Recently a "U" shaped scratch appeared on my car's front tire. It looks like someone rolled past a curb and scraped it. This has led me to the following problem:

Generate a plot (ideally polar) of a scratch pattern on a tire given tire radius and the ground height of a protrusion. Assume the protrusion is a single point and remains stationary while the tire rolls past. Also assume that the tire rolls straight past the protrusion.

If I wasn't so rusty on this material I would have taken a stab at it myself. So, any help in setting up and solving this problem would be greatly appreciated.