Please imagine that we have a circular segment with some arc length 's' and chord length 'a' (using notation from http://mathworld.wolfram.com/CircularSegment.html).
Provided only 'a' and 's', and placing the left-hand-side point of the chord at the origin of the Euclidean plane (or a more convenient point), is there sufficient information to write an expression for the height of the circular segment (i.e. the y-axis/"vertical" distance between the chord on the x-axis and the circular arc) as a function of a position on the chord?
It's a simple matter to express the chord length in terms of the arc length and 'theta': a = (s) * 2(Sin[theta/2])/theta, or an expression for the arc length in terms of the chord length and theta: s = a/(2(Sin[theta/2])/theta). And one can write an express for the maximum height as: h = (R - (1/2)*((-a)^2+4*R^2)^(1/2)), where the radius of the circle, 'R' is related to theta as: R = 1/2 Sqrt[a^2/(-1 + Cos[theta/2]^2)].
If there is insufficient information to accomplish the above, I would love to have an intuitive explanation for why this is so.