Voronoi Diagrams

Understanding Voronoi Diagrams

• Voronoi Diagrams were named after Russian mathematician Georgy Voronoi.
• They are graphical representations of the division of a space into regions according to the distance from a given set of points.
• The lines drawn in these diagrams are solutions of the equation of equal distance to the two nearest points.
• Each region of a Voronoi diagram is associated with a specific seed point, and all locations within that region are closer to that seed point than to any other.

Structure of Voronoi Diagrams

• The several regions that make up a Voronoi Diagram are called Voronoi Cells. Each Voronoi cell is a polygon that encloses one seed point.
• The points forming a Voronoi diagram are called Voronoi vertices and the lines are called Voronoi edges.
• If extended to a 3D space, the diagram consists of Voronoi polyhedra instead of polygons.

Applications of Voronoi Diagrams

• Voronoi diagrams have various practical applications. They are widely used in science, technology, and even arts.
• They’ve been used in geography to determine areas of influence around cities and in astronomy to identify empty spaces between stars.
• In the domain of computer science, they help in solving problems related to data structures, clustering algorithms, and pathfinding in game development.
• In the telecommunications field, Voronoi diagrams are used to decide the location of mobile phone masts to provide ample service coverage.

Constructing Voronoi Diagrams

• While constructing Voronoi diagrams, start by defining a finite set of seed points.
• Draw the perpendicular bisectors of the lines joining every pair of points.
• The intersection of these bisectors forms the edges and vertices of the Voronoi polygons.
• Remember, each polygon contains only one seed point and every point inside that polygon is closer to its own seed point than any other.

Properties of Voronoi Diagrams

• Voronoi Cells that share an edge are called Neighbouring Voronoi Cells - they are the closest to each other.
• The intersection of two Voronoi edges - Voronoi Vertex, is the point that is equidistant from three or more seed points.
• Voronoi edges are equidistant from the two seed points that generated the two neighbouring Voronoi cells.

Studying Voronoi Diagrams

• Thoroughly understand how to construct Voronoi Diagrams and interpret their components.
• Practice drawing Voronoi Diagrams from given sets of points.
• Understand how the concept of Voronoi Diagrams can be applied in real-life situations, particularly in technology and physical sciences.
• Apply Voronoi Diagrams concept in solving complex mathematical problems that involve the location of points in a region.
• Ensure you can calculate distances within Voronoi Diagrams, determining which points are closest to others.
• Explore multidimensional concept of Voronoi Diagrams, transitioning from planar Voronoi diagrams to 3-Dimensional Voronoi Diagrams.