Metric-Driven Voronoi Diagrams: A Comprehensive Mathematical Framework

Abstract

Voronoi partitioning is a fundamental geometric concept with applications across computational geometry, robotics, optimization, and resource allocation. While Euclidean distance is the most commonly used metric, alternative distance functions can significantly influence the shape and properties of Voronoi cells. This paper presents a comprehensive mathematical analysis of various distance metrics used in Voronoi partitioning, including Euclidean, Manhattan, Minkowski, weighted, anisotropic, and geodesic metrics. We analyze their mathematical formulations, geometric properties, topological implications, and computational complexity. This work aims to provide a theoretical framework for selecting appropriate metrics for Voronoi-based modeling in diverse applications.

1. Introduction

Voronoi partitioning is a fundamental concept in computational geometry, topology, and applied mathematics, with deep connections to diverse fields such as physics, robotics, geophysics, image processing, epidemiology, and network optimization [1,2,3,4,5,6]. At its core, a Voronoi diagram divides a given space into distinct regions based on proximity to a finite set of generator points, commonly referred to as “sites.” For each site, the corresponding region consists of all points in the domain that are closer to that site than to any other. This construct, originally introduced by Georgy Voronoi in 1908, provides a mathematically elegant method for spatial decomposition and is foundational in both theoretical and applied studies of spatial structures.

While the classical Voronoi diagram is built upon the Euclidean distance metric, the generality of the Voronoi construction allows for the incorporation of alternative distance functions, thereby yielding what are known as generalized Voronoi diagrams. The choice of metric profoundly affects the geometry, topology, and combinatorial structure of the resulting Voronoi cells. In particular, metrics such as the Manhattan ($L_1$), Chebyshev ($L_{\infty }$), Minkowski ($L_p$), weighted, anisotropic, and geodesic distances yield non-Euclidean Voronoi diagrams that exhibit a rich diversity of properties. These metrics offer powerful modeling tools for applications where Euclidean assumptions are inadequate—for example, in urban environments constrained by rectilinear movement, on curved manifolds such as planetary surfaces, or in anisotropic media like crystals or directional sensors. Mathematically, the study of Voronoi diagrams under various metrics requires a rigorous understanding of metric space theory, convex analysis, and differential geometry. From a geometric standpoint, changing the metric alters the shape of the bisectors, which in turn affects the boundaries and connectivity of the Voronoi cells. From a computational perspective, the complexity of constructing Voronoi diagrams and analyzing their stability is closely tied to the algebraic and topological properties of the chosen metric. For instance, while Euclidean Voronoi diagrams in ${\mathbb{R}}^2$ can be constructed in $O(nlogn)$ time, non-Euclidean versions often necessitate specialized algorithms or approximation schemes.

The aim of this paper is to provide a comprehensive and mathematically rigorous analysis of the impact of different distance metrics on Voronoi partitioning. We begin by formalizing the definitions of metrics and Voronoi cells in arbitrary metric spaces. We then systematically examine the most prevalent families of distance functions—highlighting their mathematical structure, geometric implications, and topological distinctions. In addition to presenting analytical comparisons, we include illustrative diagrams and algorithmic sketches to highlight key features and computational considerations. This unified mathematical treatment seeks to fill a critical gap in the literature by providing clarity on how metric selection influences the qualitative and quantitative nature of Voronoi diagrams.

Ultimately, this paper is intended as a foundational reference for mathematicians, computational scientists, and engineers interested in employing Voronoi structures under non-standard metrics for modeling, analysis, and algorithm development. The theoretical insights presented here can also serve as a springboard for further research into adaptive, hybrid, and higher-dimensional Voronoi constructions.

2. Preliminaries

In this section, we provide the necessary background and definitions to set the stage for the subsequent analysis of Voronoi partitioning under various metrics. We begin with a review of basic metric space theory, followed by the formal construction of Voronoi diagrams. Additionally, we introduce the various distance metrics commonly used in Voronoi partitioning and highlight their respective mathematical properties.

2.1. Metric Spaces

A metric space is a pair $(X,d)$, where X is a set and $d:X\times X\to {\mathbb{R}}_{\ge 0}$ is a distance function satisfying the following axioms:

• Non-negativity: $d(x,y)\ge 0$ for all $x,y\in X$;
• Identity of indiscernibles: $d(x,y)=0\iff x=y$;
• Symmetry: $d(x,y)=d(y,x)$;
• Triangle inequality: $d(x,z)\le d(x,y)+d(y,z)$ for all $x,y,z\in X$.

The concept of a metric space generalizes the notion of distance in Euclidean space to more abstract settings, such as spaces with non-Euclidean geometries or spaces with additional structure (e.g., weighted metrics or anisotropic spaces) [7,8,9,10,11]. A critical component of Voronoi partitioning is the use of various distance functions on these spaces.

2.2. Voronoi Partitioning

Given a set of sites $\mathcal{P}=\{p_1,p_2,\dots ,p_n\}\subseteq X$, the Voronoi partition of X is a decomposition of the space into regions based on proximity to these sites. For each site $p_i\in \mathcal{P}$, its corresponding Voronoi cell is defined as follows:

$$V\left(p_i\right)=\{x\in X\mid d(x,p_i)\le d(x,p_j)\forall j\ne i\}.$$

(1)

The set of all Voronoi cells forms the Voronoi diagram $V\left(\mathcal{P}\right)$, which partitions the space X into n distinct regions, each containing the set of points closest to a particular site $p_i$. This partitioning structure is highly sensitive to the choice of the distance function d.

2.3. Common Distance Metrics in Voronoi Partitioning

The choice of distance function plays a crucial role in the shape and properties of the Voronoi diagram. Below, we describe several common distance metrics used in the construction of Voronoi diagrams.

2.3.1. Euclidean Metric ($L_2$ Norm)

The Euclidean distance is the most commonly used metric and is defined as follows [11,12]:

$$d_E(x,y)=\sqrt{\sum _{i=1}^n{(x_i-y_i)}^2}$$

(2)

where $x=(x_1,x_2,\dots ,x_n)$ and $y=(y_1,y_2,\dots ,y_n)$ are two points in ${\mathbb{R}}^n$. The Voronoi diagram with this metric is often referred to as the Euclidean Voronoi diagram, where the bisectors of the Voronoi cells are hyperplanes equidistant from the sites.

2.3.2. Manhattan Metric ($L_1$ Norm)

The Manhattan distance (or $L_1$ distance) is defined as follows [13,14]:

$$d_M(x,y)=\sum _{i=1}^n|x_i-y_i|$$

(3)

In this case, the Voronoi cells are diamond-shaped in two dimensions and are aligned along the coordinate axes. The Manhattan distance is particularly useful in grid-based environments, such as city layouts or digital image processing.

2.3.3. Chebyshev Metric ($L_{\infty }$ Norm)

The Chebyshev distance (or $L_{\infty }$ distance) is defined as follows [15,16]:

$$d_C(x,y)=\underset{i}{max}|x_i-y_i|$$

(4)

This metric is used when movement is allowed only along the axes in the case of chessboard-like environments. In this case, the Voronoi cells are squares, and the boundaries of the cells align with the coordinate axes.

2.3.4. Minkowski Distance

The Minkowski distance generalizes the Euclidean, Manhattan, and Chebyshev distances and is defined as follows [17,18]:

$$d_p(x,y)={\left(\sum _{i=1}^n{|x_i-y_i|}^p\right)}^{1/p}$$

(5)

where $p\ge 1$. For $p=2$, it reduces to the Euclidean distance, for $p=1$, it reduces to the Manhattan distance, and for $p\to \infty$, it approaches the Chebyshev distance. The Minkowski distance allows for a continuous interpolation between these three common metrics, providing flexibility in modeling various types of spatial relationships.

