Center of circumscribed circles = Vertices of Voronoi diagram
Delaunay Triangulation Properties
Center of circumscribed circles = Vertices of Voronoi diagram
Delaunay Triangulation Properties
Example of non-Delaunay triangulation
Delaunay Triangulation Properties
Example of non-Delaunay triangulation
Delaunay Triangulation Incremental Algorithm We consider an initial triangulation and you want to add P – find the triangle containing P – destroy the triangles that pose problem – construct new triangles – add new triangles to the initial triangulation
Delaunay Triangulation Algorithm Flip – Compute any triangulation – While there exist non-Delaunay triangles • Flip edges
Delaunay Triangulation Construction by Convex Hull
Points Pi( xi, yi )
Delaunay Triangulation Construction by Convex Hull
Points Pi projected( xi, yi , Pi.Pi)
Delaunay Triangulation Construction by Convex Hull
Convex hull of Pi
Delaunay Triangulation Construction by Convex Hull
Lower Convex hull of Pi
Delaunay Triangulation Construction by Convex Hull
Project points on the plane z=0
Delaunay Triangulation Construction by Convex Hull
Elipse: project plan on convex hull
Delaunay Triangulation Construction by Convex Hull
Projected ellipse: circumscribed circle of the triangle
Delaunay Triangulation Construction by Convex Hull
Bibliography • Cours de Nicolas Szafran 2011/2012 (http://www-ljk.imag.fr/membres/Nicolas.Szafran/) • Scattered Data Interpolation and Approximation for Computer Graphics (Siggraph Asia course 2010) • Implicit surface reconstruction from point clouds (Johan Huysmans’s thesis) • http://www.cise.ufl.edu/~sitharam/COURSES/ CG/kreveldmorevoronoi.pdf
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