The arc tree: An approximation scheme to represent arbitrary curved shapes

Abstract

This paper introduces the arc tree, a hierarchical data structure to represent arbitrary curved shapes. The arc tree is a balanced binary tree that represents a curve of length l such that any subtree whose root is on the k-th tree level is representing a subcurve of length l/2k. Each tree level is associated with an approximation of the curve; lower levels correspond to approximations of higher resolution. Based on this hierarchy of detail, queries such as point search or intersection detection and computation can be solved in a hierarchical manner. We compare the arc tree to several related schemes and present the results of a practical performance analysis for various kinds of set and search operators. We also discuss several options to embed arc trees as complex objects in an extensible database management system and argue that the embedding as an abstract data type is most promising.