2.3.5. Geodesic Distance

In cases where the underlying space is curved (such as on a sphere or other manifold), the geodesic distance $d_G(x,y)$ is used [19,20]. The geodesic distance between two points is the length of the shortest path (or geodesic) between them, and it is particularly important in geographical applications. For instance, on the surface of the Earth, the geodesic distance corresponds to the shortest path along the surface of a sphere, often computed using great-circle distances.

2.4. Voronoi Diagram Construction

The computational complexity of constructing Voronoi diagrams depends on both the metric and the dimensionality of the space. In the Euclidean case, efficient algorithms, such as Fortune’s algorithm, allow for the construction of Voronoi diagrams in $O(nlogn)$ time. However, for non-Euclidean metrics, particularly those based on geodesic or anisotropic distances, specialized methods are often required. In some cases, approximations or randomized algorithms may be employed to handle the increased complexity.

A typical Voronoi diagram construction process involves the following:

• Identifying the sites: The set of generator points $\mathcal{P}=\{p_1,p_2,\dots ,p_n\}$ is given.
• Calculating the Voronoi cells: For each site $p_i$, the Voronoi cell $V\left(p_i\right)$ is determined by finding all points x that are closer to $p_i$ than to any other site.
• Constructing the diagram: The boundaries between the cells (the bisectors) are computed, and the Voronoi diagram is formed by the intersection of these boundaries.

In some cases, Voronoi diagrams can be generalized to higher dimensions, weighted sites, or even dynamic environments, where sites can move or change over time.

2.5. Topological Properties of Voronoi Diagrams

The topological properties of a Voronoi diagram depend on the underlying metric and the arrangement of the sites. In general, a Voronoi diagram divides the space into convex cells, and each cell has the following properties:

• Each cell is a convex set.
• The boundaries of the cells are defined by the bisectors of the sites.
• The number of vertices, edges, and faces in a Voronoi diagram depends on the number of sites and the dimensionality of the space.

These properties can vary significantly when the distance metric deviates from the Euclidean case. For example, in the Manhattan metric, the Voronoi cells in two dimensions are diamonds, whereas in the Chebyshev metric, they are squares. Such variations have implications for the combinatorial structure of the diagram and the algorithms used to construct and analyze it.

3. Euclidean Metric

The Euclidean distance is one of the most fundamental and widely used metrics in geometry and is defined as the straight-line distance between two points in Euclidean space [21,22,23,24,25]. This metric plays a pivotal role in Voronoi partitioning, particularly in situations where the geometry of the problem is flat and isotropic. In this section, we explore the formal definition of the Euclidean metric, its geometric interpretation, and the properties of the Voronoi diagram derived using this metric.

3.1. Definition of Euclidean Distance

Let $x=(x_1,x_2,\dots ,x_n)$ and $y=(y_1,y_2,\dots ,y_n)$ be two points in ${\mathbb{R}}^n$. The Euclidean distance $d_E(x,y)$ between these two points is given by the following formula:

$$d_E(x,y)=\sqrt{\sum _{i=1}^n{(x_i-y_i)}^2}$$

(6)

This distance function satisfies the four axioms of a metric space:

• Non-negativity: $d_E(x,y)\ge 0$ for all $x,y\in {\mathbb{R}}^n$, and equality holds if and only if $x=y$;
• Symmetry: $d_E(x,y)=d_E(y,x)$ for all $x,y\in {\mathbb{R}}^n$;
• Triangle inequality: $d_E(x,z)\le d_E(x,y)+d_E(y,z)$ for all $x,y,z\in {\mathbb{R}}^n$;
• Identity of indiscernibles: $d_E(x,y)=0\iff x=y$.

The Euclidean metric is often referred to as the $L_2$ norm, and it is the most familiar distance function for many practical applications, particularly in problems involving physical space.

3.2. Geometric Interpretation

In two-dimensional space (${\mathbb{R}}^2$), the Euclidean distance between two points corresponds to the length of the straight line segment connecting them. The Euclidean metric induces a circular symmetry, where the set of points that are equidistant from a given point form a circle. In three-dimensional space (${\mathbb{R}}^3$), this set forms a sphere. More generally, in ${\mathbb{R}}^n$, the points that are equidistant from a given point form an $(n-1)$-dimensional sphere.

This geometric property is central to the Voronoi diagram, as the bisectors between two sites are defined by the locus of points equidistant from the two sites, which are straight lines in ${\mathbb{R}}^2$ (and hyperplanes in higher dimensions).

3.3. Voronoi Diagram in Euclidean Space

Given a set of sites $\mathcal{P}=\{p_1,p_2,\dots ,p_n\}$ in ${\mathbb{R}}^n$, the Euclidean Voronoi diagram partitions the space into regions based on proximity to these sites. The Voronoi cell associated with each site $p_i\in \mathcal{P}$ is defined as follows:

$$V\left(p_i\right)=\{x\in {\mathbb{R}}^n\mid d_E(x,p_i)\le d_E(x,p_j)\forall j\ne i\}$$

(7)

This means that the Voronoi cell of a site consists of all points that are closer to that site than to any other. The boundaries of the Voronoi cells, or the Voronoi diagram, are determined by the bisectors of the Euclidean distance between pairs of sites. These bisectors are straight lines (or hyperplanes in higher dimensions) that divide the space into regions.

3.4. Properties of the Euclidean Voronoi Diagram

The Euclidean Voronoi diagram has several well-known properties that are useful for both theoretical analysis and computational applications:

• Convexity of Voronoi Cells: Each Voronoi cell is a convex set, meaning that for any two points within a cell, the line segment connecting them lies entirely within the cell.
• Edge Connectivity: The edges of the Voronoi cells are portions of hyperplanes that divide the space into distinct regions. In ${\mathbb{R}}^2$, these edges are straight lines, and in ${\mathbb{R}}^3$, they are portions of planes.
• Duality with Delaunay Triangulation: The Voronoi diagram is geometrically dual to the Delaunay triangulation; that is, the vertices of the Delaunay triangulation correspond to the sites of the Voronoi diagram, and the edges of the Delaunay triangulation are the perpendicular bisectors of the edges of the Voronoi diagram.
• Voronoi Cell Size: The size and shape of each Voronoi cell depend on the relative positions of the sites. If the sites are evenly spaced, the Voronoi cells are roughly equal in size and shape, typically taking the form of regular polygons in ${\mathbb{R}}^2$ or polyhedra in ${\mathbb{R}}^3$.

3.5. Computational Complexity

The construction of a Euclidean Voronoi diagram can be efficiently accomplished in $O(nlogn)$ time using algorithms such as Fortune’s algorithm [26]. This algorithm constructs the diagram by sweeping a horizontal line across the plane and maintaining a set of parabolic arcs that represent the boundaries of the Voronoi cells.

In addition to the computational efficiency, the Euclidean Voronoi diagram has many applications in computational geometry, robotics, geographic information systems (GISs), and other fields that involve spatial analysis.

3.6. Illustrative Example in Two Dimensions

To better understand the structure of the Euclidean Voronoi diagram, consider a set of three sites $p_1,p_2,p_3$ in ${\mathbb{R}}^2$, as illustrated in the following diagram (Figure 1).

In this diagram, the Voronoi cell of each site is bounded by straight lines, and the edges of these cells represent the bisectors of the Euclidean distance between pairs of sites. These edges divide the plane into regions such that every point in a given region is closer to the corresponding site than to any other.

3.7. Generalization to Higher Dimensions

The concept of the Euclidean Voronoi diagram can be generalized to higher-dimensional spaces. In ${\mathbb{R}}^d$, the Voronoi cells are convex polytopes, and the bisectors between two sites are hyperplanes of dimension $d-1$. While the basic principles remain the same, the construction and visualization of Voronoi diagrams become more complex as the number of dimensions increases. Nonetheless, the geometric properties and computational methods for constructing Voronoi diagrams in higher dimensions are extensions of the two-dimensional case.

