Shortest Paths in Two Intersecting Pencils of Lines

Suppose one has a line arrangement and one wants to find a shortest path from one point lying on a line of the arrangement to another such point. We look at a special case: the arrangement consists of two intersecting pencils (sets of lines where all intersect in a point), and the path endpoints are at opposite corners of the largest quadrilateral formed. The open problem was to find a shortest path in o(n2 ) time. We prove here that there are only two possible shortest paths, and the shortest path can thus be computed in O(n) time.

• Type: Conference
• Year: 2003
• Where: CCCG
• Authors: David Hart
• Keywords: CCCG 2003, CCCG 2007, Line Arrangement, Path Endpoints, Shortest Path