Differential geometry by Cordero L.A. (ed.)

By Cordero L.A. (ed.)

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The Delaunay triangulation form a planar graph since no two Delaunay edges intersect in their interiors. It follows from the property of planar graphs that the number of Delaunay edges is at most 3n − 6 for a set of n points. The number of Delaunay triangles is at most 2n − 4. This means that the dual Voronoi diagram also has at most 3n − 6 Voronoi edges and 2n − 4 Voronoi vertices. The Voronoi diagram and the Delaunay triangulation of a set of n points in the plane can be computed in O(n log n) time and O(n) space.

This subset is homeomorphic to a space which is formed by taking the product of Sk−1 with the closed unit interval I in R and then collapsing one of its boundary to a single point, that is, the quotient space (Sk−1 × I )/(Sk−1 × {0}). This space is a k-ball which contradicts the fact that B ∩ is not a k-ball to begin with. Therefore, as B is continually shrunk, it becomes tangent to at a point y = m. Apply the previous argument to claim that B has a medial axis point. 8 illustrates the different cases of Feature Ball Lemma in R2 .

8(b). If pq is not correct, γ contains a sample point, say s, between p and q inside B. Again, we reach a contradiction as p − s < p − q . Next we show that edges connecting a sample point to its half neighbors are also correct. 9 (Half Neighbor). An edge pq where q is a half neighbor of p is correct when ε < 13 . → makes Proof. Let r be the nearest neighbor of p. According to the definition − pq − → π at least 2 angle with pr . If pq is not correct, consider the correct edge ps incident to p other than → → pr for pr .

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