3.8. Applications of Euclidean Voronoi Diagrams

Euclidean Voronoi diagrams are widely used in various fields due to their simple yet powerful mathematical structure. Some notable applications include the following:

• Geographical Mapping and GISs: Voronoi diagrams are used to analyze the proximity of various geographical features, such as the location of hospitals, fire stations, or cell towers.
• Robotics and Path Planning: In robot motion planning, Voronoi diagrams help in path optimization by dividing the environment into regions based on proximity to obstacles or goals.
• Material Science: Voronoi diagrams are used to model the microstructure of materials, where sites represent the centers of grain boundaries and the cells correspond to the individual grains.
• Data Mining and Clustering: Voronoi diagrams are used in data clustering techniques, where the sites represent the centroids of clusters and the diagram helps in partitioning the data space.

3.9. Summary

In this section, we have explored the Euclidean distance and its role in Voronoi partitioning. We have discussed the formal definition of the Euclidean metric, its geometric interpretation, and the key properties of the Voronoi diagram constructed using this metric. The Euclidean Voronoi diagram serves as the foundation for many real-world applications, providing valuable insights into spatial structures. The next sections of the paper will extend this analysis to other distance metrics, examining how variations in the distance function impact the resulting Voronoi cells and their properties.

4. Manhattan (Taxicab) Metric

The Manhattan, or Taxicab, metric is an alternative to the Euclidean distance metric, particularly useful in urban planning, grid-like structures, and other settings where movement is restricted to orthogonal directions [27,28,29,30,31]. This metric, also known as the $L_1$ norm, calculates the distance between two points by summing the absolute differences in their coordinates. In this section, we define the Manhattan metric, explore its geometric interpretation, and discuss its implications in Voronoi partitioning.

4.1. Definition of the Manhattan Distance

Let $x=(x_1,x_2,\dots ,x_n)$ and $y=(y_1,y_2,\dots ,y_n)$ be two points in ${\mathbb{R}}^n$. The Manhattan distance $d_M(x,y)$ between these two points is defined as follows:

$$d_M(x,y)=\sum _{i=1}^n|x_i-y_i|$$

(8)

In contrast to the Euclidean distance, which measures the straight-line distance between two points, the Manhattan distance measures the total “city block” distance by summing the absolute differences in each coordinate. This metric reflects the restricted movement in a grid-like space where only horizontal and vertical steps are allowed.

The Manhattan distance also satisfies the four axioms of a metric space:

• Non-negativity: $d_M(x,y)\ge 0$ for all $x,y\in {\mathbb{R}}^n$, and equality holds if and only if $x=y$;
• Symmetry: $d_M(x,y)=d_M(y,x)$ for all $x,y\in {\mathbb{R}}^n$;
• Triangle inequality: $d_M(x,z)\le d_M(x,y)+d_M(y,z)$ for all $x,y,z\in {\mathbb{R}}^n$;
• Identity of indiscernibles: $d_M(x,y)=0\iff x=y$.

4.2. Geometric Interpretation

In ${\mathbb{R}}^2$, the Manhattan distance corresponds to the sum of the absolute horizontal and vertical distances between two points. Geometrically, this creates a diamond-shaped region of constant distance rather than the circular region seen with the Euclidean distance. In higher dimensions, the set of points equidistant from a given point forms a hyperoctahedron.

Consider two points $p_1=(x_1,y_1)$ and $p_2=(x_2,y_2)$ in ${\mathbb{R}}^2$. The Manhattan distance between them is as follows:

$$d_M(p_1,p_2)=|x_1-x_2|+|y_1-y_2|$$

(9)

The geometric interpretation of this is that one can only move horizontally or vertically between $p_1$ and $p_2$, without taking diagonal steps. The boundary of a fixed Manhattan distance forms a diamond shape, as illustrated in the following diagram (Figure 2).

The generalization of the Manhattan distance to higher dimensions similarly results in hyperoctahedral shapes, where the distance is calculated as the sum of the absolute differences along each axis.

4.3. Voronoi Diagram in Manhattan Space

Given a set of sites $\mathcal{P}=\{p_1,p_2,\dots ,p_n\}$ in ${\mathbb{R}}^n$, the Manhattan Voronoi diagram divides the space into regions based on the proximity of points to these sites, using the Manhattan distance. The Voronoi cell associated with each site $p_i\in \mathcal{P}$ is defined as follows:

$$V\left(p_i\right)=\{x\in {\mathbb{R}}^n\mid d_M(x,p_i)\le d_M(x,p_j)\forall j\ne i\}$$

(10)

This means that the Voronoi cell of a site contains all points that are closer to that site, according to the Manhattan distance, than to any other site. The boundaries of these cells are determined by the locus of points that are equidistant to two sites, creating straight lines or hyperplanes with axis-aligned segments.

4.4. Properties of the Manhattan Voronoi Diagram

The Voronoi diagram constructed with the Manhattan metric exhibits several distinctive properties compared to the Euclidean case:

• Convexity of Voronoi Cells: Just like the Euclidean case, the Voronoi cells in the Manhattan metric are convex sets. However, the cells are polygonal in nature, typically taking the form of rectangular or square shapes, depending on the dimensionality.
• Edge Connectivity: In the case of the Manhattan metric, the edges of the Voronoi cells are aligned with the coordinate axes. In ${\mathbb{R}}^2$, the edges of the cells are either horizontal or vertical lines.
• Rectilinear Bisectors: The bisectors between two sites are rectilinear, consisting of horizontal and vertical line segments or, more generally, axis-aligned hyperplanes in higher dimensions.
• Symmetry and Grid-like Structure: Since the Manhattan distance inherently enforces a grid-like structure, the resulting Voronoi diagram has a regular, lattice-like pattern that is especially useful in applications like urban planning and robotics.

4.5. Computational Complexity

The construction of a Voronoi diagram using the Manhattan metric can be computed efficiently using algorithms analogous to those used for the Euclidean case, but with different geometric considerations. In two-dimensional space, the complexity is still $O(nlogn)$ in terms of the number of sites, but the geometry of the diagram ensures that the edges and cells will have distinct axis-aligned properties compared to Euclidean diagrams.

For higher dimensions, the computational methods extend naturally, but the increased complexity of managing axis-aligned hyperplanes can add to the computational load in certain applications.

4.6. Illustrative Example in Two Dimensions

Consider a set of three points $p_1,p_2,p_3$ in ${\mathbb{R}}^2$. The Manhattan Voronoi diagram for these points is illustrated below in Figure 3:

In this diagram, the boundaries of the Voronoi cells are straight lines, either vertical or horizontal, reflecting the grid-like nature of the Manhattan distance. The edges of the cells are formed by the loci of points equidistant to pairs of sites.

4.7. Generalization to Higher Dimensions

Just as in the Euclidean case, the Manhattan metric can be generalized to higher-dimensional spaces. In ${\mathbb{R}}^d$, the Voronoi cells formed by the Manhattan distance will take the shape of orthotopes (generalizations of rectangles in higher dimensions). The bisectors between two sites are hyperplanes that are aligned with the coordinate axes, maintaining the grid-like structure that defines the Manhattan metric.

4.8. Applications of the Manhattan Voronoi Diagram

The Manhattan Voronoi diagram has various practical applications, especially in fields where grid-like or rectilinear structures are prevalent. Some notable applications include the following:

• Urban Planning and GISs: The Manhattan distance is often used in city grid systems to model the distance between buildings, roads, and infrastructure elements.
• Robotics and Motion Planning: In grid-based robotics, where movement is constrained to orthogonal directions, the Manhattan metric is used for motion planning and obstacle avoidance.
• Telecommunications: The Manhattan metric is used in the design of networks, such as cell tower placement, where connections between points follow a rectilinear pattern.
• Clustering and Data Mining: In data clustering, the Manhattan distance is sometimes preferred when the data structure is inherently grid-like or the features are measured along orthogonal axes.

