Continuous Spatial Indexing of Surfaces — Part I: Standard Triangulations

We show how to build a continuous, one-dimensional index of the points on a triangulated, two-dimensional manifold T . The index is constructed by quickly finding a vertex-adjacent ordering of the triangles (that is, sequential triangles share at least one vertex). Then the space within triangles is indexed with a spacefilling curve oriented so that the lowest point in each triangle is the vertex shared with the preceding triangle, and the highest point is the vertex shared with the next triangle. This index is a continuous circuit of T if the first and last points coincide, otherwise it is a continuous path. We show that when two triangulations with continuous circuits share an edge, then a continuous circuit can be created through their union. A continuous circuit is easily created through any pair or triple of edge-adjacent triangles. Therefore, if T can be partitioned into pairs and triples, a continuous circuit of T can be created by finding circuits in these sub-triangulations, and then combining them. This is possible in all but exceptional cases; in those cases, a continuous path can be found.