How do I plot sagitta versus arc length.

Question

I'm working on a curvature sensor, but I'm finding it hard to find the equation for my h (sagitta) as a function of my arc length.

In particular, I want to solve $\epsilon=\frac{Arc Length - Chord Length}{Chord Length}$ for the sagitta, knowing the Chord Length.

So the first step is to get the Arc Length as a function of Chord Length and sagitta, which I think I can do. The second step would be to plug it in the previous equation and solve it for h (as a function of Chord Length and $\epsilon$). Is that possible?

Thanks

Answer 1

Hint:

If the arc lenght is $r \theta$ (where $\theta$ is the angle subtending the arc $AB$) we have:

For the cord: $AB=2r\sin (\theta/2)$

For the sagitta: $s= r(1-\cos(\theta /2)$)

Can you prove these results using simple trigonometry on the circle of radius $r$?

If not see here.