4.9. Summary

In this section, we have examined the Manhattan metric and its application to Voronoi partitioning. The Manhattan distance, which sums the absolute differences between the coordinates, produces Voronoi cells with a grid-like structure, characterized by axis-aligned boundaries. The Voronoi diagram formed by this metric has distinct geometric and computational properties that make it suitable for various practical applications, particularly in fields such as urban planning, robotics, and telecommunications. In the next sections, we will explore other distance metrics and their effects on Voronoi diagrams.

5. Minkowski Metric

The Minkowski metric is a generalization of the Euclidean and Manhattan distances, offering a flexible means to define distances between points in ${\mathbb{R}}^n$ through the $L_p$ norm [32,33,34,35,36]. The Minkowski distance can represent a wide range of distance measures by adjusting the parameter p, which controls the emphasis on different directions in space. This metric is widely used in various areas of computational geometry, optimization, and machine learning, particularly for its versatility in modeling different types of distance relationships.

In this section, we define the Minkowski distance, analyze its geometric properties, and explore its role in Voronoi partitioning, providing insights into how the parameter p influences the geometry and computational characteristics of the resulting Voronoi diagrams.

5.1. Definition of the Minkowski Distance

Let $x=(x_1,x_2,\dots ,x_n)$ and $y=(y_1,y_2,\dots ,y_n)$ be two points in ${\mathbb{R}}^n$, and let $p\ge 1$ be a real number. The Minkowski distance $d_M(x,y)$ between these two points is defined as follows:

$$d_M(x,y)={\left(\sum _{i=1}^n{|x_i-y_i|}^p\right)}^{1/p}$$

(11)

This formula generalizes both the Euclidean distance and the Manhattan distance:

• When $p=2$, the Minkowski distance becomes the Euclidean distance:

  $$d_2(x,y)=\sqrt{\sum _{i=1}^n{(x_i-y_i)}^2}$$

  (12)
• When $p=1$, the Minkowski distance becomes the Manhattan distance:

  $$d_1(x,y)=\sum _{i=1}^n|x_i-y_i|$$

  (13)

The parameter p allows us to control the sensitivity of the distance measure to differences in individual coordinates. As p increases, the distance metric increasingly emphasizes larger deviations in any coordinate, making the metric more sensitive to “outliers” in the data.

5.2. Geometric Interpretation

The Minkowski distance $d_M(x,y)$ defines a set of points that are at a constant distance from a given point $p_0\in {\mathbb{R}}^n$. The shape of this set depends on the value of p (Figure 4):

• For $p=2$, the set of points at a fixed distance from $p_0$ forms a sphere (or circle in two dimensions) in Euclidean space.
• For $p=1$, the set of points at a fixed distance from $p_0$ forms a diamond (or octahedron in higher dimensions), characteristic of the Manhattan distance.
• For general p, the set of points forms a shape that smoothly transitions between a sphere (for $p=2$) and a diamond (for $p=1$), with the shape becoming more “squared off” as p increases.

In two dimensions, for example, when $p=2$, the boundary of the set of points at distance d from a point $p_0$ forms a circle. As p approaches 1, the boundary becomes a square, and for larger values of p, the shape becomes more angular with more sharply defined corners. This illustrates how the parameter p controls the “roundness” of the distance measure.

5.3. Voronoi Diagram in Minkowski Space

Given a set of sites $\mathcal{P}=\{p_1,p_2,\dots ,p_n\}$ in ${\mathbb{R}}^n$, the Voronoi diagram for the Minkowski distance partitions the space based on proximity to these sites, where proximity is determined by the Minkowski distance. The Voronoi cell $V\left(p_i\right)$ associated with each site $p_i$ is defined as follows:

$$V\left(p_i\right)=\{x\in {\mathbb{R}}^n\mid d_M(x,p_i)\le d_M(x,p_j)\forall j\ne i\}$$

(14)

In the case of the Minkowski metric, the Voronoi cells are bounded by loci of points that are equidistant from pairs of sites. The geometric shape of the Voronoi cells depends on the value of p. For $p=2$, the Voronoi diagram has cells that are convex, with boundaries formed by arcs of circles. For $p=1$, the cells are polygonal, formed by axis-aligned segments.

As the parameter p changes, the Voronoi cells transition from being circular in shape (for $p=2$) to square-like for smaller values of p, and to more angular and grid-like structures for higher values of p.

5.4. Properties of the Minkowski Voronoi Diagram

The Voronoi diagram constructed using the Minkowski distance possesses properties similar to those of the Euclidean and Manhattan Voronoi diagrams but with key differences in the geometric structure:

• Convexity: Like the Euclidean and Manhattan diagrams, the Voronoi cells formed using the Minkowski distance are convex sets. However, their shape varies with p, becoming more polygonal as p decreases.
• Shape of Voronoi Cells: In $L_p$ spaces, the unit ball has a characteristic shape determined by p (e.g., circular for $p=2$, diamond for $p=1$), which reflects the local geometry of the metric. However, Voronoi cells in these spaces are polygonal or polyhedral regions defined by metric bisectors between sites, and their shapes depend on both the underlying metric and the arrangement of sites.
• Edge Types: The edges of the Voronoi cells are determined by the geometry of the Minkowski distance. For $p=2$, the edges are arcs of circles. For $p=1$, the edges are straight lines. For intermediate values of p, the edges become increasingly jagged.
• Computational Complexity: Similar to the Manhattan and Euclidean cases, the complexity of constructing a Voronoi diagram with the Minkowski metric in ${\mathbb{R}}^2$ is $O(nlogn)$, where n is the number of sites. Higher dimensions lead to more complex geometric structures, but the overall complexity remains $O(nlogn)$.

5.5. Applications of the Minkowski Voronoi Diagram

The flexibility of the Minkowski distance makes it a powerful tool in various fields. Some notable applications include the following:

• Pattern Recognition: The Minkowski distance is useful in machine learning and pattern recognition, where different values of p can be chosen to model different types of data distributions.
• Robotics and Motion Planning: The Minkowski metric allows for flexible motion planning strategies in environments where movement is constrained by non-Euclidean geometry, such as grid-based or city-like environments.
• Computational Geometry: Minkowski Voronoi diagrams are used in computational geometry to analyze spatial distributions and perform tasks like clustering and nearest-neighbor searches.
• Computer Vision: The Minkowski metric can be used in computer vision applications, particularly in object detection and classification, where the geometry of the data requires the use of flexible distance metrics.

5.6. Summary

In this section, we have examined the Minkowski distance as a generalization of both the Euclidean and Manhattan distances. The Minkowski metric offers flexibility in modeling distance relationships through the parameter p, allowing for a wide range of geometries. The corresponding Voronoi diagrams exhibit distinct properties depending on the value of p, making the Minkowski metric highly adaptable for applications in computational geometry, robotics, pattern recognition, and more. In the next section, we will explore other distance metrics and their influence on Voronoi partitioning.

6. Weighted Voronoi Diagrams

In a traditional Voronoi diagram, each site is treated as an equally significant point. However, in many real-world applications, some sites may have more influence or importance than others. Weighted Voronoi diagrams extend the classical concept by assigning a weight to each site, influencing the size and shape of the corresponding Voronoi cells [37,38,39,40,41]. This extension allows for a more generalized and nuanced approach to partitioning space, which is particularly useful in fields such as physics, geography, and computational biology.

