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A power diagram is a type of Voronoi diagram defined from a set of circles using the power distance; it can also be thought of as a weighted Voronoi diagram in which a weight defined from the radius of each circle is added to the squared Euclidean distance from the circle's center.
As Fortune describes in ref., [1] a modified version of the sweep line algorithm can be used to construct an additively weighted Voronoi diagram, in which the distance to each site is offset by the weight of the site; this may equivalently be viewed as a Voronoi diagram of a set of disks, centered at the sites with radius equal to the weight of the site. the algorithm is found to have ...
The Delaunay triangulation of a discrete point set P in general position corresponds to the dual graph of the Voronoi diagram for P. The circumcenters of Delaunay triangles are the vertices of the Voronoi diagram. In the 2D case, the Voronoi vertices are connected via edges, that can be derived from adjacency-relationships of the Delaunay ...
Finding the largest empty circle using the Voronoi diagram (two solutions). In computational geometry , the largest empty sphere problem is the problem of finding a hypersphere of largest radius in d -dimensional space whose interior does not overlap with any given obstacles.
They are the geometric duals of Voronoi diagrams. The Delaunay triangulation of a set of points P {\displaystyle {\mathcal {P}}} in the plane contains the Gabriel graph , the nearest neighbor graph and the minimal spanning tree of P {\displaystyle {\mathcal {P}}} .
Animation of Fortune's algorithm, a sweep line technique for constructing Voronoi diagrams. In computational geometry, a sweep line algorithm or plane sweep algorithm is an algorithmic paradigm that uses a conceptual sweep line or sweep surface to solve various problems in Euclidean space. It is one of the critical techniques in computational ...
In many applications, one needs to determine the location of several different points with respect to the same partition of the space. To solve this problem efficiently, it is useful to build a data structure that, given a query point, quickly determines which region contains the query point (e.g. Voronoi Diagram).
Let be the Voronoi diagram for a set of sites , and let be the Voronoi cell of corresponding to a site . If V p {\displaystyle V_{p}} is bounded, then its positive pole is the vertex of the boundary of V p {\displaystyle V_{p}} that has maximal distance to the point p {\displaystyle p} .