Competitive searching for a line on a line arrangement

We discuss the problem of searching for an unknown line on a known or unknown line arrangement by a searcher S, and show that a search strategy exists that finds the line competitively, that is, with detour factor at most a constant when compared to the situation where S has all knowledge. In the case where S knows all lines but not which one is sought, the strategy is 79-competitive. We also show that it may be necessary to travel on Ω(n) lines to realize a constant competitive ratio. In the case where initially, S does not know any line, but learns about the ones it encounters during the search, we give a 414.2-competitive search strategy.