6.1. Definition of Weighted Voronoi Diagram

Given a set of n sites $\mathcal{P}=\{p_1,p_2,\dots ,p_n\}$ in ${\mathbb{R}}^n$, each site $p_i$ is associated with a weight $w_i\in {\mathbb{R}}_{\ge 0}$. The weight $w_i$ represents the “influence” or “size” of the site, and it modifies the distance metric used in the Voronoi partitioning process. The weighted Voronoi diagram partitions space based on the distance to the weighted sites rather than the sites themselves.

The weighted Voronoi cell $V\left(p_i\right)$ for site $p_i$ is defined as the set of points $x\in {\mathbb{R}}^n$ that are closer to $p_i$ than to any other site $p_j$, where the distance is modified by the weight $w_i$.

This distance is often defined as follows:

$$d(x,p,w)=\parallel x-p\parallel +w.$$

(15)

The specific form of the weighted distance can vary depending on the application and the choice of metric. For instance, in the Euclidean case, the weighted distance becomes

$$d(x,p,w)=\sqrt{\sum _{i=1}^n{(x_i-p_i)}^2}+w$$

(16)

Here, the weight w shifts the site, effectively increasing its influence on the partitioning process.

6.2. Geometric Interpretation

In a weighted Voronoi diagram, the influence of each site depends on its weight. Larger weights result in smaller Voronoi cells, while smaller weights correspond to larger cells. Geometrically, the Voronoi cells are no longer necessarily convex polygons with uniform shapes, but their boundaries are adjusted by the relative weights of the sites.

• For equal weights, the weighted Voronoi diagram reduces to the classical Voronoi diagram, where all sites are treated equally.
• For differing weights, the Voronoi cells become asymmetrical, with sites of higher weight having smaller and more dominant cells, while sites with lower weights have larger and less influential cells.
• The boundaries between the cells are no longer defined solely by the Euclidean distance but are instead adjusted by the weighted distances, causing the cells to shift and expand based on the relative magnitudes of the weights.

The Voronoi cells in the weighted case are still convex, but the specific shape of each cell depends on the interplay between the sites’ positions and their respective weights.

6.3. Applications of Weighted Voronoi Diagrams

Weighted Voronoi diagrams are used in various applications where some sites have more influence or importance than others. Some prominent applications include the following:

• Facility Location and Design: In operations research and urban planning, weighted Voronoi diagrams can model the placement of facilities (such as hospitals, schools, or warehouses), where the demand or capacity of each facility is represented by its weight. The facility’s capacity can affect the size and shape of the area it serves.
• Robotics and Motion Planning: In robotics, weighted Voronoi diagrams are used for path planning, where obstacles or goals have different levels of importance. For instance, regions with higher weights may correspond to goals or higher-priority areas that need to be reached with greater urgency.
• Geographical and Environmental Studies: In geographical sciences, weighted Voronoi diagrams can represent influence zones in environmental modeling. For example, in hydrology, the size of a watershed’s influence could be modeled with weights, with larger watersheds having more influence over the partitioned space.
• Machine Learning and Clustering: In clustering algorithms, weighted Voronoi diagrams can be used to assign different levels of importance to data points. This is useful when the data points are heterogeneous and some points (e.g., centroids in clustering problems) should have a larger effect on the resulting clusters.

6.4. Computational Aspects

Computing weighted Voronoi diagrams involves calculating the weighted distance between points and determining the corresponding Voronoi cells. While the general computational complexity of Voronoi diagram construction remains $O(nlogn)$, the added complexity of weights requires careful handling in the distance calculation and the determination of boundaries between cells.

The most common algorithm for computing Voronoi diagrams, the Fortune’s algorithm, can be adapted to handle weighted Voronoi diagrams by modifying the distance metric during the diagram’s construction. Alternatively, algorithms like the incremental algorithm or divide-and-conquer methods can also be used, but with adjustments to accommodate for the weighted distance calculation.

6.5. Illustrative Example of Weighted Voronoi Diagram

To demonstrate the concept of a weighted Voronoi diagram (WVD), consider a simplified 2D environment with three service facilities (e.g., fire stations) located at different coordinates and having varying capabilities, represented through weights.

Let the three facilities be the following:

• Station A at $(2,2)$ with weight $w_A=1$;
• Station B at $(8,3)$ with weight $w_B=2$;
• Station C at $(5,7)$ with weight $w_C=0.5$.

Here, the weight $w_i$ inversely represents the service efficiency of each station. A lower weight corresponds to a more efficient or faster facility.

In the context of a WVD, the weighted distance $d_w(p,s_i)$ between a point $p\in {\mathbb{R}}^2$ and site $s_i$ is defined as follows:

$$d_w(p,s_i)=w_i·\parallel p-s_i\parallel$$

(17)

where $\parallel p-s_i\parallel$ denotes the Euclidean distance between p and $s_i$, and $w_i$ is the associated weight of site $s_i$.

Based on the above definition, the 2D space is partitioned into three regions such that each region contains the set of points p for which a particular station offers the minimum weighted distance. Unlike classical Voronoi diagrams where boundaries are equidistant to two sites, the boundaries in a WVD are curved and skewed depending on the weights (Figure 5).

This example highlights the utility of WVDs in modeling facility placement problems with heterogeneous capabilities, such as sensor coverage, service zones in emergency response planning, or task allocation in multi-agent systems.

6.6. Additively Weighted Voronoi Diagrams

Additively weighted Voronoi diagrams are a specific type of weighted Voronoi diagram where the weight associated with each site influences the distance metric in a simple additive manner. In this case, the distance between a point $x\in {\mathbb{R}}^n$ and a site $p_i$ with weight $w_i$ is modified by adding the weight to the Euclidean distance rather than modifying the coordinates or using a different function for the distance. The additively weighted metric is straightforward but has significant implications for the resulting diagram.

6.6.1. Definition and Mathematical Formulation

Given a set of n sites $\mathcal{P}=\{p_1,p_2,\dots ,p_n\}$ with respective weights $w_1,w_2,\dots ,w_n$, the distance between any point $x\in {\mathbb{R}}^n$ and a site $p_i$ is defined by

$$d(x,p_i,w_i)=\parallel x-p_i\parallel +w_i$$

(18)

where $\parallel x-p_i\parallel$ represents the Euclidean distance between the point x and the site $p_i$, and $w_i$ is the associated weight of the site.

The Voronoi cell $V\left(p_i\right)$ for site $p_i$ is then defined as the set of points $x\in {\mathbb{R}}^n$ that are closer to $p_i$ (under the modified distance metric) than to any other site $p_j$, i.e.,

$$V\left(p_i\right)=\{x\in {\mathbb{R}}^n\mid d(x,p_i,w_i)\le d(x,p_j,w_j)\forall j\ne i\}$$

(19)

This definition ensures that the influence of each site is proportional to its weight, where larger weights make the corresponding Voronoi cells smaller and smaller weights yield larger cells.

6.6.2. Geometric Interpretation

In the additively weighted case, the influence of each site grows with the weight in an additive fashion. Geometrically, for a site $p_i$ with weight $w_i$, the influence of the site extends beyond its Euclidean reach, causing the boundaries between cells to shift. The weight acts as a constant shift, pushing the boundary of $p_i$’s Voronoi cell outward relative to the other sites with smaller weights. In particular, the following holds:

• A larger weight corresponds to a cell with a smaller effective size, as the weighted distance grows faster.
• A smaller weight corresponds to a larger cell, as the distance is less influenced by the weight.
• The cells may still be convex, but their shapes are modified by the relative distances and weights.

This additive weighting model is often simpler to compute than other forms of weighted Voronoi diagrams, such as multiplicative or logarithmic weighting. However, its simplicity makes it suitable for a variety of applications where the influence of sites is naturally additive, such as facility placement problems, where the service capacity is a straightforward additive factor.

