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Triangulation in geoscience

Triangulation in geoscience. Motivation History Delaunay triangulation Voronoi diagrams Algorithms in 2D IGMAS „semi“-3D TINs and Higher Order DT 3D Triangulations References. Motivation for triangulations. Generation of surfaces from irregular point sets Networks and space relations

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Triangulation in geoscience

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  1. Triangulation in geoscience Motivation History Delaunay triangulation Voronoi diagrams Algorithms in 2D IGMAS „semi“-3D TINs and Higher Order DT 3D Triangulations References

  2. Motivation for triangulations • Generation of surfaces from irregular point sets • Networks and space relations • Interpolation • Computer graphics • Modelling of data (e.g. IGMAS) • Molecule physics / Cristallography Curso Caracas, 2006

  3. Definition of „Triangulation“ The triangulation T of the point set S consist of the maximum amount of not crossing lines connecting two points p of S with each other. This leads directly to a mesh of triangles. Often it is desirable to maximize the smallest occuring angle in the triangles of T – the triangles shall be “well shaped”! Curso Caracas, 2006

  4. History • Star map of (Descartes 1644) • Dirichlet (2D & 3D) (1850) • Voronoi (Rm) (1907,1908,1909) • Delone (frz: Delaunay) “empty sphere” (1924,1928,1929,1932,1934) • Snow (1855) • Boldyrev Boreholes (1909) • Wiegner & Seitz Chemistry (1933) • Shannon Maximum Likelyhood decoding (1959) • Hoofd Medicine “capillary domains” (1985) • Icke Astronomy (1987) • Severel further ones in 1994, 1995, 1997 etc. ... … … Curso Caracas, 2006

  5. First known application (1855) Though they were reluctant to believe him, they agreed to remove the pump handle as an experiment. When they did so, the spread of cholera dramatically stopped. A sophisticated “network Voronoi-area diagram”, rediscovered 150 years later again! from http://www.soi.city.ac.uk/~dk708/pg1_1.htm Curso Caracas, 2006

  6. Good and bad triangulation Planless triangulation: Very long triangles with small angles exist. Here: “as equilateral as possible” or “maximize smallest angle”. Curso Caracas, 2006

  7. Delaunay triangulation Construction of the circle through all three points of an triangle. In Zero Order Delauney it must not contain further points! Delauney criteria test in a small triangulation. Voronoi cell in red. Curso Caracas, 2006

  8. Voronoi diagrams Real Voronoi experts! Construction of the first Voronoi cell around point 7. Further Voronoi edges forming further cells successively. Curso Caracas, 2006

  9. Voronoi diagrams vs. Delaunay triangulation? They are so called dual graphs in mathmatical graph theory. That means: Both hold the same information content! Curso Caracas, 2006

  10. Algorithms in 2D #1 1) 2) Curso Caracas, 2006

  11. Algorithms in 2D #2 1) 2) 3) Curso Caracas, 2006

  12. Algorithms in 2D #3 a) b) c) VoroGlide (Source: Praktische Informatik VI, FernUniversität Hagen) Curso Caracas, 2006

  13. Triangulation and Isolines First we construct the Delaunay Triangulation and afterwards the isolines shown as red lines in the pictures. Curso Caracas, 2006

  14. IGMAS „semi“-3D In IGMAS the triangulation is carried out exclusively between parellel planes, on which structures are modeled by polygons under „minimal area“ condition“. Curso Caracas, 2006

  15. IGMAS - triangulation „trap“ Curso Caracas, 2006

  16. Triangulated Irregular Network Direct triangulation between the points defining the isolines Delauney triangulation, here with a smoothness effect! Curso Caracas, 2006

  17. Higher Order Delaunay Triangulation The solution is a constrained “Heigher Order Delaunay”-Triangulation (HOD). Triangulated valley with a river river The dam is cutting the river line. Why does this happen? Curso Caracas, 2006

  18. Triangulation in 3D space Crystall structure and Wigner-Seitz-Cell of bcc and fcc lattice. Source: PhysNet Uni Hamburg Curso Caracas, 2006

  19. Triangulation in 3D space Calculated valence electron density of a silicon nanocrystal. Source: Zack Helms, NCSA “Buckyball” – a C60 Fullerene Macromolecule. Source: Rayshade, Carnegie Mellone SCS Wigner-Seitz-Cell of bcc and fcc lattice. Source: PhysNet Uni Hamburg Curso Caracas, 2006

  20. References • Okabe, Boots, Sugihara, Chiu: Spatial Tessellations, concepts and applications of Voronoi diagrams, 2nd Ed., 1998, JOHN WILEY & SONS, LTD, Chichester, GB (ISBN: 0-471-98635-6) • Gudmudsson: Geometric Decompositions and Networks, Approximation Bunds and Algorithms, 2000, Lund University, Sweden (ISBN: 91-7874-098-3) • Shewchuk: Triangle: Engeneering a 2D Quality Mesh Generator and Delaunay Triangulator, Carnegie Mellon University, Pittsburgh, USA • Cignoni, Montani, Scopigno: DeWall: A Fast Divide & Conguer Delaunay Triangulation Algorithm in Ed, 1997, Pisa, Italy • Barrio, Gangui, Götze, Schmidt, Viramonte, Omarini: Curso de postgrado: Aplicaciones de la Computación Gráfica en Geología y Geofísica, 1992, Universidad Nacional de Salta, Argentinia Curso Caracas, 2006

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