Elliptical Arc Length

Date: 11/17/1999 at 04:48:36
From: Steve Malmberg
Subject: Numerical methods for calculating elliptical arc length?

I know elliptic integrals are used for calculating the arc length of 
an ellipse, but is there a series approximation or some other 
numerical method for calculating the arc length (not necessarily the 
perimeter) of an ellipse? I'm looking for something that's reasonable 
to implement in software.

The problem I'm ultimately trying to solve is this: starting at an 
arbitrary angle theta on the ellipse, what is the angular extent alpha 
of an arc whose length is given?

Thanks.

Date: 11/17/1999 at 06:44:45
From: Doctor Mitteldorf
Subject: Re: Numerical methods for calculating elliptical arc length?

Dear Steve,

Doing the integral numerically is going to be much easier than coming 
up with a series representation for the answer you want. Here are the 
steps:

1) Write down the expression for the ellipse: x^2/a^2 + y^2/b^2 = 1. 
Solve for y.

2) Differentiate your expression for y with respect to x to get the 
slope y'.

3) The arclength formula is ds = sqrt(1+y'^2)dx. Substitute your 
expression for y' into this to give a formula for ds.

4) Integrate this with respect to x using Simpson's rule or Runge-
Kutta or any integration routine you're comfortable with. (I must 
confess that in practice for small problems I often use straight 
addition of the integrand - computers are so fast now that I just let 
it run with a very small dx to save me programming time, and to avoid 
the kinds of errors that can come when I try to apply someone else's 
canned routine.)

5) Continue the integration until you reach the target arclength. Then 
note what x is at that point, and use x to calculate y.

6) Apply tan(theta) = y/x at the beginning and endpoints of the 
integral to obtain the start and ending angles.

Why don't you wrestle with this for a while, and feel free to write 
again and ask for more details if you get stuck.

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/