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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. [15] The Voronoi diagram of points in -dimensional space can ...
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.
Largest empty circle: Given a set of points, find a largest circle with its center inside of their convex hull and enclosing none of them. Euclidean shortest path: Connect two points in a Euclidean space (with polyhedral obstacles) by a shortest path. Polygon triangulation: Given a polygon, partition its interior into triangles; Mesh generation
Voronoi diagram; Extremal shapes ... 2-D case: Maximum empty circle (largest empty circle) Interaction/search. Collision detection; Line segment intersection;
The case when the sought rectangle is an axis-oriented square may be treated using Voronoi diagrams in metrics for the corresponding obstacle set, similarly to the largest empty circle problem. In particular, for the case of points within rectangle an optimal algorithm of time complexity Θ ( n log n ) {\displaystyle \Theta (n\log n)} is known.
Instead, each new point should be selected as the center of the largest empty circle defined by the previously-selected point set. [12] This center will always lie on a vertex of the Voronoi diagram of the already selected points, or at a point where an edge of the Voronoi diagram crosses the domain boundary.
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).
Largest empty rectangle; Largest empty sphere; ... Midpoint circle algorithm; ... Weighted Voronoi diagram; Wiener connector