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Exploring Curved Objects in Euclidean Space for Proximity Solutions

Discover the significance of curves in three-dimensional space and explore proximity queries through Voronoi diagrams and Delaunay graphs. Learn about finding nearest neighbors and quantifying object relations.

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Exploring Curved Objects in Euclidean Space for Proximity Solutions

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  1. 1.1 The motivation • Most of the objects we encounter in the real world are curved objects in the three dimensional Euclidean space. Curves are frequently encountered in maps. • Many problems from different fields involve proximity queries like finding the nearest neighbour, finding all the neighbours, quantifying or qualifying the neighbour-liness of two objects. • The Voronoi diagram (of semi-algebraic sets) is the solution to those proximity problems.

  2. 1.2 The Voronoi diagram

  3. CIRCLE CUBIC 1.4 The Delaunay graph

  4. 1.5 The Delaunay empty circle criterion

  5. 1.7 The problem

  6. (x,y) r R C4

  7. 6.2 An example

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