Hypergeometric function – Perimeter of an Ellipse and other applications (wp34s)
01-14-2020, 11:47 PM
Post: #8
 Albert Chan Senior Member Posts: 2,142 Joined: Jul 2018
RE: Hypergeometric function – Perimeter of an Ellipse and other applications (wp34s)
(01-14-2020 05:23 PM)Gerson W. Barbosa Wrote:  Here is still another approximation to the perimeter of the ellipse:

p ≈ 2π[(aˢ + bˢ)/2]¹ᐟˢ

where

s = 3/2 + 1/(32/h² - ⅗h² - 4)

and where

h = [(a - b)/(a + b)]

How did you get "⅗" from the keyboard ?
BTW, if you change "⅗" to "217/360", above formula is always better than Ramanujan II

Let ellipse perimeter = Pi (a+b) corr(h)
To simplify, assume a=1, we have h = (1-b)/(1+b) → b = (1-h)/(1+h)

Below Mathematica code tried to compare error of corr(h):

In:= corr[h_] := 4*EllipticE[1-b^2] / (Pi(1+b)) /. b->(1-h)/(1+h)
In:= ramj2[h_]:= 1 + 3h^2/(10 + (4-3h^2)^(1/2))
In:= Series[corr[h] - ramj2[h], {h,0,12}] // Simplify
﻿ ﻿ ﻿ ﻿ → (3/131072) h^10 + (79/2097152) h^12 + O(h^14)

In:= muir[b_,s_] := 2*((1+b^s)/2)^(1/s) / (1+b)
In:= muir2[h_]:= muir[(1-h)/(1+h), 3/2+1/(32/h^2 - 3/5*h^2 - 4)]
In:= Series[corr[h] - muir2[h], {h,0,12}] // Simplify
﻿ ﻿ ﻿ ﻿ → (1/737280) h^8 + (41/13762560) h^10 + (-31319/825753600) h^12 + O(h^14)

In:= muir3[h_]:= muir[(1-h)/(1+h), 3/2+1/(32/h^2 - 217/360*h^2 - 4)]
In:= Series[corr[h] - muir3[h], {h,0,12}] // Simplify
﻿ ﻿ ﻿ ﻿ → (181/61931520) h^10 + (-16717/440401920) h^12 + O(h^14)

New constant removed h^8 term.
Perimeter error is smaller until h > 0.3946 (eccentricity > 0.9009).
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 Messages In This Thread Hypergeometric function – Perimeter of an Ellipse and other applications (wp34s) - Gerson W. Barbosa - 12-10-2019, 11:50 PM RE: Hypergeometric function – Perimeter of an Ellipse and other applications (wp34s) - Paul Dale - 12-11-2019, 01:47 AM RE: Hypergeometric function – Perimeter of an Ellipse and other applications (wp34s) - Gerson W. Barbosa - 12-13-2019, 04:08 AM RE: Hypergeometric function – Perimeter of an Ellipse and other applications (wp34s) - Ángel Martin - 01-29-2020, 11:44 AM RE: Hypergeometric function – Perimeter of an Ellipse and other applications (wp34s) - Paul Dale - 12-13-2019, 05:04 AM RE: Hypergeometric function – Perimeter of an Ellipse and other applications (wp34s) - Gerson W. Barbosa - 12-13-2019, 05:16 AM RE: Hypergeometric function – Perimeter of an Ellipse and other applications (wp34s) - Gerson W. Barbosa - 01-11-2020, 01:20 AM RE: Hypergeometric function – Perimeter of an Ellipse and other applications (wp34s) - Gerson W. Barbosa - 01-14-2020, 05:23 PM RE: Hypergeometric function – Perimeter of an Ellipse and other applications (wp34s) - Albert Chan - 01-14-2020 11:47 PM RE: Hypergeometric function – Perimeter of an Ellipse and other applications (wp34s) - Gerson W. Barbosa - 01-15-2020, 12:59 AM RE: Hypergeometric function – Perimeter of an Ellipse and other applications (wp34s) - Gerson W. Barbosa - 01-16-2020, 04:18 AM RE: Hypergeometric function – Perimeter of an Ellipse and other applications (wp34s) - Gerson W. Barbosa - 01-18-2020, 01:14 AM RE: Hypergeometric function – Perimeter of an Ellipse and other applications (wp34s) - grsbanks - 01-18-2020, 10:25 AM RE: Hypergeometric function – Perimeter of an Ellipse and other applications (wp34s) - Gerson W. Barbosa - 01-18-2020, 03:41 PM RE: Hypergeometric function – Perimeter of an Ellipse and other applications (wp34s) - Gerson W. Barbosa - 01-21-2020, 09:01 PM RE: Hypergeometric function – Perimeter of an Ellipse and other applications (wp34s) - Gerson W. Barbosa - 01-25-2020, 02:53 AM RE: Hypergeometric function – Perimeter of an Ellipse and other applications (wp34s) - Paul Dale - 01-29-2020, 12:29 PM RE: Hypergeometric function – Perimeter of an Ellipse and other applications (wp34s) - Gerson W. Barbosa - 02-08-2020, 11:17 PM

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