I'm not sure you've defined "silly" in a useful way. For example, here's a case where we know how to compute the two numbers to gazillions of decimal places and yet it's still pretty silly: A = π, B = π with the n'th bit in its binary expansion flipped if Goldbach's conjecture fails at n.

The two wikipidia pages contradict each other-- either the definition of Schlegel diagram is wrong, or the 600-cell picture isn't a Schlegel diagram (or both). In any case, that shadow is what you get when the candle is just outside a vertex (exactly analogous to @TomGoodwillie's description of the "20-hedron corner view poking you in the eye"), so doesn't that picture complete the answer to the question?

@TonyK please see previous few comments, I couldn't cc more than one person at a time.

@GerryMyerson Yes I know, I purposely put a space in the @ TonyK (and doing it again now) to denote my failed intent there. Frustrating. Okay, I thought "normal and offset" was clarifying, but I guess not. I'll replace it with something explicit like "Ax+By=C with A,B,C integers". That will make it more like the stackoverflow phrasing, which I found to be quite good and clear in this regard.

@GerhardPaseman and @ TonyK I edited the question to fix the problems we've been complaining about, and removed three of my previous comments about them.

I just edited the question, hopefully fewer red herrings now: the lines no longer need to pass through integer points, and the statements about NP hopefully actually make sense.

You say "I don't believe that the problem is NP-complete, because you're working in a fixed dimension." I don't understand what one has to do with the other. You agree a naive implementation will take exponential time (i.e. linear time in the magnitudes of the input numbers) despite having polynomial sized output, right? So it's not obvious that it can be done in polynomial time. Why then is it unreasonable to think the problem might be NP-complete?

I've posted a solid solution, which I implemented and tested exhaustively on small inputs (the original formulation, lines don't have to hit grid points) so I'm confident I didn't miss anything. The testing did reveal a bug in the case enumeration, easily fixed. @WadimZudilin yes, it's a version of EA. Now I'm trying to think of how to generate some killer test cases, that is, examples that wouldn't be solvable in one's lifetime using a naive implementation. Best I can think of is run the algorithm backwards, but I wonder if there's a clever simpler method for generating challenging examples.