A Continuous Spatial Index of a Triangulated Surface

We show how to build a continuous, one-dimensional index of the points on a triangulated twodimensional manifold such as a surface map of a geographic information system. The method involves creating an adjacency-preserving ordering of the triangular regions and then indexing each region with a space lling curve oriented in a special way. We prove that a continuous path can be created through any triangulation T , and that a continuous circuit can be constructed through all but a certain family of triangulations. We provide a construction algorithm requiring O(n) steps, when T has n faces. We characterize the family of noncircuit-indexable triangulations, and provide a simple O(n) necessary and su cient test condition for circuit-indexability. We show how continuous paths and circuits can be extended to hierarchical triangulations, requiring only local changes to existing indices. When a cell in T is itself triangulated withm triangles, the current index of T can be augmented in O(m) steps.