On Constructing Binary Space Partitioning
Trees
Ravinder Krishnaswamy
Ghascm S. Alijani
Auto Trol Technology
ShyhChang 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 npolygon 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 npolygon 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 realtime
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 0897913485/90/0002/0230
$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 heightbalanced.
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 :
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 Staff
 Computer Science, Graph Theory, Polygons, Collision Detection, Binary space partitioning

Click to edit the document details