Claire Dufourd
MIGS 2
Triangulation and Meshing
Outline
Introduction and definitions
Delaunay Triangulation
Voronoï Diagram
Applications
Delaunay Triangulation Algorithm
Introduction and definitions
Triangulation:
Process to mesh the
convex hull of a set of
points in the plane with
triangles.
Introduction and definitions
Delaunay
triangulation:
All the circumcircles of
all the triangles are
empty.
It maximizes the
minimum angle of all the
angles of the triangles in
the triangulation.
Introduction and definitions
Voronoï Diagram:
Collection of regions that divide
up the plane.
Each region corresponds to
one of the”site”(= point of the
set), and all the points in one
region are closer to the
corresponding site than to any
other site.
Introduction and definitions
Relationship between
Delaunay triangulation
and Voronoï diagram:
One is the dual graph of the
other.
Each vertex of the Voronoï
diagram is the center of a
circumcircle of a triangle.
Each edge is on the
perpendicular bisector of an
edge of a triangle.
(Perpendicular bisector = médiatrice)
Introduction and definitions
Applications:
•Geometric Modeling
•Anthropology and Archeology
•Geography
•Zoology
Delaunay triangulation algorithm
Principle:
Incremental algorithm, one
vertex is added at a time.
Initialization:
3 points-> Unique triangulation,
it is a Delaunay triangulation.
Delaunay triangulation algorithm
Case 1:
The new point is inside the
convex hull of the previous set.
Delaunay triangulation algorithm
Case 2:
The new point is outside all of the circumcircles of the
triangles.
Delaunay triangulation algorithm
Case 3:
The new point is outside the convex hull but inside at
least one circumcircle of the triangles.