1. 1 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac

2. 2 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Lecture 15: COMPUTATIONAL GEOEMTRY: Delaunay triangulations Jarek Rossignac CS1050: Understanding and Constructing Proofs Spring 2006

3. 3 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Lecture Objectives Learn the definitions of Delaunay triangulation and Voronoi diagram Develop algorithms for computing them

4. 4 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Find the place furthest from nuclear plants Find the point in the disk that is the furthest from all blue dots

5. 5 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac The best place is … The green dot. Find an algorithm for computing it. Teams of 2.

6. 6 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Center of largest disk that fits between points

7. 7 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac 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?

8. 8 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac 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); }; Circumcenter A B C AC AB AB.left AC.left X 2AB  AX=AB  AB 2AC  AX=AC  AC

9. 9 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Delaunay triangles 3 sites form a Delaunay triangle if their circumscribing circle does not contain any other site.

10. 10 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Inserting points one-by-one

11. 11 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac The best place is a Delaunay circumcenter Center of the largest Delaunay circle (stay in convex hull of cites)

12. 12 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac 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

13. 13 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Applications of Delaunay triangulations Find the best place to build your home Finite element meshing for analysis: nice meshes Triangulate samples for graphics

14. 14 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac 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

15. 15 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Voronoi diagram = dual of Delaunay http://www.cs.cornell.edu/Info/People/chew/Delaunay.html

16. 16 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Assigned Reading http://www.voronoi.com/applications.htm http://www.ics.uci.edu/~eppstein/gina/voronoi.html http://www.cs.berkeley.edu/~jrs/mesh/

17. 17 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Assigned Homework Definition and algorithms for computing Delaunay triangulation Duality with Voronoi diagram