The On the Perimeter of an Ellipse - Wolfram ResearchOn the Perimeter of an Ellipse Paul Abbott Computing accurate approximations to the perimeter of an ellipse is a fa-vorite problem
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The Mathematica®Journal
On the Perimeter of an EllipsePaul AbbottComputing accurate approximations to the perimeter of an ellipse is a fa-vorite problem of mathematicians, attracting luminaries such as Ramanu-jan [1, 2, 3]. As is well known, the perimeter of an ellipse with semimajoraxis a and semiminor axis b can be expressed exactly as a complete ellipticintegral of the second kind.
-
Approximate formulas can, of course, be obtained by truncating the seriesrepresentations of exact formulas. For example, Kepler used the geometric
mean, º 2 p a b , as a lower bound for the perimeter. In this article, weexamine the properties of a number of approximate formulas, using seriesmethods, polynomial interpolation, rational polynomial approximants, andminimax methods.
‡ IntroductionThe well-known formula for the perimeter of an ellipse with semimajor axis aand semiminor axis b can be expressed exactly as a complete elliptic integralof the second kind, which can also be written as a Gaussian hypergeometricfunction,
4 a E 1 -b2
a2 2 p a 2F1
1
2, -
1
2; 1; 1 -
b2
a2.
The quadratic hypergeometric transformations [4, 5] lead to additional identities,including a particularly elegant formula, symmetric in a and b,
What is less well known is that the various exact forms attributed toMaclaurin, Gauss|Kummer, and Euler are related via quadratic hypergeo-metric transformations. These transformations lead to additional identi-ties, including a particularly elegant formula symmetric in a and b.
‡ Cartesian EquationThe Cartesian equation for an ellipse with center at H0, 0L, semimajor axis a, andsemiminor axis b reads
In[1]:= Ix_, y_M =x
a
2
+y
b
2
1;
Introducing the parameter j into the Cartesian coordinates, as Hx = a sinHjL,y = b cosHjLL, we verify that the ellipse equation is satisfied.
In[2]:= Simplify@Ha sinHjL, b cosHjLLD
Out[2]= True
‡ ArclengthIn general, the parametric arclength is defined by
(1) = ‡j1
j2 ∂x
∂j
2
+∂ y
∂j
2
„j.
The arclength of an ellipse as a function of the parameter j is an (incomplete)elliptic integral of the second kind.
That is, = 4 a EIe2M, where EHmL is the complete elliptic integral of the secondkind.
· Alternative Expressions for the Perimeter
The given expression for the perimeter of the ellipse is unsymmetrical with re-spect to the parameters a and b. This is “unphysical” in that both parameters, be-ing lengths of the (major and minor) axes, should be on the same footing. Wecan expect that a symmetric formula, when truncated, will more accurately approx-imate the perimeter for both a ¥ b and a § b.
Noting that the complete elliptic integral is a Gaussian hypergeometric function,
In[7]:= 2F1
1
2, -
1
2; 1; z
Out[7]=
2 EHzL
p
we obtain Maclaurin’s 1742 formula [2]
In[8]:= 1Ha, bL 2 p a 2F1
1
2, -
1
2; 1; eHa, bL2
Out[8]= True
Equivalent alternative expressions for the perimeter of the ellipse can be ob-tained from quadratic transformation formulas for Gaussian hypergeometricfunctions. For example, using functions.wolfram.com/07.23.17.0106.01,
The hidden symmetry with respect to the interchange a ¨ b is revealed.
In[15]:= FullSimplify@%, b > a > 0D
Out[15]= b E 1 -a2
b2 a E 1 -
b2
a2
Defining
In[16]:= 4Ha_, b_L = p 2 Ia2 + b2M 2F1
1
4, -
1
4; 1;
a2 - b2
a2 + b2
2
;
we can directly check the formula.
