On Envelopes of Arrangements of Lines

[Gr72] Graham R., " An efficient algorithm for determining the convex hull of a planar set, " In-A simple algorithm for determining the envelope of a set of lines, " Information [To91] Toussaint G.T., " Computing morphological properties of arrangements of lines, " Proc. [Ve87] Vegter G., " Computing the bounded region determined by finitely many lines in the plane, " Technical Report 87-03, [Za75] Zaslavsky T., " Facing up to arrangements: face-count formulas for partitions of space by hyperplanes, " Mem.-31-tight upper bounds for general polygons can be solved in linear time for envelope polygons. A few examples of such open problems include finding the longest or shortest diagonal of an envelope, or computing the geodesic diameter and the geodesic center of an envelope. We introduced a hierarchy of classes of arrangements of lines based on the number of convex vertices of their envelopes. In particular, we looked at a class called sail arrangements: given a sail arrangement A of n lines, we can find the three convex vertices of E(A) and therefore the convex hull, the diameter and the points with minimum or maximum x-coordinate of the arrangement in O(n) time. Curiously, however, the complexity of constructing the remainder of the envelope of a sail arrangement remains Ω(n log n). We also showed that 1-sail arrangements admit hamiltonian circuits, and that a streamlined hamiltonian circuit of a 1-sail arrangement of n lines can be computed in time Θ(n log n). It remains, however, an open problem as to whether or not 2 and 3-sail arrangement graphs admit hamiltonian circuits. Owing to the well defined geometrical structure of sail arrangements, we conjecture that 2 and 3-sail arrangements are hamiltonian. Another open problem is to see if we can determine in O(n) time if a given arrangement of lines is a sail or not. Finally, it would be of interest to study further the hierarchy E c , especially for values of c greater than 3, or to define additional new classes of arrangements for which interesting results can be obtained.