Geometric computing and uniform grid technique

Abstract

If computational geometry should play an important role in the professional environment (e.g. graphics and robotics), the data structures it advocates should be readily implemented and the algorithms efficient. In the paper, the uniform grid and a diverse set of geometric algorithms that are all based on it, are reviewed. The technique, invented by the second author, is a flat, and thus non-hierarchical, grid whose resolution adapts to the data. It is especially suitable for telling efficiently which pairs of a large number of short edges intersect. Several of the algorithms presented here exist as working programs (among which is a visible surface program for polyhedra) and can handle large data sets (i.e. many thousands of geometric objects). Furthermore, the uniform grid is appropriate for parallel processing; the parallel implementation presented gives very good speed-up results. © 1989

Parallel object-space hidden surface removal

“…The uniform grid technique has been used to exploit parallelism in the geometric aspects of hidden surface removal. The uniform grid technique could be successfully used for parallelizing other geometric problems [3], [5]. A conflict-detection and back-off strategy has been introduced to achieve parallelism in the visible region reconstruction which is related to the topological aspects of the problem.…”

“…One objection against the uniform grid technique could be that it is not suitable for irregular scenes and that hierarchical methods such as quadtrees need to be used. But this has not been a problem in practice [3]. From the parallel processing viewpoint, implementation becomes a lot easier with a flat data structure since the overhead on scheduling can be reduced by using a static scheduling scheme.…”

“…From the parallel processing viewpoint, implementation becomes a lot easier with a flat data structure since the overhead on scheduling can be reduced by using a static scheduling scheme. Readers are referred to [3] for fiarther details on the uniform grid technique and its applications to geometric computing.…”

Fast exact parallel map overlay using a two-level uniform grid

“…The uniform grid works well even for uneven data for various reasons [2,8,14,15]. First, the total time is the sum of one component (inserting edges into cells) that runs slower with a finer grid, plus another component (intersecting edges in cells) that runs faster.…”

Simplification of tetrahedral meshes with accurate error evaluation

“…The intrinsic non-locality of this kind of degeneration makes it difficult to efficiently and correctly prevent it without using auxiliary structures. To speedup self-intersection checks (a quadratic problem in its naive implementation) a uniform grid [1] could be adopted, to store all the vertices of the current boundary of the mesh. For each edge collapse (v s,vd) that involves a boundary edge, we should check whether after the collapse, all the edges on the boundary incident in v d do not intersect the mesh boundary.…”