Exploring Triangulated Networks in Geoinformation Technology

1. Geoinformation Technology: lecture 9b Triangulated Networks Prof. Dr. Thomas H. Kolbe Institute for Geodesy and Geoinformation Science Technische Universität Berlin Credits: This material is mostly an english translation of the course module no. 2 (‘Geoobjekte und ihre Modellierung‘) of the open e-content platform www.geoinformation.net.

2. Excursion: Voronoi Diagrams • Given: a set M of n points in a plane • The Voronoi diagram of the point set divides the plane into n disjoint areas (Voronoi regions). • The Voronoi region of one point p contains exactly one of the points of M as well as all points q, which lie closer to p than to every other point p‘M with p≠p‘ (“areas of same nearest neighbours”).

3. Voronoi Diagram & Delaunay Triangulation • the Voronoi diagram immediately provides the Delaunay triangulation • connect the nodes of neighbouring faces by (yellow) edges • the yellow edges constitute the wanted Delaunay TIN • note: the yellow Delaunay edges stand perpendicularly on the dashed Voronoi edges • the Delaunay triangulationis the “dual graph” of the Voronoi diagram

4. TINs with Break Lines • problem: The edges of topographic objects should be considered within the triangulation • aim: break lines are aggregations of triangle edges • inserting break lines leads to a finer triangle structure • In general, this triangulation does not fulfill the Delaunay criterion

5. Constrained Delaunay Triangulation • „Visibility“ of points: • P is visible from Q, if the straight connection PQ does not intersects a break line. • The constrainedcircle criterion: • no visiblefourth node lies in the perimeter of a triangle • Constrained Delaunay triangulations fulfill the constrainedcircle criterion • This criterion provides an algorithm for the insertion of break lines to a (constrained) Delaunay triangulation ( exercise).

6. Triangulated Networks - Example „Siebengebirge“ Rhineriver Bonn

7. Traingulated Networks - Example „Siebengebirge“

8. Application Example for TINs • Analysis of differences in height (water flow) leads to 3 edge types: • transfluent edge: water flows from neighbouring triangle over the edge away • confluent edge (drain): water from at least one triangle flows off along the edge • diffluent edge (watershed): neither diffluent nor confluent

9. Simple Drainage Model • simplifying assumption: the earth's surface is impermeable • confluent edges form the hydrography • diffluent edges form water sheds transfluent diffluent: border of a catchment area confluent: direction of water drain

10. Triangle networks Literature • Lenk, Ulrich: 2.5D-GIS und Geobasisdaten-Integration von Höheninformationen und Digitalen Situationsmodellen. PhD thesis, Institute for Photogrammetry and Geoinformation, University of Hannover, 2001 • Worboys, Michael F.: GIS: A Computing Perspective. Taylor & Francis Inc., London 1995