Arc Length of an Ellipse: Formula & Calculation

[natski]

Is there a general formula for calculating the arc length between two points along an ellipse?

 

[nicksauce]

You either have to use elliptical integrals or numerical methods. See:
http://mathforum.org/library/drmath/view/51945.html

 

[natski]

Is there any Fortran code around to do this?

 

[mathwonk]

it depends what you mean by "calculate". i.e. almost all interesting integral calculations use numerical methods.

just saying an integral equals pi/4 is not calculating it, since you have not calculated pi. you are merely relating the length to the length of an arc of a circle by saying this.

so even to calculate arc length on a circle requires numerical methods.

on an ellipse these same numerical methods work equally well (Simpsons rule, etc..).

 

[natski]

Ok, got my Simpson's rule going on in a code and getting some reasonable results now... thanks

 

[natski]

It strikes me that using the method provided by Dr Math, you get an equation where you must integrate f(x) between x2 and x1 by dx to get the arc length, where x is the Cartesian coordinate system.

But one can imagine two points on the ellipse with the same position in x but different position in y. So the arc length would not be zero but Dr Math's method but give a zero value nontheless...

 

[natski]

To solve this problem, I tried to calculate the arclength as L = L1 + L2
where L1 is between x=x2 and x=a and L2 is between x=a and x=x1.

But the denominator in the integrand is x^2 - a^2 so we get some crazy infinite stuff going on there messing up everything...