6.6.3. Applications of Additively Weighted Voronoi Diagrams

Additively weighted Voronoi diagrams are particularly useful in scenarios where the influence of sites is directly proportional to some additive factor. Some notable applications include the following:

• Geographical and Urban Planning: In urban planning, the service areas of facilities (e.g., schools, hospitals) can be modeled using additively weighted Voronoi diagrams. Here, the weight could represent factors like the size or capacity of the facility, with larger facilities having a larger influence on the surrounding area.
• Robotic Coverage and Exploration: In robotic exploration, the additively weighted Voronoi diagram can be used to model coverage areas where robots of different capacities are deployed. A robot with greater coverage capability would have a larger weight, resulting in a smaller Voronoi cell, signifying its higher capacity to cover a region.
• Resource Allocation Problems: In resource allocation problems, additively weighted Voronoi diagrams can model the distribution of resources among different sites. The weight could represent the amount of resource at a particular site, and the Voronoi cells can then show how the resource is distributed across the region.
• Wireless Networks: In wireless network design, the signal strength or capacity of each transmitter can be modeled with weights, and the resulting Voronoi diagram shows the coverage area for each transmitter. Sites with larger transmission power will have a larger influence, and thus a smaller cell.

6.6.4. Computational Aspects

From a computational standpoint, additively weighted Voronoi diagrams are relatively straightforward to compute. The construction process generally involves calculating the Euclidean distance between all points in the space and then adjusting these distances by the weight of each site. Once the weighted distances are computed, the Voronoi cells can be determined by comparing distances from each point to the sites.

The algorithm for constructing additively weighted Voronoi diagrams is similar to that for traditional Voronoi diagrams, with the primary modification being the distance function. Popular algorithms such as Fortune’s algorithm, the incremental algorithm, and divide-and-conquer methods can be adapted to handle additively weighted distances.

6.6.5. Example of Additively Weighted Voronoi Diagram

Consider three sites $p_1,p_2,p_3$ in ${\mathbb{R}}^2$ with positions $p_1=(1,1),p_2=(3,3),p_3=(5,1)$ and corresponding weights $w_1=1,w_2=2,w_3=0.5$. The additively weighted Voronoi diagram will show how the larger weight of site $p_2$ results in a smaller Voronoi cell, while the smaller weight of site $p_3$ results in a larger cell. In an additively weighted Voronoi diagram, each site $p_i\in {\mathbb{R}}^2$ is assigned a weight $w_i\in \mathbb{R}$. The additively weighted distance from a point $x\in {\mathbb{R}}^2$ to a site $p_i$ is defined as follows:

$$d(p_i,x)=\parallel x-p_i\parallel -w_i$$

(20)

Unlike standard Voronoi diagrams where regions are determined solely based on Euclidean distances, the weights here modify the effective proximity. A higher weight $w_i$ results in a smaller Voronoi region for site $p_i$, as it effectively reduces the distance needed for other points to be closer to competing sites.

Consider three sites:

• $p_1=(1,1)$ with weight $w_1=1$;
• $p_2=(3,3)$ with weight $w_2=2$;
• $p_3=(5,1)$ with weight $w_3=0.5$.

In the resulting diagram (Figure 6), the following occurs:

• Site $p_2$, having the highest weight ($w_2=2$), yields the smallest Voronoi cell.
• Site $p_3$, with the lowest weight ($w_3=0.5$), covers a significantly larger region.
• Site $p_1$ occupies an intermediate region proportional to its weight.

This example illustrates how additive weights influence the spatial dominance of each site. As the weight increases, the influence of a site decreases in spatial terms, resulting in smaller Voronoi regions, and vice versa.

6.6.6. Summary

Additively weighted Voronoi diagrams are a special case of weighted Voronoi diagrams in which the influence of a site is added to the Euclidean distance. This model is simple and intuitive, making it suitable for a wide range of applications where the influence of a site is directly proportional to its weight. The resulting Voronoi cells are modified by the additive influence of each site’s weight, leading to partitions that reflect the relative importance of each site. The computational complexity of constructing additively weighted Voronoi diagrams is comparable to traditional Voronoi diagrams, making them an efficient tool for solving real-world spatial problems.

6.7. Multiplicatively Weighted Voronoi Diagrams

Multiplicatively weighted Voronoi diagrams are another extension of the classical Voronoi diagram, where each site is assigned a weight, but this time the weight influences the distance metric in a multiplicative manner. Instead of simply adding the weight to the Euclidean distance, as in the additively weighted case, the weight is multiplied by the distance between the point and the site. This approach results in a different geometric interpretation of the Voronoi cells, particularly in scenarios where the influence of sites varies in a more complex, non-linear way.

6.7.1. Definition and Mathematical Formulation

Given a set of n sites $\mathcal{P}=\{p_1,p_2,\dots ,p_n\}$ with respective weights $w_1,w_2,\dots ,w_n$, the distance between a point $x\in {\mathbb{R}}^n$ and a site $p_i$ with weight $w_i$ is defined by

$$d(x,p_i,w_i)=w_i\parallel x-p_i\parallel$$

(21)

where $\parallel x-p_i\parallel$ represents the Euclidean distance between x and $p_i$, and $w_i$ is the weight of site $p_i$. The weight $w_i$ acts as a multiplicative factor, stretching or shrinking the distance from the site, depending on whether the weight is greater or less than 1.

The Voronoi cell $V\left(p_i\right)$ for site $p_i$ is then defined as the set of points $x\in {\mathbb{R}}^n$ that are closer to $p_i$ (under the multiplicatively weighted distance) than to any other site $p_j$, i.e.,

$$V\left(p_i\right)=\{x\in {\mathbb{R}}^n\mid d(x,p_i,w_i)\le d(x,p_j,w_j)\forall j\ne i\}$$

(22)

This definition ensures that the influence of each site is proportional to the multiplicative factor introduced by its weight.

6.7.2. Geometric Interpretation

The multiplicative nature of the weight means that the Voronoi cells behave differently compared to the additively weighted or unweighted cases. Specifically:

• A larger weight $w_i$ causes the Voronoi cell of site $p_i$ to become smaller, as the distance between the points in $V\left(p_i\right)$ is effectively scaled down by the factor $w_i$.
• Conversely, a smaller weight $w_i$ causes the Voronoi cell of site $p_i$ to become larger, as the distance is stretched by the factor $w_i$.
• The boundaries of the cells are no longer straight lines (in higher dimensions), but instead are affected by the multiplicative factor, causing the cells to deform in a more non-linear way compared to the additively weighted case.

Thus, the multiplicative weight introduces a scaling effect that significantly changes the shape of the Voronoi cells, and the result is a more flexible model for representing influence in spatial partitions.

6.7.3. Applications of Multiplicatively Weighted Voronoi Diagrams

Multiplicatively weighted Voronoi diagrams are used in various applications where the influence of sites varies according to a multiplicative factor, which could be based on size, capacity, or other factors that scale in proportion to distance. Some notable applications include the following:

• Wireless Networks: In wireless network design, the transmission power of each transmitter is often modeled with a multiplicative weight. Sites with higher transmission power will have smaller Voronoi cells, meaning their signal coverage area is larger compared to sites with lower power.
• Facility Location Problems: In facility location problems, the capacity of a facility may be modeled as a multiplicative factor, where facilities with larger capacities will have smaller cells, thus serving a larger area more effectively.
• Geographical Distribution Modeling: In ecology and geography, multiplicatively weighted Voronoi diagrams can model the territorial influence of species or environmental factors that have a scaling effect. For example, a species with a greater range or influence could be modeled with a larger weight, thus having a larger area in the Voronoi diagram.
• Robotics and Sensor Networks: In robotics, sensor coverage and range can be modeled multiplicatively. Sensors with larger ranges have smaller Voronoi cells, covering a larger area of the environment. This is particularly useful in applications like path planning and area coverage.

