UNC Chapel Hill M. C. Lin Delaunay Triangulations Reading: Chapter 9 of the Textbook Driving Applications –Height Interpolation –Constrained Triangulation

UNC Chapel Hill M. C. Lin Height Interpolation A terrain is a 2D surface in 3D space with a special property: every vertical line intersects it in a point. That is, it is the graph of a function f: A  R 2  R that assigns a height f(p) to every point p in the domain. Problem: From the height of the sample points we somehow have to approximate the height at the other points in the domain.  Triangles with small angles are undesirable. So, rank the triangulations by their smallest angles.

UNC Chapel Hill M. C. Lin Triangulations of Planar Point Sets Let P := {p 1, p 2, …, p n } be a set of points in the plane. A triangulation of P is the maximal planar subdivision S whose vertex set is P, s.t. no edge connecting 2 vertices can be added to S without destroying its planarity. Let P be a set of n points in the plane, not all collinear, and let k denote the number of points in P that lie on the boundary of the convex hull of P. Then any triangulation of P has 2n-2-k triangles and 3n-3-k edges.

UNC Chapel Hill M. C. Lin Basic Properties Let C be a circle, l a line intersecting C in points a and b, and p, q, r, and s points lying on the same side of l. Suppose that p and q lie on C, that r lies inside C, and that s lies outside C, then  arb >  apb =  aqb >  asb An edge is illegal if we can locally increase the smallest angle by flipping that edge. q l a b C p r s

UNC Chapel Hill M. C. Lin Basic Properties Let T be a triangulation with an illegal edge e, and T’ be a triangulation obtained from T by flipping e. Then A(T’) > A(T), where A(.) stands for the smallest angle. Let edge p i p j be incident to triangles p i p j p k and p i p j p l, and let C be the circle through p i, p j, p k and p l. The edge is illegal if and only if the point p l lies in the interior of C. Furthermore, if the four points form a convex quadrilateral and do not lie on a common circle, then exactly one of p i p j and p k p l is an illegal edge.

UNC Chapel Hill M. C. Lin Legal Triangulation A legal triangulation is a triangulation that do not contain any illegal edge. Any angle-optimal triangulation is legal. We can compute legal triangulation by simply flip illegal edges till all edges are legal, given a triangulation. (see next) –there are only finite number of triangulations and every iteration the smallest angle increases. So, it works, but too slow!

UNC Chapel Hill M. C. Lin LegalTriangulation(T) Input: Some triangulation T of a point set P. Output: A legal triangulation T. 1. while T contains an illegal edge p i p j 2. do (* Flip p i p j *) 3. Let p i p j p k and p i p j p l be the two triangles adjacent to p i p j 4. Remove p i p j from T, and add p k p l instead 5. return T

UNC Chapel Hill M. C. Lin Voronoi Diagram & Dual Graph The Voronoi diagram of P, Vor(P), is the subdivision of the plane into n regions, one for each site in P, s.t. the region of a site p  P contains all points in the plane for which p is the closest site. The region of a site p is called the Voronoi cell of p, denoted by V(p). The dual graph of Vor(P), G, has a node for every Voronoi cell (or every site). G has an arc between 2 nodes if the corresponding cells share an edge.

UNC Chapel Hill M. C. Lin Delaunay Graph & Triangulation Delaunay graph of P, DG(p) is the straight-line embedding of G, where the node corresponding to the Voronoi cell V(p) is the point p, and the arc connecting the nodes of V(p) is the segment pq. If P is in general position (i.e. no 4 points lie on a circle), then all vertices of the Voronoi diagram have degree three, and consequently all bounded faces of DG(p) are triangles. In this case, DG(p) is the Delaunay triangulation of P and unique.

UNC Chapel Hill M. C. Lin Basic Properties of Delaunay Triangulations Let P be a set of points in the plane. –3 points are vertices of the same face of the DG(P) iff the circle thru them contains no point of P in interior. –2 points form an edge of DG(P) iff there is a closed disc C that contains them on its boundary and doesn’t contain any other point. Let P be a set of points in the plane, and let T be a triangulation of P. Then T is a Delaunay triangulation of P, iff the circumcircle of any triangulation of T doesn’t contain a point of P in its interior.

