Computational Geometry Questions Medium
The Voronoi diagram is a fundamental concept in computational geometry that is used to partition a given space into regions based on the proximity to a set of points. It is named after the Russian mathematician Georgy Voronoi, who introduced the concept in 1908.
In a Voronoi diagram, a set of points, also known as sites or generators, are given as input. The diagram then divides the space into regions, where each region consists of all points that are closer to a particular site than to any other site. These regions are called Voronoi cells or Voronoi polygons.
To construct a Voronoi diagram, several algorithms can be used. One common approach is the Fortune's algorithm, which has a time complexity of O(n log n), where n is the number of input points. This algorithm starts by sorting the points in a specific order and then iteratively builds the diagram by sweeping a line across the plane.
The Voronoi diagram has numerous applications in various fields, including computer graphics, computer vision, geographic information systems, and robotics. It can be used for proximity analysis, nearest neighbor search, spatial clustering, and spatial interpolation, among others.
In addition to its practical applications, the Voronoi diagram also has several interesting properties. For example, the edges of the Voronoi cells are equidistant between the two nearest sites, and the vertices of the diagram correspond to the circumcenters of the triangles formed by three neighboring sites.
Overall, the Voronoi diagram is a powerful tool in computational geometry that allows for efficient spatial analysis and provides valuable insights into the proximity relationships between points in a given space.