Voronoi Diagrams: How to Create Stunning Voronoi Diagrams

Voronoi Diagrams

How to Create Stunning Voronoi Diagrams

Voronoi diagrams are a fascinating and versatile concept used in various fields, such as mathematics, computer graphics, and geographical information systems. These diagrams partition a plane into regions based on the distance to a specific set of points, known as seeds or sites. Each region contains all points closest to one seed, creating a visually captivating tessellation. In this article, we will delve into the intricacies of Voronoi diagrams, explore their applications, and provide tips for creating your own stunning Voronoi diagrams.

Understanding Voronoi Diagrams

A Voronoi diagram divides space into several regions. Each region consists of all points closer to one particular seed point than to any other. This method of partitioning space has applications in many fields, such as meteorology, astronomy, ecology, and urban planning. Understanding the principles behind Voronoi diagrams can enhance your ability to create these beautiful and useful patterns.

The Mathematics Behind Voronoi Diagrams

Voronoi diagrams are constructed using a set of seed points. The process involves:

  1. Defining Seed Points: Select a set of points in a plane. These points are the seeds around which the Voronoi cells will form.
  2. Calculating Distances: For each point in the plane, calculate the distance to each seed point.
  3. Assigning Regions: Assign each point in the plane to the region of the closest seed point. This creates a partition where each region contains all the points closer to one seed than any other.

Mathematically, for a given set of seed points \( P = { p_1, p_2, \ldots, p_n } \), the Voronoi cell \( V(p_i) \) for a seed \( p_i \) is defined as:

\( V(p_i) = { x \in \mathbb{R}^2 \ | \ \forall j \neq i, \ d(x, p_i) < d(x, p_j) } \)

where \( d(x, p_i) \) denotes the Euclidean distance between point \( x \) and seed \( p_i \).

Applications of Voronoi Diagrams

Voronoi diagrams are widely used in various disciplines due to their unique properties. Here are some notable applications:

Geography and Urban Planning

In geography, Voronoi diagrams help understand the spatial distribution of features. For example, they model the area of influence of different cities or service centres. Urban planners use them to allocate resources and optimise the placement of facilities like hospitals and schools.

Computer Graphics and Gaming

Voronoi diagrams are used in computer graphics for texture generation and procedural content creation. They help in creating realistic patterns and terrains. In gaming, they assist in generating maps and managing territories.

Biology and Ecology

Biologists use Voronoi diagrams to study cellular structures and the spatial organisation of tissues. Ecologists apply them to analyse the territories of animals and the distribution of plant species.

Meteorology

Meteorologists use Voronoi diagrams to model weather patterns and predict the influence of different weather stations. They help in visualising areas affected by various meteorological phenomena.

Tips for Creating Stunning Voronoi Diagrams

Creating visually appealing Voronoi diagrams requires attention to detail and a good understanding of design principles. Here are some tips to enhance the aesthetic appeal of your diagrams:

Use Contrasting Colours

Select a colour scheme that provides good contrast between regions. This helps in distinguishing different areas clearly.

Incorporate Background Textures

Adding subtle background textures can make your Voronoi diagrams more visually interesting. Ensure the textures do not overpower the main diagram.

Experiment with Line Styles

Vary the line styles and thicknesses to add depth and hierarchy to your diagrams. Use dashed or dotted lines to differentiate between different types of boundaries.

Utilise Gradients

Applying gradients within the regions can add a sophisticated touch to your diagrams. Gradients can represent varying intensities or densities within each region.

Add Annotations

Label important regions or seed points to provide additional context. Annotations can include information such as coordinates, weights, or significance of the points.

Advanced Features

To create truly stunning Voronoi diagrams, explore advanced features such as:

Weighted Voronoi Diagrams

Assign weights to seed points to create diagrams where regions are influenced by the weight of the seeds. This adds another layer of complexity and usefulness.

3D Voronoi Diagrams

Extend the concept to three dimensions for more complex applications. 3D Voronoi diagrams can be used in fields like geology and material science.

Voronoi Diagrams on Different Surfaces

Apply Voronoi diagrams on spherical surfaces or other non-Euclidean spaces. This can be particularly useful in astronomy and global mapping.

What to do with Voronoi Diagrams

Voronoi diagrams are a powerful tool for partitioning space in a visually captivating manner. Whether used in scientific research, urban planning, computer graphics, or art, these diagrams offer endless possibilities for exploration and creativity.

Understanding Voronoi Diagrams

By understanding their mathematical foundations and exploring their applications, you can create stunning Voronoi diagrams for various purposes. Remember to experiment with different tools, customisations, and advanced features to refine your diagrams and make them truly unique.

Creating Voronoi Diagrams: A Step-by-Step Guide

Voronoi Diagrams-01

This diagram shows the locations of the three hospitals, $A$, $B$ and $C$, in a city.

When a car accident occurs, it is important to locate the nearest hospital.

How could we improve the diagram to make it easier to identify the closest hospital to any given location?

If it is decided that a new hospital should be built as close as possible which is furthest from all three existing hospitals, where should a new hospital be built?

To solve these issues, we can construct a Voronoi diagram to determine the best solutions.

Consider the diagram here, for each hospital, there is a region which contains all the points that are closer to that hospital than to any other hospital.

Notice each region contains one hospital.

For instance, the point \( K \) lies in the region containing hospital \( A \), so hospital \( A \) is the closest one to \( K \).

In this case, the point \( K \) lies in the region containing hospital \( B \), so hospital \( B \) is the closest one to \( K \).

Now, the point \( K \) lies in the region containing hospital \( C \), so hospital \( C \) is the closest one to \( K \).

Point \( K \) is equally close to hospital \( A \) and \( B \).

Point \( K \) is equally close to hospital \( A, B \) and \( C \).

Terminology

Significant points are called sites.

This diagram illustrates site \( A \), site \( B \) and site \( C \).

Each site is surrounded by a region or cell with points closer to that site than any other site.

This diagram illustrates region \(A\) or cell \(A\).

Construction

The line segments which separate the regions or cells are called edges.

The point(s) at which the edges meet are called vertex or vertices.

All points on the perpendicular bisector are equidistant from point \( A(-5,2) \) and point \( B(5,4) \).

The midpoint of \( AB \) is \( \displaystyle \left( \frac{-5+5}{2}, \frac{2+4}{2} \right) = \left( 0,3 \right) \)

The gradient of \( AB \) is \( \displaystyle \frac{4-2}{5+5} = \frac{1}{5} \).

Thus, the gradient of the perpendicular bisector is
\( \begin{align} \displaystyle \frac{1}{5} \times m &=-1 \\ \therefore m &=-5 \end{align} \).

So, the perpendicular bisector of \( AB \) has gradient \( -5 \) and passes through \( (0,3) \).

The equation of the perpendicular bisector of \( AB \) is,
\( \require{AMSsymbols} \begin{align} y-y_1 &= m(x-x_1) \\ y-3 &=-5(x-0) \\ \color{red}{\therefore y} &\color{red}{=-5x + 3} \end{align} \)

Similarly, the equation of the perpendicular bisector of \( BC \) is \( \displaystyle \color{blue}{y=-\frac{1}{2}x + \frac{3}{2}} \),
and the equation of the perpendicular bisector of \( AC \) is \( \require{AMSsymbols} \color{green}{y = x+1} \),

Remove all unnecessary lines but edges.

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