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Aug 16, 2022 β€’145 likes β€’1.46k views

CS133 Computational Geometry Voronoi Diagram Delaunay Triangulation 5/17/2018 1 Nearest Neighbor Problem Given a set of points and a query point , find the closest point to , ,

  1. Handle Circle Event ( 𝜹 ) Delete the leaf 𝛿 that corresponds to the disappearing arc 𝛽 from 𝜐 Add the center of the circle event as a vertex in VD. This center is one side of two half- edges Check for any new circle events caused by the now adjacent triples of arcs Running time: 𝑃 π‘œ log π‘œ 5/17/2018 47

  2. Delaunay Triangulation 5/17/2018 48

  3. Delaunay Triangulation A Delaunay triangulation can be defined as the (unique) triangulation in which the circumcircle of each triangle has no other sites NaΓ―ve algorithm: Consider all possible triangles 𝑃 π‘œ 3 Check if the circumcircle of the triangle is empty 𝑃 π‘œ Running time 𝑃 π‘œ 4 5/17/2018 49

  4. Guibas and Stolfi’s Algorithm A divide and conquer algorithm 5/17/2018 50

  5. Algorithm Outline DelaunayTriangulation(P) If (|P| <= 3) return TrivialDT(P) Split P into P1 and P2 DT1 = DelaunayTriangulation(P1) DT2 = DelaunayTriangulation(P1) Merge(DT1, DT2) 5/17/2018 51

  6. Split Pre-sort by x 5/17/2018 52

  7. TrivialDT(P) P TrivialDT(P) 5/17/2018 53

  8. Merge(P1, P2) 5/17/2018 54

  9. Merge(P1, P2) 5/17/2018 55

  10. Merge(P1, P2) 5/17/2018 56

  11. Merge(P1, P2) 5/17/2018 57

  12. Find the First LR edge Base LR edge Upper tangent of π’Ÿβ„‹ 𝑄 1 , π’Ÿβ„‹ 𝑄 2 5/17/2018 58

  13. Rising Bubble 5/17/2018 59

  14. Rising Bubble 5/17/2018 60

  15. Rising Bubble 5/17/2018 61

  16. Rising Bubble New Base LR edge 5/17/2018 62

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