Voronoi diagrams and Delaunay triangulations are two fundamental concepts in computational geometry that have numerous applications in various fields such as computer graphics, pattern recognition, and geographic information systems.
In this article, we will explore how to derive a Voronoi diagram from a given point set by using the Delaunay triangulation. But before we dive into the details, let's understand what these two concepts mean.
A Voronoi diagram is a partitioning of a plane into regions based on a set of points. Each region contains all the points that are closer to a particular point than any other point in the set. This creates a series of polygons around each point, known as Voronoi cells. These cells can be used to perform various spatial analysis tasks, such as finding the nearest neighbor of a point or calculating the shortest path between two points.
On the other hand, a Delaunay triangulation is a way to connect a set of points to form a triangulated network, where each triangle is formed by connecting three non-collinear points, and no point lies inside the circumcircle of any triangle. Delaunay triangulations have the property that they minimize the maximum angle of all the triangles in the network, making them ideal for various applications, including finite element analysis and terrain modeling.
Now, let's see how we can combine these two concepts to derive a Voronoi diagram from a point set.
Step 1: Create the Delaunay Triangulation
To start with, we need a set of points. Let's take a simple example of five points in a 2D plane: A, B, C, D, and E. The first step is to create the Delaunay triangulation of these points. This can be done using various algorithms, such as the Bowyer-Watson algorithm or the Delaunay refinement algorithm. For simplicity, we will use the Bowyer-Watson algorithm, which is relatively easy to understand.
First, we create a supertriangle that encloses all the points. This supertriangle will be removed from the final triangulation. In our example, the supertriangle can be formed by connecting points A, B, and C. Next, we add each point in the set one by one and update the triangulation accordingly. This means that after adding point A, we will have a single triangle ABC. Then, we add point B, and the triangulation will be updated to include two new triangles: ABD and BCD. Similarly, we add points C, D, and E, and the triangulation will be updated accordingly.
Step 2: Identify Voronoi Edges
Once we have the Delaunay triangulation, we can start deriving the Voronoi diagram. To do this, we need to identify the Voronoi edges, which are the perpendicular bisectors of the Delaunay edges. In our example, the Voronoi edges will be the lines connecting the midpoints of AB, BC, CD, DE, and EA. These lines will form the boundaries of the Voronoi cells.
Step 3: Create Voronoi Cells
Next, we need to create the Voronoi cells by connecting the Voronoi edges. To do this, we start from any point in the set and connect it to the intersection points of the Voronoi edges surrounding it. For example, starting from point A, we connect it to the intersection points of AB and EA, and the Voronoi cell for point A is formed. Similarly, we create the Voronoi cells for points B, C, D, and E.
Step 4: Remove Outer Cells
In the final step, we remove the outer cells, which are the cells that extend to infinity. These cells can be identified by checking if they contain any points from the original set. In our example, the outer cells will be the cells around points A and C. Once we remove these cells, we are left with the desired Voronoi diagram.
Conclusion
In this article, we have seen how to derive a Voronoi diagram from a point set by using the Delaunay triangulation. This method is not only simple but also efficient, as it takes advantage of the properties of Delaunay triangulations to create accurate Voronoi cells. Voronoi diagrams and Delaunay triangulations are powerful tools that have numerous applications in various fields, and understanding how to derive one from the other can be valuable for anyone working with spatial data.
