Perimeter of the Ellipse (HP-15C)

[Gerson W. Barbosa]

(05-27-2021 11:28 PM)PedroLeiva Wrote:  I am using Ramanujan II-Cantrell: here a & b are radius (in my PDF the input is diameter)
H= [(a- b) / (a + b)]^2
P= π * (a+b) * [ 1 + 3H / (10+ SQRT(4-3H)) + (4/π - 14/11) * H^12]

For ellipses with H>0.6, here the examples:
a= 20 ----H= 0.546313800
b= 3 ----P= 82.52178335

a= 20 ---H= 0.669421488
b= 2 ---P= 81.27883093

a= 20 ---H= 0.904818560
b= 0.5 ---P= 80.11412754

For the other combinations of the radius, the results are:
a= 20 ---H= 1
b= 0 ---P= 80

a= 20 ---H= 0.0006557462
b= 19 ---P= 122.5422527

a= 20 ---H= 0.00277083
b= 18 ---P= 119.4632087

Please let me know your conclusions,

I was thinking of h = [(a - b) / (a + b)], not h = [(a - b) / (a + b)]^2. That is, I was actually interested in ellipses with h greater than 0.36, not 0.6.

Ramanujan-Cantrell is more accurate than Ramanujan-II, but only for very eccentric ellipsis (h close to 1).

I should have included a direct comparison between Ramanujan-Cantrell and Modified Muir approximations, but the spreadsheets below might give an idea on how these three approximations perform when compared to the exact results.