Other Topological Models

1. Other Topological Models C8 • Triangular Irregular Network (TIN) S14 S15 C1 T3 C3 S9 C8 C8 T2 S2 T4 T10 S8 S3 T9 S10 S13 S1 T11 C2 S16 C7 S4 S7 T7 C4 T1 T12 S6 C8 S5 S11 T8 T5 C5 S12 C8 C6 T6 S18 S17 C8

2. A TIN Representation Bounding Connections Cobounding Control Points Adjacent Triangles Triangles T1 S13 S18 S7 C2 C8 C5 T2 T6 T7 T2 S14 S13 S8 C1 C8 C2 T3 T1 T9 T3 S15 S14 S9 C3 C8 C1 T4 T2 T10 T4 S16 S15 S10 C7 C8 C3 T5 T3 T11 T5 S17 S16 S11 C6 C8 C7 T6 T4 T12 T6 S18 S17 S12 C5 C8 C6 T1 T5 T8 T7 S7 S6 S1 C2 C5 C4 T1 T8 T9 T8 S6 S12 S5 C4 C5 C6 T7 T6 T12 T9 C1 C2 C4 T2 T7 T10 S8 S1 S2 T10 S9 S2 S3 C3 C1 C4 T3 T9 T11 T11 S10 S3 S4 C7 C3 C4 T4 T10 T12 T12 S11 S4 S5 C6 C7 C4 T5 T11 T8

3. Thiessen Polygons or Voronoi Diagram V3 B3 B2 Thiessen Centroid E14 E15 V2 C1 E9 E8 Boundary Thiessen Vertex E2 C3 V10 V4 E13 C2 E10 V9 E3 E1 Interior Thiessen Vertex C4 E7 V11 E4 V7 E6 C7 V1 E16 V12 C5 E5 V8 C6 E11 Thiessen Edge E12 P E18 E17 P' V5 B1 V6 B4

4. Conventions and Definitions (1) • A background Thiessen centroid exists such that each bounding edge is equidistant between it and the respective interior centroids • Each portion of the convex boundary that is connected by two boundary Thiessen vertices is called a boundary Thiessen edge • Each Thiessen vertex is directly connected to three other Thiessen vertices by either a boundary or interior Thiessen edge • Each boundary Thiessen vertex lies on a bounding edge that connects two sequentially numbered bounding vertices (B1,B2,B3,B4, e.g.)

5. Conventions and Definitions (2) • One centroid is called a Thiessen neighbor of another centroid if the polygons associated with the two centroids cobound a common Thiessen edge • If two centroids only cobound a common Thiessen vertex (a degenerated edge), they are called Thiessen half-neighbors. However, an additional vertex having the same location can be added so that each vertex will cobound exactly three Thiessen edges

6. Relations among Numbers of Thiessen Centroids, Thiessen Vertices and Thiessen Edges For a Thiessen diagram containing n centroids, there are 3(n-1) Thiessen edges and 2(n-1) Thiesen vertices. For example, 7 centroids are given, the resulted number of Thiessen edges is 3(7-1)=18; and the number of Thiessen vertices is 2(7-1)=12.

7. Delaunay Triangulation • Duality to Thiessen Diagram • Each Thiessen vertex is the circumcenter of a circle that inscribes each Delaunay triangle • Delaunay triangulation can be made by searching such three points among the given point set, that a circle passing through these three points contains no other point

8. A Thiessen Diagram Representation Cobounding Edges Cobounding Control Points Adjoining Vertices Boundary Vertex Vertices V1 E13 E18 E7 C2 C8 C5 V2 V6 V7 B1 V2 E14 E13 E8 C1 C8 C2 V3 V1 V9 B1 V3 E15 E14 E9 C3 C8 C1 V4 V2 V10 B2 V4 E16 E15 E10 C7 C8 C3 V5 V3 V11 B3 V5 E17 E16 E11 C6 C8 C7 V6 V4 V12 B4 V6 E18 E17 E12 C5 C8 C6 V1 V5 V8 B4 V7 E7 E6 E1 C2 C5 C4 V1 V8 V9 0 V8 E6 E12 E5 C4 C5 C6 V7 V6 V12 0 V9 E8 E1 E2 C1 C2 C4 V2 V7 V10 0 V10 E9 E2 E3 C3 C1 C4 V3 V9 V11 0 V11 E10 E3 E4 C7 C3 C4 V4 V10 V12 0 V12 E11 E4 E5 C6 C7 C4 V5 V11 V8 0

9. 100 random points Delaunay Triangulation Voronoi Diagram