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JonLeonard-OnConstructingBinarySpacePartitioningTrees - On...

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On Constructing Binary Space Partitioning Trees Ravinder Krishnaswamy Ghascm S. Alijani Auto Trol Technology Shyh-Chang Su Research and Development Computer Science Department 12500 North Washington University of Wyoming Denver, CO 80233 Laramie, WY 82071 Abstract Binary Space Partitioning Trees have several applications in computer graphics. We prove that there exist n-polygon problem instances with an O(n2) lower bound on tree size. We also show that a greedy algorithm may result in constructing a tree with O(n2) nodes, while there exist a tree for the same n-polygon instance with only O(n) nodes. Finally, we formulate six different heuristics and test their performance. 1. Introduction The hidden line elimination problem is of fundamental importance in computer graphics [4,6,7]. The Binary Space Partitioning (BSP) tree is one approach to the problem [l], and has received recent attention in [2.3,8]. An attractive aspect of the BSP tree is that once the tree corresponding to a collection of objects is created, the scene represented by the tree can be displayed with hidden line elimination in time that is linear in the number of nodes of the tree. This efficiency inherent in BSP tree based hidden line elimination is cause for serious consideration of BSP trees to be used in parallel real-time scene generation [5]. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy other- wise, or to republish, requires a fee and/or specific permission. 0 1990 ACM 089791-348-5/90/0002/0230 $1.50 230 In general, the tree representing a set of polygons is not unique, in fact, the number of nodes in trees representing the same scene may vary substantially depending on heuristics used in constructing the tree. The focus of this research is to evaluate several heuristics used to construct BSP trees according to the following two criteria : (a) The size (number of nodes) of the tree. @J) The extent to which the tree is height-balanced. The order in which polygons in the left subtree of a node are processed is independent of the order in which polygons of the right subtree of the node are processed. This observation is the motivation for including (b) as a method of judging the potential for parallelism (balanced subdivision of labour) reflected in the tree structure 2. Definition and Notation Definition : A BSP tree is a binary tree constructed from a polygon list. The basic notion is that given a plane in a three dimensional scene and a viewing point, no polygon on the view point side of the plane can be obstructed by any polygon on the far side. The tree can be recursively constructed as follows :
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A polygon is selected from the list and placed at the root, Each remaining polygon is tested to see which side of the plane containing the root polygon it lies in, and is placed in the appropriate side list. A polygon that intersects the
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