Geoinformation Technology: lecture 9b Triangulated Networks

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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.
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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).

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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

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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

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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).

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Triangulated Networks - Example Siebengebirge
Rhineriver
Bonn
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Traingulated Networks - Example Siebengebirge
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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

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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
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Triangle networks Literature

• Lenk, Ulrich 2.5D-GIS und Geobasisdaten-Integrat
  ion 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