Perimeter of Ellipse
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12-07-2019, 07:57 PM
(This post was last modified: 01-05-2020 01:03 AM by Gerson W. Barbosa.)
Post: #15
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RE: Perimeter of Ellipse
(12-04-2019 10:27 PM)Gerson W. Barbosa Wrote: ———- 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 |
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