6.7.4. Computational Aspects

Computing multiplicatively weighted Voronoi diagrams is more complex than computing traditional Voronoi diagrams due to the multiplicative nature of the distance metric. However, the construction process still follows the same general approach, with the primary modification being the weighted distance calculation.

• Complexity: The computational complexity remains $O(nlogn)$ for most algorithms, as the core structure of the algorithm remains the same as for the classical Voronoi diagram. The complexity arises from the need to compute distances considering the multiplicative weight.
• Algorithms: Popular algorithms such as Fortune’s algorithm or divide-and-conquer approaches can be adapted to handle multiplicative weighting by modifying the distance function. The key challenge is ensuring that the multiplicative factor is correctly applied when comparing distances between points and sites.
• Numerical Stability: Care must be taken in numerical methods to avoid issues when dealing with very small or very large weights. Precision in computing distances becomes important, especially when the weight values are significantly large or small.

6.7.5. Example of Multiplicatively Weighted Voronoi Diagram

Consider three sites $p_1=(1,1),p_2=(4,4),p_3=(7,1)$ in ${\mathbb{R}}^2$, with corresponding weights $w_1=1,w_2=2,w_3=0.5$. The multiplicatively weighted Voronoi diagram will result in site $p_2$, with the largest weight, having the smallest Voronoi cell, and site $p_3$, with the smallest weight, having the largest cell.

6.7.6. Summary

Multiplicatively weighted Voronoi diagrams are an extension of the classical Voronoi diagram, where each site’s weight influences the partition by scaling the Euclidean distance between points and sites. The multiplicative nature of the weight results in Voronoi cells that are stretched or compressed depending on the size of the weight. These diagrams are particularly useful in applications where the influence of a site scales with distance, such as wireless networks, facility location problems, and robotics. The computational complexity remains manageable, but careful handling of numerical precision is necessary due to the multiplicative nature of the weights. The resulting Voronoi partitions offer a more nuanced model for spatial influence, providing valuable insights in many practical problems.

7. Anisotropic Metrics

Anisotropic metrics extend the classical notion of distance by allowing for different scaling factors along different directions. Unlike isotropic metrics (such as the Euclidean and Manhattan metrics), which treat all directions equally, anisotropic metrics provide a more flexible framework by enabling varying distance metrics along different axes or directions [42,43,44,45,46]. This can be particularly useful in scenarios where the underlying space or environment has inherent directional biases, such as in image processing, geographic modeling, and material science.

7.1. Definition and Mathematical Formulation

An anisotropic metric can be defined as a distance function that applies different scaling factors along different coordinate axes. A general anisotropic metric can be represented as follows:

$$d(x,p)=\sqrt{\sum _{i=1}^n{a}_i{(x_i-p_i)}^2}$$

(23)

where $x=(x_1,x_2,\dots ,x_n)$ and $p=(p_1,p_2,\dots ,p_n)$ are two points in ${\mathbb{R}}^n$, and $a_i$ is a scaling factor for the i-th coordinate direction. The scaling factors $a_i$ control how the distance is measured along each axis and allow for differential treatment of the coordinates, resulting in an anisotropic (direction-dependent) distance.

In particular, when the scaling factors $a_i$ are equal for all directions, the metric reduces to the Euclidean distance. Conversely, when different scaling factors are applied along different axes, the resulting distance metric becomes anisotropic. This metric is useful for modeling systems where some directions have different lengths or importance compared to others.

7.2. Geometric Interpretation

The key feature of anisotropic metrics is that they allow for different degrees of stretching or compression along various axes or directions. In this sense, anisotropic metrics create a space where the shape of the Voronoi cells or distance contours is not necessarily spherical (as in the Euclidean case) but may instead take the form of ellipsoids or other shapes, depending on the specific values of the scaling factors $a_i$.

• If the scaling factor $a_i>1$, the distance along the i-th direction is stretched, resulting in elongated contours along that axis.
• If the scaling factor $a_i<1$, the distance along the i-th direction is compressed, resulting in more compact contours along that axis.
• When all scaling factors are equal, the distance metric reduces to the Euclidean metric, and the contours are circular or spherical.

The anisotropy thus introduces a directional dependence to the space, where the geometry of the distance metric is governed by the scaling factors. This can be visualized as a deformation of the space, where distances in certain directions are either exaggerated or diminished compared to others.

7.3. Examples of Anisotropic Metrics

Several well-known anisotropic metrics can be derived from the general form described above. These include the following:

7.3.1. Mahalanobis Distance

The Mahalanobis distance is a popular example of an anisotropic metric, especially used in statistics and pattern recognition. The Mahalanobis distance between two points x and p is given by

$$d(x,p)=\sqrt{{(x-p)}^T{\Sigma }^{-1}(x-p)}$$

(24)

where $\Sigma$ is the covariance matrix of the data. The covariance matrix encapsulates the anisotropy by defining the scaling along different axes. In this case, the anisotropy depends on the correlation between the different dimensions, with larger correlations leading to more stretched distances along certain axes. When the covariance matrix is the identity matrix, the Mahalanobis distance reduces to the Euclidean distance.

7.3.2. Affine-Invariant Distance

The affine-invariant distance is another example of an anisotropic metric commonly used in computer vision and image processing. It is used to measure distances between points under affine transformations, which include scaling, rotation, translation, and shearing. The affine-invariant distance is defined in terms of the matrix representation of the affine transformation and is sensitive to the anisotropic scaling introduced by such transformations.

7.3.3. Directional Metrics

In some applications, it is desirable to measure distance in a way that is sensitive to specific directions. Directional metrics can be considered anisotropic metrics where the scaling factors vary depending on the direction in which the distance is being measured. One common example is the weighted ($L_2$) metric, where different weights are applied along different axes of the space.

$$d(x,p)=\sqrt{\sum _{i=1}^n{w}_i{(x_i-p_i)}^2}$$

(25)

where $w_i$ are weights that scale the distance along the i-th direction. By adjusting the weights, it is possible to emphasize or de-emphasize certain directions in the distance calculation, which can be useful in applications like directional filtering or anisotropic image processing.

7.4. Applications of Anisotropic Metrics

Anisotropic metrics are widely used in various fields, especially in situations where spatial or directional relationships are not uniform. Some prominent applications include the following:

• Geographical Modeling and Geographic Information Systems (GISs): Anisotropic metrics are used in geographical modeling to account for variations in terrain, land use, or climate across different directions. For example, the distance between two points in a mountainous region may be different depending on whether the path follows a ridge or a valley.
• Image Processing and Computer Vision: In computer vision, anisotropic metrics are often used to measure distances between points or regions in images. For instance, the metric may be used to account for directional information when performing edge detection or object recognition in images with anisotropic features.
• Pattern Recognition and Classification: Anisotropic metrics, such as the Mahalanobis distance, are widely used in statistical classification tasks where the underlying data exhibit directional correlations. These metrics allow for a better representation of the structure of the data compared to isotropic metrics.
• Material Science and Engineering: In material science, anisotropic metrics are used to model the properties of materials that have different strengths or behaviors along different axes. For instance, composite materials or crystalline structures often exhibit directional dependence in their mechanical properties, which can be modeled using anisotropic metrics.
• Robotics and Path Planning: Anisotropic metrics are also applied in robotic path planning, especially in environments where the cost of movement differs in various directions. For example, if a robot is operating in a terrain with varying friction or obstacles, an anisotropic metric can model the cost of moving in different directions.

