Polygon comparison using a graph representation

Abstract

A b s t r a c tAll of the information necessary to perform the polygon set operations (union, intersection, and difference) and therefore polygon clipping can be generated by a single application of a process called polygon comparison. This process accepts two or more input polygons and generates one or more polygons as output. These output polygons contain unique homogenous areas, each falling within the domain of one or more input polygons. Each output polygon is classified by the list of input polygons in which its area may be found. The union contour of all input is also generated, completing all of the information necessary to perform the polygon set operations. This paper introduces a polygon comparison algorithm which features reduced complexity due to its use of a graph data representation. The paper briefly introduces some of the possible approaches to the general problem of polygon comparison including the polygon set and clipping problems. The new algorithm is then introduced and explained in detail.The algorithm is sufficiently general to compare sets of concave polygons with holes. More than two polygons can be compared at one time; all information for future comparisons of subsets of the original input polygon sets is available from the results of the initial application of the process.The algorithm represents polygons using a graph of the boundaries of the polygons. These graphs are imbedded in a two dimensional geometric space.The use of the graph representation simplifies the comparison process considerably by eliminating many special cases from explicit consideration. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct co~ercial advantage, the ACM copyright notice and O]980 ACM 0-8979]-021-4/80/0700-0010 $00.75 10 Polygon operations like the ones described above are useful in a variety o! application areas, especially those which deal with problems involving two dimensional or projected two dimensional geometric areas. Examples include VLSI circuit design, cartographic and demographic applications, and polygon clipping for graphic applications such as viowporl clipping, hidden surface and line removal, detailing, and shadowing. I n t r o d u c t i o nPolygon comparison is the process where several input polygons are compared and output polygons are generated such that the area of each output polygon is unique (non-overlapping) and completely homogenous in terms of the number and identity of the input polygons which contain the output area (see figure 2-1). These results provide six common polygon manipulation operations including the polygon set operations: union, intersection, and difference (see figures 2-2a through 2-2d), and the polygon clipping operations (see figures 2-2e and 2-2f). Polygon clipping, as normally used in computer graphics, is a combination of the difference and intersection of the input polygons. This paper introduces a polygon comparison algorithm which represents an advance over its predece...

Boundary evaluation of non-convex primitives to produce parametric trimmed surfaces

“…The nonconvex planar facet requires a special 2-D processor in which the facet is bounded by curves rather than halfspaces. This processor is based on 2-D ray firing with 2-D local transition detection [23].…”

Domain composition methods for associating geometric modeling with finite element modeling

“…Such a representation contains sufficient information Figure 10 shows the BReps of two geometric primitives before and after they are merged into a single BRep (see [36] for a description of the topological merge operation). The heavy lines represent edges, the large dots represent vertices, and the shaded areas represent faces.…”

The Brush-Trajectory Approach to Figure Specification: Some Algebraic Solutions

“…Then a postprocessing phase which generates as its output one or more closed nonoverlapping and nonintersecting contours is needed. For this, one can make use of the winding number concept [3] or the graph manipulation approach common in geometric modeling [8].…”

“…It is important to note that, in general, we may therefore have more than two outline curves; some of them possibly intersecting with one another. As noted earlier the actual boundary has to be derived from these outline curves using the winding number concept [3], or graph manipulation approach [8].…”