lpetrich
Contributor
Why is there no equation for the perimeter of an ellipse‽ - YouTube
Stand-Up Maths, by Matt Parker
Mentions several approximations.
For semi-axis lengths a, b, with a >= b:
Eccentricity:
\( e = \frac{\sqrt{a^2-b^2}}{a} \)
Relative difference squared:
\( h = \left( \frac{a-b}{a+b} \right)^2 \)
He mentioned these approximations:
Arithmetic mean:
\( \pi (a + b) \)
Geometric mean:
\( 2\pi \sqrt{a b} \)
Root mean square:
\( 2\pi \sqrt{\frac{a^2 + b^2}{2}} \)
Srinivasa Ramanujan has two formulas:
\( \left( 3(a+b) - \sqrt{(3a+b)(a+3b)} \right) \)
(best linear + sqrt(quadratic))
\( \pi (a+b) \left( 1 + \frac{3h}{10 + \sqrt{4-3h}} \right) \)
(best of this form)
Matt Parker also mentioned some asymmetric formulas that have slightly erroneous small-eccentricity limits:
\( \pi \left( \frac{6}{5} a + \frac{3}{4} b \right) \)
\( \pi \left( \frac{53}{3} a + \frac{717}{35} b - \sqrt{269 a^2 + 667 a b + 371 b^2} \right) \)
The exact formula is a complete Jacobi elliptic integral of the second kind
\( 4 a E(e) \ / \ 4 a E(e^2) \)
Using both argument conventions:
\( E(k) = \int_0^{\pi/2} \frac{d\phi}{\sqrt{1 - k^2 \sin^2 \phi}} \)
\( E(m) = \int_0^{\pi/2} \frac{d\phi}{\sqrt{1 - m \sin^2 \phi}} \)
I used this page to test my formulas:
Interactive LaTeX Editor
Stand-Up Maths, by Matt Parker
Mentions several approximations.
For semi-axis lengths a, b, with a >= b:
Eccentricity:
\( e = \frac{\sqrt{a^2-b^2}}{a} \)
Relative difference squared:
\( h = \left( \frac{a-b}{a+b} \right)^2 \)
He mentioned these approximations:
Arithmetic mean:
\( \pi (a + b) \)
Geometric mean:
\( 2\pi \sqrt{a b} \)
Root mean square:
\( 2\pi \sqrt{\frac{a^2 + b^2}{2}} \)
Srinivasa Ramanujan has two formulas:
\( \left( 3(a+b) - \sqrt{(3a+b)(a+3b)} \right) \)
(best linear + sqrt(quadratic))
\( \pi (a+b) \left( 1 + \frac{3h}{10 + \sqrt{4-3h}} \right) \)
(best of this form)
Matt Parker also mentioned some asymmetric formulas that have slightly erroneous small-eccentricity limits:
\( \pi \left( \frac{6}{5} a + \frac{3}{4} b \right) \)
\( \pi \left( \frac{53}{3} a + \frac{717}{35} b - \sqrt{269 a^2 + 667 a b + 371 b^2} \right) \)
The exact formula is a complete Jacobi elliptic integral of the second kind
\( 4 a E(e) \ / \ 4 a E(e^2) \)
Using both argument conventions:
\( E(k) = \int_0^{\pi/2} \frac{d\phi}{\sqrt{1 - k^2 \sin^2 \phi}} \)
\( E(m) = \int_0^{\pi/2} \frac{d\phi}{\sqrt{1 - m \sin^2 \phi}} \)
I used this page to test my formulas:
Interactive LaTeX Editor
