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

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 kth tree level is representing a subcurve of length $\text{l}\text{2}{}^{\mathrm{k}}$. 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. Algorithms start out near the root of the tree and try to solve the queries at a very coarse resolution. If that is not possible, the resolution is increased where necessary. We describe and analyze several such algorithms to compute a variety of set and search operators. Various related approximation schemes to represent curved shapes are also discussed.