Why don't you take the original equation for your parabola and calculate the height from there?

You can do this with good ol' calculus by finding the extrema for the equation by calculating all zero-values for the first derivative of your equation within your given range. You can then use the second derivative to determine whether each extremum is a local maximum, a local minimum or an inflection point.

Since you're working with a parabola you'll only find one extremum which will be the x-value for your highest point. Evaluate this x in your original function and you'll get your height.

EDIT:

Easier yet, if your parabola is of the form y = a*x^2 + b, where a < 0 (otherwise you can't find a maximum) then your height is simply the value of b

L = 0.5?16h²+w² + [w²/(8h)][Ln(4h + ?16h²+w²) - Ln(w)]

This equation cannot be rearranged in terms of height with elementary algebra, because it has the [eqn]h[/eqn] both inside and outside a transcendental function (here [eqn]\ln[/eqn]). You can write it out as a closed-form recursive process or a recurrence and then approximate it numerically (or directly approximate it via Newton-Raphson with two variables) - wouldn't know how to do it right then though, sorry.

So if I understand you correctly, you need to keep the length of the parabolic arc constant, while modifying its base width, so you need to obtain the correct height for it to have the same length?

OK, I think I got it. I would expect the length of the parabola to increase monotonically as the height increases, for a same base width. In this case, you can use good old bisection. This will allow you to converge on the correct height required to obtain a given arc length, knowing the base width. I would expect the algorithm to converge pretty quickly on the solution, which means it should be fast if you only need, say, 5 digits of precision, but I can't say anything about performance. I believe it would be correct though.

The code would look something like this (may have some edge cases but it's the idea):


input: base length W, required length L.
output: approximate correct H such that Length(W, H) = L

1. Let lo = 0, hi = 4h + c (twice the limit form plus some constant, try various c)
2. Compute H_mid = (lo + hi) / 2
3. Compute L' = Length(W, H_mid)
4. If L' < L, let lo = H_mid and goto 2
5. If L' > L, let hi = H_mid and goto 2
6. If L' is close to L, return H_mid


Though note that if arc length doesn't strictly increase with height with constant base width as I think it does, this might not work.

EDIT:

If w is a constant, then:

O(0.5?16h²+w² + [w²/(8h)][ln(4h + ?16h²+w²) - ln(w)]) = O(0.5?16h² + [1/(8h)][ln(4h + ?16h²)]) = O(2h + ln(8h)/(8h)) = O(2h + ln(8h)/(8h))

Which tends to 2h in the limit, so the arc length does increase monotonically with h, this should work.

You can also use Newton-Raphson instead by differentiating (have you seen this method at high school?), it would converge faster but each iteration takes more work, and the differentiated expression may not be so nice to compute.

Just out of interest, how is the parabola being used? If you're talking about the motion of a projectile under the force of gravity, there are other equations you can use that are more suited to the task.