怎樣計萛隋圓形的周邊?

Question

怎樣能計萛隋圓形的周邊?
十萬個唔該,不過最好有step
因為我要教我弟弟

thx

Answer 1

As answered above, there is no closed form equation for the perimeter, however, there are approximations.
Define:
a, b semi-axes (as defined above)
h = (a-b)^2/(a+b)^2

FOR LOW ECCENTRICITIES: (a approx. equal to b)
A very approximate formula:
P approx.= pi*sqrt(2(a^2+b^2) - (a-b)^2/2)

An excellent approximation due to Ramanujan
P=pi* (a+b)* [ 3 - sqrt(4-h) ]

FOR HIGH ECCENTRICITIES (flat):

Lindner's formula:
P= pi* (a+b) [ 1 + h/8 ]^2

This formula is also due to Ramanujan
P=pi* (a+b) [ 1 + 3h / ( 10+ sqrt(4-3h) ) ]

For more information, there are plenty of articles on the subject at the following site:

http://home.att.net/~numericana/answer/#ellipse

Answer 2

橢圓周界是沒有公式的, 寫成積分的那個東西叫橢圓分, 沒法in出一個closed form 來的.

但橢圓面積是有公式的, 而且十分簡單:
一般橢圓形的parametrization係(x/a)^2 + (y/b)^2 = 1
其中a,b 分別係個橢圓既major/minor axis 的長度

咁個橢圓的面積就是 a*b*pi

這個很簡單的formula可以用integration證出來.