Fun story. I was in college taking differential equations. And I was faithfully attempting to apply what I'd learned in class, and having a somewhat hard time at it (this was before I realized how useful matlab was). When I was doing some research on the side, and I learned that the vast majority of differential equations are unsolvable, you can only approximate an answer. After that, I was thouroughly annoyed at wasting time in a class doing all this math by hand, and having to assume the answer was of a form that was solvable (and, yes, many useful engineering diff eq are of a solvable form).
So, now to answer your question. I thought this backboard seemed like a problem that dish makers should be able to trivially solve, but getting to know the math, and modifying it enough to solve this particular problem can take more time than setting up a simple probabilistic simulation. And after my experience with diffeq, I'm happy with a quick and dirty approximate solution, and then moving on with your life. Then maybe you run into someone who knows the math and is motivated enough to apply it, and maybe you update your project then.
Thanks for a great anecdote. While I feel I'm pretty good at recognising that "done is better than perfect", I'm not as good at realising when "near enough is good enough". For some reason approximations usually don't sit well with me... something I can work on for sure.
Sorry, I didn't mean to imply it required diff eq to solve these parabolic equations. I merely meant to give an anecdote about one of the first times I felt truly betrayed by mathematics. A subject I'd previously held in the highest regard.
No it’s not. Parabolic antennas work well for a single particular parallel input beam. That’s why they’re on gimbals, and directed at the anticipated direction, or scanned back and forth. Here, he is attempting to image many different parabolic paths onto a small circle in the plane of the rim.
I commented something similar on the video, but I realized you would have to define the problem much more rigorously to get an analytical solution. The ball can hit a point from many angles, so doesn’t that mean there can’t be a perfect solution? If you define the problem with precise constraints that roughly match the paths that will tend to occur from real people shooting baskets, and ignore variables like ball spin that probably don’t matter much in practice, I would imagine there would be an analytical solution.
So, now to answer your question. I thought this backboard seemed like a problem that dish makers should be able to trivially solve, but getting to know the math, and modifying it enough to solve this particular problem can take more time than setting up a simple probabilistic simulation. And after my experience with diffeq, I'm happy with a quick and dirty approximate solution, and then moving on with your life. Then maybe you run into someone who knows the math and is motivated enough to apply it, and maybe you update your project then.