I'm searching for a type of function, which fulfills the following criteria:
The function must have a symmetry axis, two "zeros" and most important, there must be an algebraic expression for h depending on the arc length and the chord.
Furthermore, the tangent at 0 and s must not be vertical like for an ellipse.
I have already tried parabolas, circle segements and sinus functions, but for none of them there seems to be an algebraic expression for "h" (the result is always a transcendental equation with ArcSin, ArcSinh or an integral equation). The only way is to solve the equation numerically, but due to the way the function will be used this is not an option.