Department of Geoinformation Science Technische Universität Berlin WS 2006/07 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

2 T. H. Kolbe – Geoinformation Technology: lecture 9 Department of Geoinformation Science WS 2006/07 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 pp (areas of same nearest neighbours).

3 T. H. Kolbe – Geoinformation Technology: lecture 9 Department of Geoinformation Science WS 2006/07 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 triangulation is the dual graph of the Voronoi diagram

4 T. H. Kolbe – Geoinformation Technology: lecture 9 Department of Geoinformation Science WS 2006/07 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 T. H. Kolbe – Geoinformation Technology: lecture 9 Department of Geoinformation Science WS 2006/07 Constrained Delaunay Triangulation Visibility of points: P is visible from Q, if the straight connection PQ does not intersects a break line. The constrained circle criterion: no visible fourth node lies in the perimeter of a triangle Constrained Delaunay triangulations fulfill the constrained circle criterion This criterion provides an algorithm for the insertion of break lines to a (constrained) Delaunay triangulation ( exercise).

6 T. H. Kolbe – Geoinformation Technology: lecture 9 Department of Geoinformation Science WS 2006/07 Triangulated Networks - Example Siebengebirge Rhine river Bonn

7 T. H. Kolbe – Geoinformation Technology: lecture 9 Department of Geoinformation Science WS 2006/07 Traingulated Networks - Example Siebengebirge

8 T. H. Kolbe – Geoinformation Technology: lecture 9 Department of Geoinformation Science WS 2006/07 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 T. H. Kolbe – Geoinformation Technology: lecture 9 Department of Geoinformation Science WS 2006/07 Simple Drainage Model simplifying assumption: the earth's surface is impermeable confluent edges form the hydrography diffluent edges form water sheds transfluent confluent: direction of water drain diffluent: border of a catchment area

10 T. H. Kolbe – Geoinformation Technology: lecture 9 Department of Geoinformation Science WS 2006/07 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