Distance problems in computational geometry with fixed orientations

• Authors: P. Widmayer;Y. F. Wu;C. K. Wong
• Venue: SCG '85 Proceedings of the first annual symposium on Computational geometry
• Year: 1985

In computational geometry, problems involving only rectilinear objects with edges parallel to the x -and y-axes have attracted great attention. They are often easier to solve than the same problems for arbitrary objects, and solutions are of high practical value, for instance in VLSI design. This is because in VLSI design technology requirements often dictate the use of only two orthogonal orientations for the boundary edges of objects as well as wires.The restriction on the boundary edges motivates the study of rectilinear objects, while the restriction on wires brings the focus on the well-known L1-metric (the Manhattan distance). In short, given the two orthogonal orientations, both the shape of objects and the distance function are determined in a natural way.More recent VLSI fabrication technology is capable of creating edges and wires in both the orthogonal and diagonal orientations. This motivates us to study more general polygons, and to generalize the distance concept to the case where any fixed set of orientations is allowed. We introduce a family of naturally induced metrics, and the subsequent generalization of geometrical concepts. A shortest connection between two points is in this case a path composed of line segments with only the given orientations. We derive optimal solutions for various basic planar distance problems in this setting, such as the computation of a Voronoi diagram, a minimum spanning tree, and the (minimum and maximum) distance between two convex polygons. Many other theoretically interesting and practically relevant problems remain to be solved. In particular, the new family of metrics may help bridge the gap between the L1- and the L2-metrics, as those are the limiting cases for two and infinitely many regularly distributed orientations.