What is the formula for the perimeter of an ellipse?

[joecoss]

Could anyone direct me to an analytically correct algebraic formula for the Perimeter of an Ellipse based on either the eccentricity or the Semi-Major and Semiminor Axes other than the Elliptic Integral ? If so, how accurate will it be for relatively high eccentricities such as 0.9-1.0 ? Thanks.

 

[AKG]

Given those properties of a given ellipse, you should be able to define a function, f(x) which describes the top half of the ellipse. The perimeter of the ellipse would be twice the length of f(x) on the interval on which it exists. The length of f(x) on that interval, let's call it [-a,a], is:

$$L = \int _{-a} ^a \sqrt{1 + [f'(x)]^2}dx$$

 

[AKG]

To see why this works, think of f'(x) as dy/dx. Now, put the "dx" under the square root, and you'll get:

$$L = \int _{-a} ^a \sqrt{{dx}^2 + {dy}^2}$$

Now, if you consider an infinitessimal piece of the function, you can treat it as a straight line segment. If you think of this segment as the hypoteneuse of a triangle with sides dx and dy, then clearly, the length of this hypoteneuse is the integrand. Sum the lengths of these tiny segments over the desired interval, and you get the length of the function on that interval.

 

[TALewis]

I have a handbook that lists the perimeter of an ellipse as approximately:

$$2\pi\sqrt{\frac{1}{2}(a^2+b^2)}$$

a and b are the semi-major and semi-minor axes, respectively. No idea on the accuracy.

 

[joecoss]

OK I am with you on the Arclength Formula, do you know the function if given a (S-maj), b (S-minor), or c (Focus to center) ? Thanks a lot.

 

[TALewis]

Won't the arclength formula lead to the elliptic integral? You already said you don't want that.

Edit: There is no exact solution for the perimeter of an ellipse. You either have to numerically integrate this:

$$4a\int_0^{\pi/2}\sqrt{1-e^2\sin^2 t}\,dt}$$

(where e is the eccentricity)

Or use an approximation like the one I gave in my earlier post.

Edit Edit: This page seems to have some better approximations listed at the bottom:

http://mathforum.org/dr.math/faq/formulas/faq.ellipse.circumference.html