I just installed a track lighting rail on my ceiling so I can illuminate my new wall art! The track rail runs perpendicular to the wall surface, and as I attach the 25 degree beam angle track light to the rail, I slowly move it down the rail to find the right position. I start off close to the wall, where the flood of the beam is too small and can't illuminate the whole art piece. As I move the light backwards away from the wall, I keep the light angled so that it's pointed on the art piece until the flood of the beam angle on the wall is the exact height of the art piece, with the art piece precisely lit by the track light. If I had keep going further backwards away from the wall, the flood of the light would start to illuminate empty wall above and below the art piece, so it's good that I stopped when I did! I wonder if I could have calculated how far away from the wall the light would need to be, though?

The distance of the track light from the vertical slice wall with the art piece, the height of the art piece, the distance of the art piece below the top of the ceiling, and the beam angle of the light all together have a physical solution. There is one specific distance from the wall where the light perfectly floods the wall art, that can be determined only from these other known inputs, and requires no further information and has no other solutions. It should be perfectly calculable.

However, actually calculating this value has been an absolute monster and I'm finding I need a math Witcher to deal with it. No it is not unsolvable, it has an extremely clear and apparent physical solution, it is just a touch non-trivial to figure out a general solution for. It is sufficiently non-trivial that I've put this past 4 different friends with undergrad math degrees, and it has bested all of them.

Here's a diagram, although you may want to draw your own:

C
___________
\ • Φ |
\ Θ • | B
\ • |
\ •|
\ ||
\ ||
\ || A
\ ||
\ ||

A is the height of the wall art - known
B is the distance from the top of the wall art to the ceiling - known
Θ is the beam angle of the light - known
Φ reflects the angle the track light is pointed downwards - can be anything
C is the distance of the track light from the wall - solve for this