Maybe, but as he explains a normal parabola (like a satellite dish) wouldn't work, since the shots themselves are arced because of pesky gravity. I thought using a Monte Carlo analysis and finding a fixed point was actually a very clever way to solve it.
Yeah but the acceleration from gravity is constant so maybe there is just a (possibly non linear) change of coordinates that can be applied to the regular paraboloid.
Also a paraboloid would work only for rays that are parallel to one another.
Depends on the importance of a couple effects that light is immune to (at least for an optics equation)
1) gravity post impact with backboard
2) impact angle for a given (start, backboard) tuple depend on velocity and launch angle for a basketball, but are always the same for light. Also the impact angle for the basketball and light are very different
I think it's safe to ignore both of these factors at this scale. 1) The effects of gravity is bounded and negligible on the surface of the Earth. 2) (I suspect) the wideness of the launch angle and hoop size are probably directly proportional, so you could shape the back board based on a direct impact to get a point to center the hoop on. The hoop can only get so big and still be considered reasonable.
The effect of gravity on the surface of the Earth is hardly negligible!
And your intuition on the launch angle is just flat wrong. In fact, I believe it's one of the primary predictive factors in a shots success with a normal backboard!
I think you gotta take the radius of the ball into account - that's the only bit he messed up on. (And if we can just have any-old assumptions, then it seems like the optimal solution is to make the entire backboard a hoop!)
I think within a range velocity and mass can be ignored ie. if the velocity is too small the ball won't reach the board to begin with and beyond a threshold most likely the bounce curve will depend almost exclusively on incident angle.