Triangulation in geoscienceTriangulation in geoscience

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

Curso Caracas, 2006

Motivation for triangulationsMotivation 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

Definition of „Triangulation“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

HistoryHistory

•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

A sophisticated “network Voronoi-area diagram”, rediscovered 150 years later again!

First known application (1855)First known application (1855)

from http://www.soi.city.ac.uk/~dk708/pg1_1.htm

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.

Curso Caracas, 2006

Good and bad triangulationGood and bad triangulation

Here: “as equilateral as possible” or “maximize smallest angle”.

Planless triangulation: Very long triangles with small angles exist.

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Delaunay triangulationDelaunay triangulation

Delauney criteria test in a small triangulation. Voronoi cell in red.

Construction of the circle through all three points of an triangle. In Zero Order Delauney it must not contain further points!

Curso Caracas, 2006

Voronoi diagramsVoronoi diagramsFurther Voronoi edges forming further cells successively.

Construction of the first Voronoi cell around point 7.

Real Voronoi experts!

Curso Caracas, 2006

Voronoi diagrams vs. Delaunay triangulation?

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

Algorithms in 2D #1Algorithms in 2D #1

1) 2)

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Algorithms in 2D #2Algorithms in 2D #21) 2) 3)

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Algorithms in 2D #3Algorithms in 2D #3

VoroGlide (Source: Praktische Informatik VI, FernUniversität Hagen)

a) c)b)

Curso Caracas, 2006

Triangulation and IsolinesTriangulation and Isolines

First we construct the Delaunay Triangulation and afterwards the isolines shown as red lines in the pictures.

Curso Caracas, 2006

IGMAS „semi“-3DIGMAS „semi“-3DIn IGMAS the triangulation is carried out exclusively between parellel planes, on which structures are modeled by polygons under „minimal area“ condition“.

Curso Caracas, 2006

IGMAS - triangulation „trap“IGMAS - triangulation „trap“

Curso Caracas, 2006

Triangulated Irregular NetworkTriangulated Irregular Network

Direct triangulation between the points defining the isolinesDirect triangulation between the points defining the isolines

Delauney triangulation, here with a smoothness effect!Delauney triangulation, here with a smoothness effect!

Curso Caracas, 2006

Higher Order Delaunay TriangulationHigher Order Delaunay Triangulation

The dam is cutting the river line. Why does this happen?

Triangulated valley with a river

river

The solution is a constrained “Heigher Order Delaunay”-Triangulation (HOD).

Curso Caracas, 2006

Triangulation in 3D spaceTriangulation in 3D space

Crystall structure and Wigner-Seitz-Cell of bcc and fcc lattice.

Source: PhysNet Uni Hamburg

Curso Caracas, 2006

Triangulation in 3D spaceTriangulation in 3D space

Wigner-Seitz-Cell of bcc and fcc lattice.

Source: PhysNet Uni Hamburg

Calculated valence electron density of a silicon nanocrystal. Source: Zack Helms, NCSA

“Buckyball” – a C60 Fullerene Macromolecule. Source: Rayshade, Carnegie Mellone SCS

Curso Caracas, 2006

References

1. 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)

2. Gudmudsson: Geometric Decompositions and Networks, Approximation Bunds and Algorithms, 2000, Lund University, Sweden (ISBN: 91-7874-098-3)

3. Shewchuk: Triangle: Engeneering a 2D Quality Mesh Generator and Delaunay Triangulator, Carnegie Mellon University, Pittsburgh, USA

4. Cignoni, Montani, Scopigno: DeWall: A Fast Divide & Conguer Delaunay Triangulation Algorithm in Ed, 1997, Pisa, Italy

5. Barrio, Gangui, Götze, Schmidt, Viramonte, Omarini: Curso de postgrado: Aplicaciones de la Computación Gráfica en Geología y Geofísica, 1992, Universidad Nacional de Salta, Argentinia