A Note on Some Applications of Interval Arithmetic in Hierarchical Solid Modeling

Abstract

Techniques of reliable computing like interval arithmetic can be used to
guarantee a reliable solution even in the presence of numerical round-off
errors. The need to trace bounds for the error function separately can be 
eliminated using these techniques. In this talk, we focus on some 
demonstrations how the techniques and algorithms of reliable computing 
can be applied to the construction and further processing of hierarchical 
solid representations using the octree model as an example.

An octree is a common hierarchical data structure to represent 3D 
geometrical objects in solid modeling systems or to reconstruct a real 
scene. The solid representation is based on recursive cell decompositions 
of the space. Unfortunately, the data structure may require a large amount 
of memory when it uses a set of very small cubic nodes to approximate a 
solid.

In this talk, we present a novel generalization of the octree model created 
from a CSG object that uses interval arithmetic and allows us to extend the 
tests for classifying points in space as inside, on the boundary or outside 
the object to handle whole sections of the space at once. Tree nodes with 
additional information about relevant parts of the CSG object are 
introduced in order to reduce the depth of the required subdivision. 
Furthermore, this talk is concerned with interval-based algorithms for 
reliable proximity queries between the extended octrees and with further 
processing of the structure. We conclude the talk with some examples of 
implementations.