Assume a height value is associated with each point. " A triangulation of the points defines a piecewiselinear surface of triangular patches.

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1 A triangulation of set of oints in the lane is a artition of the convex hull to triangles whose vertices are the oints, and are emty of other oints." There are an exonential number of triangulations of a oint set." Assume a height value is associated with each oint. " A triangulation of the oints defines a iecewiselinear surface of triangular atches." The height of a oint inside a triangle is determined by the height of the triangle vertices, and the location of." The result deends on the triangulation." > > > 2D 3D Any oint inside a triangle can be exressed uniuely as a convex combination of the triangle vertices:" v 3 A 1 A 2 A 3 v 2 v (x 3,y 3,z 3 ) (x 2,y 2,z 2 ) (x 1,y 1,z 1 ) (x 5,y 5,z 5 ) (x 4,y 4,z 4 ) (x 1,y 1,z 1 ) 68.

2 Reeat until imossible:" " Select two sites." " If the edge connecting them does not intersect revious edges, keet." Construct the convex hull, and connect one arbitrary vertex to all others." Insert the other sites one after the other." Two ossibilities:" " Inside a triangle (one triangle becomes three)." " On an edge (two triangles become four)" The number of triangles in a triangulation of n oints deends on the number of vertices h on the convex hull." t = 2n-h-2 Let α(t) = (α 1, α 2,.., α 3t ) be the vector of angles in the triangulation T in increasing order." A triangulation T 1 will be better than T 2 if α(t 1 ) > α (T 2 ) lexicograhically." The Delaunay triangulation is the best." n = 8 h = 6 t = 8 h = 5 t = 9 good bad Let C be a circle, and l a line intersecting C at oints a and b. Let,, r and s be oints lying on the same side of l, where and are on C, r inside C and s outside C. Then:" In any convex uadrangle, an edge flis ossible. If this flimroves the triangulation locally, it also imroves the global triangulation." s a l r b If an edge flimroves the triangulation, the first edge is called illegal."

3 Lemma: An edge is illegal iff one of its oosite vertices is inside the circle defined by the other three vertices." roof: By Thalesʼ theorem." Reeat until imossible:" " Select a trile of sites." " If the circle through them is emty of other sites, kee the triangle whose vertices are the trile." Theorem: A Delaunay triangulation does not contain illegal edges." Corollary: A triangle is Delaunay iff the circle through its vertices is emty of other sites (the emty-circle condition). Corollary: The Delaunay triangulation is not uniue if more than three sites are co-circular Theorem: If a,b,c,d form a CCW convex olygon, then d lies in the circle determined by a, b and c iff: roof: We rove that euality holds if the oints are co-circular. There exists a center and radius r such that: Start with an arbitrary triangulation. Fli any illegal edge until no more exist." Reuires roof that there are no local minima." Could take a long time to terminate." and similarly for b, c, d: So these four vectors are linearly deendent, hence their det vanishes. Corollary: d circle(a,b,c) iff b circle(c,d,a) iff c circle(d,a,b) iff a circle(b,c,d) General osition assumtion: There are no four co-circular oints." Draw the dual to the Voronoi diagram by connecting each two neighboring sites in the Voronoi diagram." Corollary: The DT may be constructed in O(nlogn) time." It is easy to see that any resulting triangle is Delaunay, but can these triangles intersect?" Let S be a set of sites, and let DT(S) be the dual of the Voronoi diagram." Assume by contradiction that j intersects another edge in DT(S)." and j are neighbors, hence there exists an emty circle through them centered on their bisector." j demo

4 j k Case A:" " Imossible, because the circle is emty, hence also the triangle j o." Case B:" " Each yellow edge must intersect a white one two white edges also intersect, but these edges are in disjoint Voronoi cells. Contradiction." o l o 2 Case A Case B Theorem:" "Let S be a set of oints in the lane. Then," "(i), j, k S are vertices of a triangle (face) of DT(S)" " " "The circle assing through, j, k is emty;" "(ii), j (for, j S) is an edge of DT(S)" " " "There exists an emty circle assing " " "through, j." roof: Dualize the Voronoi-diagram theorem." Case C is ossible." Case C 198. Corollary:" "A triangulation T(S) is DT(S)" " " "Every circumscribing circle of " " " "a triangle Δ T(S) is emty." 28. Randomized incremental algorithm: " Form bounding triangle which encloses all the sites." Add the sites one after another in random order and udate triangulation." If the site is inside an existing triangle:" " Connect site to triangle vertices." " Check if a 'fli' can be erformed on one of the triangle edges. If so check recursively the neighboring edges." If the site is on an existing edge:" " Relace edge with four new edges." " Check if a 'fli' can be erformed on one of the oosite edges. If so check recursively the neighboring edges." A new vertex r is added, causing the creation of the edges r and j r." The legality of the edge j (with oosite vertex) k is checked." If j is illegal, erform a fli, and recursively check edges k and j k, the new edges oosite r." Notice that the recursive call for k cannot eliminate the edge r k." Note: All edge flis relace edges oosite the new vertex by edges incident to it!" k r j r j Theorem: The exected number of edges flis made in the course of the algorithm (some of which also disaear later) is at most 6n." roof:" "During insertion of vertex, k i new edges are created: 3 new initial edges, and k i -3 due to flis. " "Backward analysis: E[k i ] = the exected degree of after the insertion is comlete = 6 (Euler)." k

5 oint location for every oint: O(log n) time." Flis: Θ(n) exected time in total (for all stes)." Total exected time: O(n log n)." Sace: Θ(n)." Euclidean Minimum Sanning Tree (EMST): A tree of minimum length connecting all the sites." Relative Neighborhood Grah: Two sites, are connected iff! Gabriel Grah (GG): Two sites, are connected iff the circle whose diameter is is emty of other sites." Theorem: EMST RNG GG DT demo roject the 2D oint set onto the 3D araboloid Comute the 3D lower convex hull roject the 3D facets back to the lane. The intersection of a lane with the araboloid is an ellise whose rojection to the lane is a circle." s lies within the circumcircle of,, r iff sʼ lies on the lower side of the lane assing through ʼ, ʼ, rʼ.",, r S form a Delaunay triangle iff ʼ, ʼ, rʼ form a face of the convex hull of Sʼ." r s r s The 2D triangulation is Delaunay! Given a set S of oints in the lane, associate with each oint =(a,b) S the lane tangent to the araboloid at :" z = 2ax+2by-(a 2 +b 2 )." VD(S) is the rojection to the (x,y) lane of the 1-skeleton of the convex olyhedron formed from the intersection of the halfsaces above these lanes. " 298.