Closure Spaces in the PlaneReport
When closure operators are defined over figures in the plane, they are normally defined with respect to convex closure in the Euclidean plane. This report concentrates on discrete closure operators defined over the discrete, rectilinear plane. Basic to geometric convexity is the concept of a geodesic, or shortest path. Such geodesics can be regarded as the closure of two points. But, given the usual definition of geodesic in the discrete plane, they are not unique. And therefore convex closure is not uniquely generated. We offer a different definition of geodesic that is uniquely generated, although not symmetric. It results in a different geometry; for example, two unbounded lines may be parallel to a third, yet still themselves intersect. It is a discrete geometry that appears to be relevant to VLSI design.
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Pfaltz, John. "Closure Spaces in the Plane." University of Virginia Dept. of Computer Science Tech Report (1999).
University of Virginia, Department of Computer Science