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Lecture 15: COMPUTATIONAL GEOEMTRY: Delaunay triangulations. CS1050: Understanding and Constructing Proofs . Spring 2006. Jarek Rossignac. Lecture Objectives. Learn the definitions of Delaunay triangulation and Voronoi diagram Develop algorithms for computing them.

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Lecture 15 computational geoemtry delaunay triangulations l.jpg

Lecture 15: COMPUTATIONAL GEOEMTRY: Delaunay triangulations

CS1050: Understanding and Constructing Proofs

Spring 2006

Jarek Rossignac


Lecture objectives l.jpg
Lecture Objectives

Learn the definitions of Delaunay triangulation and Voronoi diagram

Develop algorithms for computing them


Find the place furthest from nuclear plants l.jpg
Find the place furthest from nuclear plants

Find the point in the disk that is the furthest from all blue dots


The best place is l.jpg
The best place is …

The green dot. Find an algorithm for computing it.

Teams of 2.



Algorithm for best place to live l.jpg
Algorithm for best place to live

max=0;

foreach triplets of sites A, B, C {

(O,r) = circle circumscribing triangle (A,B,C)

found = false;

foreach other vertex D {if (||OD||<r) {found=true;};};

if (!found) {if (r>max) {bestO=O; max=r;};

}

return (O);

Complexity?


Circumcenter l.jpg
Circumcenter

pt centerCC (pt A, pt B, pt C) { // circumcenter to triangle (A,B,C)

vec AB = A.vecTo(B);

float ab2 = dot(AB,AB);

vec AC = A.vecTo(C); AC.left();

float ac2 = dot(AC,AC);

float d = 2*dot(AB,AC);

AB.left();

AB.back(); AB.mul(ac2);

AC.mul(ab2);

AB.add(AC);

AB.div(d);

pt X = A.makeCopy();

X.addVec(AB);

return(X);

};

2ABAX=ABAB

2ACAX=ACAC

AB.left

C

AC.left

AC

X

A

AB

B


Delaunay triangles l.jpg
Delaunay triangles

  • 3 sites form a Delaunay triangle if their circumscribing circle does not contain any other site.



The best place is a delaunay circumcenter l.jpg
The best place is a Delaunay circumcenter

Center of the largest Delaunay circle (stay in convex hull of cites)


Assigned project l.jpg
Assigned Project

  • Implement Delaunay triangulation

    • I provide graphics, GUI, circumcenter

    • You provide loops and point-in-circle test

  • Report time cost as a function of input size


Applications of delaunay triangulations l.jpg
Applications of Delaunay triangulations

  • Find the best place to build your home

  • Finite element meshing for analysis: nice meshes

  • Triangulate samples for graphics


School districts and voronoi regions l.jpg
School districts and Voronoi regions

Each school should serve people for which it is the closest school. Same for the post offices.

Given a set of schools, find the Voronoi region that it should serve.

Voronoi region of a site = points closest to it than to other sites


Voronoi diagram dual of delaunay l.jpg
Voronoi diagram = dual of Delaunay

http://www.cs.cornell.edu/Info/People/chew/Delaunay.html


Assigned reading l.jpg
Assigned Reading

  • http://www.voronoi.com/applications.htm

  • http://www.ics.uci.edu/~eppstein/gina/voronoi.html

  • http://www.cs.berkeley.edu/~jrs/mesh/


Assigned homework l.jpg
Assigned Homework

  • Definition and algorithms for computing Delaunay triangulation

  • Duality with Voronoi diagram


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