Perimeter of Ellipse

[Gerson W. Barbosa]

(12-04-2019 10:27 PM)Gerson W. Barbosa Wrote:  ———-

This first Ramanujan approximation brings the error down to under one meter:

π[3(a + b) - √((3a + b)(a + 3b))]

His second approximation is even better, with an error less than one micrometer²:

https://www.johndcook.com/blog/2013/05/0...-ellipse/

Instead of

p ~ 2π[3*agm(a,b) - 2√(a*b)]

I will suggest a better approximation involving the arithmetic-geometric mean and the geometric mean:

p ~ 2π{a + b - a*b/agm(a,b) - 2[agm(a,b) - √(a*b)]}

Let’s use it to compute the length of the orbit of Pluto¹:

——

wp34s program

001 LBL  A
002 ©ENTER
003 STO+  T
004 RCL×  Y
005 √
006 STO  I
007 x⇆  L
008 ⇆  ZYXT
009 AGM
010 STO/  Y
011 RCL- I
012 STO+  X
013 +
014 -
015 #  π
016 ×
017 STO+ X
018 END

——

5906376272 ENTER 5720637952.8 A ->

36529672878.01583848170946635744574

Exact result:

36529672878.01583840603514193230844 km

Difference:

0.000000075674324425137 km,

or

75.674 μm

——


¹ Assuming the parameters are exact and the orbit is perfectly elliptical.

——-
01-03-2020 11:57 PM

PS:

I present another formula involving AGM:

p ~ 2π[agm(a,b)(192(1 - h) - h²) - 128(1 - h)√(ab)]/[64(1 - h) - h²]

where h = [(a - b)/(a + b)]²

Error: 32.59 nm (orbit of Pluto)

Further improvement is still possible .



² The actual error produced by Ramanujan’s second formula is slightly less than one nanometer, not one micrometer.

——-
01-05-2020 01:03 AM

PSS:

Just a small refinement – more are still possible – and the overall error in the length of the orbit of Pluto is only 19.34 fm (femtometers!).

p ~ 2π{agm(a,b)[77h² - 768(h - 1)] - [54h² - 512(h - 1)]√(ab)}/[23h² - 256(h - 1)]

where h = [(a - b)/(a + b)]²

Example:

a = 5906376272 km

b = 5720637952.8 km

p ~ 36529672878.01583840603514191296851 km

Exact:

p = 36529672878.01583840603514193230844 km