In[17]:= SimplifyAFunctionExpand@4Ha, bL 1Ha, bLD, a > b > 0E
Out[17]= True
· Other Identities
There are many other possible transformation formulas that can be appliedto obtain alternative expressions for the perimeter. For example, usingfunctions.wolfram.com/07.23.17.0054.01 we obtain the following formula
The perimeter can also be expressed in terms of Legendre functions (see Sec-tions 8.13 and 15.4 of [6]). For example, using 15.4.15 of [6] we obtain the ele-gant and simple symmetric formula
Out[25]= 84.45496 a, 4.45496 a, 4.45496 a, 4.45496 a, 4.45496 a, 4.45496 a, 4.45496 a<
In[26]:= Equalûû%
Out[26]= True
‡ Numerical ApproximationAt www.ebyte.it/library/docs/math05a/EllipsePerimeterApprox05.html [1] weare encouraged to search for “…an efficient formula using only the four algebraicoperations (if possible, avoiding even square-root) with a maximum error below10 ppm. It would also be nice if such a formula were exact for both the circle andthe degenerate flat ellipse”.
The Gauss|Kummer series expressed as a function of the homogeneous variableh 1 - 4 a b
Ia+bM2, reads
In[27]:= GaussKummer@h_D =2Ha, bL
a + bê. a + b Æ
2 a b
1 - h
Out[27]= p 2F1 -1
2, -
1
2; 1; h
· Series Expansions
The series expansion about h = 0 is useful for small h.
Using the linear approximant 4 h + p H1 - hL and noting that h H1 - hL vanishes atboth h = 0 and h = 1 leads to an optimal @N + 2, MD extreme perfect approxi-mant of the form
p 2F1 -1
2, -
1
2; 1; h º 4 h + p H1 - hL + a h H1 - hL
¤i=1N Hh - piL
¤j=1M Ih - q jM
,
where the parameters a, 8pi<i=1,… ,N , and 9q j= j=1,… ,M need to be determined. Im-plementation of the approximant is immediate.
‡ ConclusionsComputing the perimeter of an ellipse using a simple set of approximants demon-strates that Mathematica is an ideal tool for developing accurate approximants toa special function. In particular:
Ë All special functions of mathematical physics are built in and can be eval-uated to arbitrary precision for general complex parameters andvariables.
Ë Standard analytical methods~such as symbolic integration, summation,series and asymptotic expansions, and polynomial interpolation~areavailable.
Ë Properties of special functions~such as identities and transformations~are available at MathWorld [8] and The Wolfram Functions Site [9] and,because these properties are expressed in Mathematica syntax, they canbe used directly.
Ë Relevant built-in numerical methods include rational polynomial ap-proximants, minimax methods, and numerical optimization for arbitrarynorms.
Ë Visualization of approximants can be used to estimate the quality ofapproximants.
Ë Combining these approaches is straightforward and naturally leads tooptimal approximants.
‡ References[1] S. Sykora. “Approximations of Ellipse Perimeters and of the Complete Elliptic Integral
[3] R. R. Simha, “Perimeter of Ellipse and Beyond” (lecture, Indian Institute of Technology,Bombay, February 2, 2000). www.math.iitb.ac.in/news/simha.html.
[4] G. Almkvist and B. Berndt, “Gauss, Landen, Ramanujan, the Arithmetic-Geometric Mean,Ellipses, p, and the Ladies Diary,” The American Mathematical Monthly, 95(7),1988pp. 585|608.
[5] R. W. Barnard, K. Pearce, and L. Schovanec, “Inequalities for the Perimeter of an El-lipse.” www.math.ttu.edu/~pearce/papers/schov.pdf.
[6] M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formu-las, Graphs, and Mathematical Tables (AMS55), 10th ed., Washington, D.C.: UnitedStates Department of Commerce, National Bureau of Standards, 1972. www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP.
[7] P. Kahlig, “A New Elliptic Mean,” Sitzungsber. Abt. II, 211, 2002 pp. 137|142. hw.oeaw.ac.at/?arp=x-coll7178b/2003-7.pdf.
[8] E. W. Weisstein, “Arithmetic-Geometric Mean” from Wolfram MathWorld~A WolframWeb Resource. mathworld.wolfram.com/Arithmetic-GeometricMean.html.
[9] M. Trott, The Wolfram Functions Site~A Wolfram Web Resource. functions.wolfram.com.