Embeddability of pseudoline arrangements and point configurations to Eucliean plane

The present thesis explores embeddability (realizability) properties of pseudoline arrangements, perhaps, the most important mathematical structures in computational geometry. The underlying theme is the use of combinatorial representation of arrangements vs. usual geometric representation to approach the embeddability problem. The combinatorial representation chosen is that of counterclockwise systems (CCsystems) introduced by D.E. Knuth in 1992. CC systems were defined as sets of ordered triples of points that incorporate and obey counterclockwise relations on points in the plane. If a CC system can arise from actual points in the plane, it is called realizable. Not all CC-systems are realizable in plane, since they are defined by axioms that involve at most five points. The most famous nonrealizable configuration on nine points corresponds to so-called theorem of Pappus, which is a legal CC-system. Any heuristic solution to realizability question is proven to be NP-hard. The primary goal of this thesis is to consider embeddings of CC-systems to the Euclidean plane and to exploit properties coherent to embeddable systems. As a major effort in this task we have developed and implemented algorithms (brute force as well as breadth-first search) for finding the convex hull of a CC-system and geometrically testing the realizability of the system using linear programming methods. Recently, Goodman, Pollack, Wenger, and Zamfirescu1 2 have proved two conjectures of Grunbaum, showing that any arrangement of pseudolines on 8 points in the plane can be embedded into a projective plane, and that there exists a universal topological projective plane in which every arrangement of pseudolines is stretchable (realizable). Thus, we have conducted our realizability tests on systems arising from 9 or more points. Pappus system mentioned above and other non-realizable extensions of it on more than 9 points were studied as well. We have also developed a graphical interface (Java application) for creating, dynamically updating, visualizing and storing CC-systems. Visual geometric embedding of a particular system point-by-point is another feature provided by this interface