# 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**.