# Solving for arc length of an ellipse

1. Feb 24, 2010

### redeemer90

1. The problem statement, all variables and given/known data
The task is to solve for the arc length of an ellipse numerically. a & b are given for an ellipse centered at the origin and a value for x is given.

2. Relevant equations

Equation of ellipse is given to be
$$x^{2}/a^{2} + y^{2}/b^{2} = 1$$
and the equation to solve for the arc length is given as
$$a \int^{\theta}_{0}\sqrt{1-k^{2} sin^{2}t} dt$$
Assuming a is the major axis

3. The attempt at a solution
$$-a \leq x\leq a$$, so $$\theta$$ can be $$\ge 0.5 \pi$$
Since $$\int^{\theta}_{0}\sqrt{1-k^{2} sin^{2}t} dt$$ does not seem to work when $${\theta} \ge 0.5 \pi$$

The only solution I can think of is as follows
1. If x < 0
2. pb4 = quarter the perimeter of the ellipse
3. Set x= -x (reflect about the x axis)
4. ptemp = arc length for the positive x
5. The final answer would be p=pb4+(pb4-ptemp)

This would mean evaluating equation $$\int^{\theta}_{0}\sqrt{1-k^{2} sin^{2}t} dt$$ twice.
Is there a better solution to this problem?

Thanks,

- Sid