponding optimal value of p (0.410117...) to claim an overall accuracy of 4.2 ppm. Lu Chee Ket (Malaysia. 2004-06-23 e-mail, and 2004-06-25) C.K. Lu's rediscovery (2003) of an expansion due to Euler (1773). In 1773, Leonhard Euler (1707-1783) gave an exact expansion for the perimeter of an ellipse which may be expressed as a power series of the quantity d defined below. By itself, the first term of this expansion gives Euler's crude approximation, which has already been discussed above...
where d = [ (a2 - b2 ) / (a2 + b2 ) ] 2 = 4h / (1+h)2
For a flat ellipse, expanding up to d n
yields a relative error of 1/16n+O(1/n2). (2004-07-06) Exact Expansions for the Perimeter of an Ellipse : Some of the above exact formulas for the circumference of an ellipse may be expressed using Gauss's (1812) hypergeometric function F (also denoted 2F1 ). 
(*) The Gauss-Kummer series first appeared in the earliest memoir ever published by the Scottish mathematician Sir James Ivory (1765-1842; knighted in 1831): A New Series for the Rectification of the Ellipse, Transactions of the Royal Society of Edinburgh, volume 4, part II, pp.177-190 (1796). Credit is given to Eduard Kummer (1810-1898) because he established (in 1836) a few general transformations of the hypergeometric function F, which may be used to prove the equivalence of the three hypergeometric formulas, namely:
F(u,v;2v;z) =
(1-z/2)-u F( u/2, (u+1)/2; v+½; [ z / (2-z) ] 2 ) To apply these two transforms here, the reader may want to remark that: d = 4h / (1+h) 2 = [ e2 / (2-e2 ) ] 2  |
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