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 ScienceTechnische 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.

WS 2006/072 T. H. Kolbe – Geoinformation Technology: lecture 9

Department of Geoinformation Science

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”).

WS 2006/073 T. H. Kolbe – Geoinformation Technology: lecture 9

Department of Geoinformation Science

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

WS 2006/074 T. H. Kolbe – Geoinformation Technology: lecture 9

Department of Geoinformation Science

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

WS 2006/075 T. H. Kolbe – Geoinformation Technology: lecture 9

Department of Geoinformation Science

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).

WS 2006/076 T. H. Kolbe – Geoinformation Technology: lecture 9

Department of Geoinformation Science

Triangulated Networks - Example „Siebengebirge“

Rhineriver

Bonn

WS 2006/077 T. H. Kolbe – Geoinformation Technology: lecture 9

Department of Geoinformation Science

Traingulated Networks - Example „Siebengebirge“

WS 2006/078 T. H. Kolbe – Geoinformation Technology: lecture 9

Department of Geoinformation Science

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

WS 2006/079 T. H. Kolbe – Geoinformation Technology: lecture 9

Department of Geoinformation Science

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

WS 2006/0710 T. H. Kolbe – Geoinformation Technology: lecture 9

Department of Geoinformation Science

Triangle networks Literature

Lenk, Ulrich: 2.5D-GIS und Geobasisdaten-Integration von Höheninformationen und Digitalen Situationsmodellen. PhD thesis, Institute for Photogrammetry andGeoinformation, University of Hannover, 2001

Worboys, Michael F.: GIS: A Computing Perspective. Taylor & Francis Inc., London 1995