UNC Chapel Hill M. C. Lin Basic Properties of Delaunay Triangulations Let P be a set of points in the plane. A triangulation T of P is legal if and only if T is a Delaunay triangulation of P. Let P be a set of points in the plane. Any angle-optimal triangulation of P is a Delaunay triangulation of P. Any Delaunay triangulation of P maximizes the minimum angle over all triangulations of P.

UNC Chapel Hill M. C. Lin Computing Delaunay Triangulation Use Voronoi diagram to get Delaunay Triangulation. Use Randomized Incremental Construction: –Start with a big triangle that contains all the points. The vertices of this big triangle should not lie in any circle defined by 3 points in P. –Add one point p r at a time, then add edges from p r to the vertices of the existing triangle. –There are 2 cases to consider: p r lies in the interior of a triangle p r falls on an edge (need to make sure new edges are legal by flipping edges if necessary).

UNC Chapel Hill M. C. Lin DelaunayTriangulation(P) Input: A set P of n points in the plane Output: A Delaunay triangulation of P 1. Let p -1, p -2 & p -3 be 3 point s.t. P is contained in triangle p -1 p -2 p Initialize T as triangulation consisting of a single triangle p -1 p -2 p Compute a random permutation p 1, …, p n of P 4. for r  1 to n 5. do (* Insert p r into T : *) 6. Find a triangle p i p j p k  T containing p r 7. if p r lies in the interior of the triangle p i p j p k 8. then Add edges from p r to 3 vertices of p i p j p k and split it 9. LegalizeEdge( p r, p i p j, T ) 10. LegalizeEdge( p r, p j p k, T ) 11. LegalizeEdge( p r, p k p i, T )

UNC Chapel Hill M. C. Lin DelaunayTriangulation(P) 12. else (* p r lies on an edge of p i p j p k, say the edge p i p j *) 13. Add edges from p r to p k and to the third vertex of other triangle that is incident to p i p j, thereby splitting 2 triangles incident to p i p j into 4 tri’s 14. LegalizeEdge( p r, p i p l, T ) 15. LegalizeEdge( p r, p l p j, T ) 16. LegalizeEdge( p r, p j p k, T ) 17. LegalizeEdge( p r, p k p i, T ) 18. Discard p -1, p -2 and p -3 with all their incident edges from T 19. return T

UNC Chapel Hill M. C. Lin LegalizeEdge(p r, p i p j, T) 1. (* The point being inserted is p r, and p i p j is the edge of T that may need to be flipped *) 2. if p i p j is illegal 3. then Let p i p j p k be triangle adjacent to p r p i p j along p i p j 4. (* Flip p i p j *) Replace p i p j with p r p k 5. LegalizeEdge( p r, p i p k, T ) 6. LegalizeEdge( p r, p k p j, T )

UNC Chapel Hill M. C. Lin Find Triangle Containing P r While we build the Delaunay triangulation, we also build a point location structure D. The leaves of D correspond to the triangles of current triangulation T. We maintain cross pointers between the leaves and the triangulation. The internal nodes of D correspond to triangles that were in triangulation at some earlier stage, but have been destroyed. We initialize D as a DAG with a single leaf node that corresponds to the big triangle. For the rest, see the pictorial example.

UNC Chapel Hill M. C. Lin How to Pick a Big Triangle Let M be the maximum value of any coordinate of a point in P. Then, the triangle has the vertices at ( 3M, 0), (0, 3M ), and (- 3M, - 3M ). Then, handle the cases where the edge that P r lies on has one vertex with a negative index.

UNC Chapel Hill M. C. Lin Algorithm Analysis The Delaunay triangulation of a set of n points in the plane can be computed in O(n log n) expected time, using O(n) expected storage –see the proof in 9.4 (similar to previous analysis for randomized algorithms)