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Delaunay Triangulations - MIT

March 3, 2005 Lecture 9: Delaunay Triangulations Delaunay Triangulations (slides mostly by Glenn Eguchi)March 3, 2005 Lecture 9: Delaunay Triangulations Motivation: Terrains Set of data points A R2 Height (p) defined at each point p in A How can we most naturally approximate height of points not in A?March 3, 2005 Lecture 9: Delaunay Triangulations Option: Discretize Let (p) = height of nearest point for points not in A Does not look naturalMarch 3, 2005 Lecture 9: Delaunay Triangulations Better Option: Triangulation Determine a triangulation of A in R2, then raise points to desired height triangulation: planar subdivision whose bounded faces are triangles with vertices from AMarch 3, 2005 Lecture 9: Delaunay Triangulations Triangulation: Formal Definition maximal planar subdivision: a subdivision S such that

Proof: •Assume there are other points inside the circle. •Choose one point p inside the circle, and remove all other points but p i, p j, p k.Note that, after the removal of points, p i, p j, p k remains a triangle. •Assume lies opposite p j. • p is closer to the center than are p i, p j, p k.

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