Could someone who groks this math tell me why not buy the skis once you’ve paid half their price on rentals?
Paying at 1/2 will be optimal if it ends before you buy, very bad (3x optimal) if it ends right after you buy, and slightly better than the solution in the post if it lasts at least twice that long (1.5x optimal vs e/(e-1)).
The metric in the post is just the worst of those ratios. Assuming the unproven statement in the post (that the solution which is a constant factor worse than optimal is best), any solution of the form you suggest is going to have similar tradeoffs. If we had a distribution, we could choose.
Lines in 3D remaining straight in a photo is unrelated and not actually demonstrated by the image. I'm having trouble imagining why this matters - you're trying to find the intersection of two lines in an image without drawing anything?