Spatial Data Structures

This chapter presents an overview of several spatial decomposition techniques, as well as their associated data structures. We assume that the reader is familiar with some basic concepts of set theory, topology and geometry.

Spatial decompositions apply to both ambient spaces and their subspaces. In this textbook, we will focus on particular subspaces, say implicit curves and surfaces. Spatial decompositions of these subspaces are here called object decompositions. For example, the resolution of singularities of a level set (e.g. implicit surface in ℝ³) gives rise to its decomposition into manifolds. These object decompositions are particularly useful for rendering implicit curves and surfaces through continuation algorithms (see Chapter 6 for further details).

Decompositions that cover all the ambient space (e.g. a bounding box or even the whole ℝⁿ) containing an embedded object are called space decompo- sitions. These decompositions are used by space-partitioning algorithms for implicit objects (see Chapter 7 for more details).