First Try:
Using your measurements, unfortunately they do not suggest a perfect ellipse. If it were, 'h' would have to be in the neighborhood of 48.6'.
As I recall, the top of "the" aviary is not flat, it comes to a point. Also, the edge of the canopy is not vertical, suggesting only sort of an ellipse, or maybe just a portion of the to, rather than all of it.
So, there are several options. We can use various shapes to determine minimum and maximum arc lengths for the four sections. An ellipse may not be the best choice. You haven't built it, already, have you? If you have, maybe you can just go throw a cable over the top and measure it. It's been done?
Oh, there is one other important factor. You provided only one cross section of the structure. Is that the only one getting a pipe? Does the rafter stay the same size the entire length or does it taper? As I recall, "the" aviary stays the same width most of the way, but has the ends closed off differently.
What's the chance we're talking about the same aviary?
So, where does that leave us. This is an excellent example for showing the difference between textbook problems and real-world problems. If you're actually going to build it, you have to abandon all your simplifying assumptions.