7.5. Computational Aspects of Anisotropic Metrics

Computing distances using anisotropic metrics generally involves adjusting the classical algorithms for Voronoi diagram construction or nearest-neighbor search to account for the directional scaling factors. The complexity of computing anisotropic distances can vary depending on the specific metric used. For instance,

• Mahalanobis Distance:Involves matrix inversion or solving systems of linear equations, which can be computationally expensive, especially for large data sets or high-dimensional spaces.
• Weighted ($L_2$) Metric: Involves simple element-wise multiplication with the weights, which is computationally efficient.
• Affine-Invariant Distance:Requires knowledge of affine transformations and matrix operations, which can be more complex but still manageable with appropriate numerical techniques.

For computational efficiency, approximate methods or dimensionality reduction techniques such as principal component analysis (PCA) may be used in certain applications to reduce the complexity of anisotropic distance computations.

7.6. Summary

Anisotropic metrics generalize the classical notion of distance by introducing directional scaling factors. These metrics are essential in modeling scenarios where the underlying space exhibits inherent biases along certain axes, making them valuable in a variety of fields, including image processing, pattern recognition, geographic modeling, and robotics. The anisotropic nature of these metrics allows for a more nuanced representation of distances, offering greater flexibility in applications where uniform distance measures are insufficient. The computational techniques for anisotropic metrics depend on the specific metric employed, but are generally efficient and well-supported by existing numerical methods.

8. Geodesic Distance on Manifolds

In differential geometry, the concept of geodesic distance is fundamental to understanding the shortest path between two points on curved surfaces or more general manifolds [47,48,49,50,51]. Unlike Euclidean space, where the shortest path between two points is a straight line, the shortest path on a manifold may curve due to the underlying geometry of the space. Geodesic distance provides a way of measuring the distance between two points by following the curve that minimizes the distance, termed a geodesic.

8.1. Definition of Geodesic Distance

The geodesic distance between two points p and q on a manifold M is defined as the length of the shortest curve (geodesic) connecting these points. A geodesic is a curve that locally minimizes distance and is a generalization of the concept of a straight line in Euclidean space. Geodesics on manifolds satisfy a second-order differential equation known as the geodesic equation.

Mathematically, the geodesic distance $d(p,q)$ is given by

$$d(p,q)=\underset{\gamma }{inf}{\int }_0^1\parallel \dot{\gamma }\left(t\right)\parallel dt$$

(26)

where $\gamma$ is a curve connecting p to q, $\dot{\gamma }\left(t\right)$ is the velocity vector of the curve, and $\parallel \dot{\gamma }\left(t\right)\parallel$ is the norm of the velocity vector with respect to the metric of the manifold. The infimum is taken over all possible curves connecting p and q. In practice, the geodesic distance is computed along the geodesic that minimizes this integral.

8.2. Geodesics and the Metric Tensor

On a Riemannian manifold, distances are measured with respect to a metric tensor g, which encodes the intrinsic geometry of the manifold. The metric tensor g allows for the computation of lengths and angles on the manifold. The geodesic equation, which governs the path of a geodesic, is derived from the metric tensor and the connection on the manifold.

The geodesic equation can be written in terms of the Christoffel symbols ${\Gamma }_{\mu \nu }^{\lambda }$ of the connection as

$${\displaystyle \frac{d^2{x}^{\lambda }}{d{t}^2}}+{\Gamma }_{\mu \nu }^{\lambda }{\displaystyle \frac{d{x}^{\mu }}{dt}}{\displaystyle \frac{d{x}^{\nu }}{dt}}=0$$

(27)

where $x^{\lambda }\left(t\right)$ represents the coordinates of the geodesic curve and the Christoffel symbols ${\Gamma }_{\mu \nu }^{\lambda }$ encode information about the curvature and torsion of the manifold.

8.3. Computational Methods for Geodesic Distance

Computing geodesic distances on manifolds is generally more complex than in Euclidean space due to the curvature of the manifold. However, several numerical methods can be employed to approximate geodesic distances:

• Riemannian Gradient Descent: This approach uses the concept of gradient descent on manifolds to iteratively find the geodesic between two points. Starting from an initial guess, the algorithm moves along the geodesic direction, adjusting the path to minimize the distance between the points.
• Fast Marching Method: This is a popular algorithm for computing geodesic distances in manifolds that are embedded in higher-dimensional spaces. It is based on solving the eikonal equation and is particularly efficient for computing geodesic distances in computational geometry and computer vision.
• Geodesic Shooting Method: In this method, a geodesic is shot from one point, and the distance is computed by adjusting the initial conditions of the geodesic until it reaches the second point. This technique is widely used in computer graphics and computational anatomy.

These methods generally involve solving the geodesic equation numerically, which can be computationally expensive, especially in high-dimensional spaces or for complex manifolds. Efficient algorithms and optimization techniques are essential for handling such computations in practical applications.

8.4. Applications of Geodesic Distance

Geodesic distances are widely used in various fields that involve the analysis of curved spaces or manifolds. Some prominent applications include the following:

• Image and Shape Analysis: In image processing and computer vision, geodesic distances are used to measure distances between points on the surface of a 3D shape or between curves on a manifold. This is crucial for tasks such as surface matching, shape recognition, and object detection in non-Euclidean spaces.
• Geodesic Mesh Processing: In computer graphics, geodesic distance is used to measure the distance between points on the surface of 3D meshes, which is critical for mesh segmentation, texture mapping, and surface parametrization.
• Medical Imaging: In medical imaging, geodesic distance is used for anatomical shape analysis. It is employed to compare anatomical structures that are modeled as 3D manifolds, such as in the analysis of brain surfaces in functional neuroimaging.
• Robotics and Path Planning: In robotics, especially in robotic navigation on curved surfaces or terrains, geodesic distance is used for path planning to determine the shortest route between two points, while considering the geometry of the environment.
• Data Analysis and Machine Learning: Geodesic distances are used in manifold learning techniques, where data points are assumed to lie on a lower-dimensional manifold embedded in a higher-dimensional space. The geodesic distance can help in analyzing the structure of the manifold and can be used in clustering and classification tasks.
• Astronomy and Cosmology: In cosmology, geodesic distance is used to measure the separation between points in curved spacetime. This is crucial in general relativity for understanding the geometry of the universe and the movement of celestial bodies.

8.5. Summary

Geodesic distance provides a way to measure the shortest path between two points on a manifold, generalizing the concept of straight-line distance from Euclidean space to curved spaces. It is defined as the length of the shortest curve connecting the points, and it is governed by the geometry of the manifold, encapsulated in the metric tensor. Computing geodesic distances can be computationally challenging, especially in high-dimensional spaces, but several numerical methods, such as Riemannian gradient descent and fast marching, offer efficient solutions. Geodesic distance is widely applied in fields such as computer vision, robotics, medical imaging, and cosmology, making it a critical tool for analyzing and understanding complex geometric spaces.

9. Topological and Geometric Properties

The study of Voronoi partitions on manifolds is intrinsically tied to both topological and geometric properties. These properties govern the structure of the Voronoi diagram, its cells, and the behavior of distances within the partitioned space. Understanding these properties is crucial for analyzing the nature of the Voronoi regions, their shapes, and the interactions between neighboring regions. In this section, we delve into the topological and geometric aspects that influence Voronoi partitions, focusing on concepts such as the curvature of the underlying manifold, the connectivity of Voronoi cells, and the impact of the metric on the overall geometry of the diagram.

9.1. Topological Properties

Topological properties of a space are concerned with its inherent shape and connectivity, without regard to distances or angles. These properties focus on the qualitative characteristics of the space that remain unchanged under continuous transformations, such as stretching or bending, but not tearing or gluing.

9.1.1. Connectivity of Voronoi Cells

One of the fundamental topological aspects of a Voronoi diagram is the connectivity of its cells. In Euclidean space, Voronoi cells are typically convex, meaning that any two points within a cell can be connected by a straight line without leaving the cell. However, on curved manifolds, the cells may lose their convexity, depending on the curvature of the underlying space.

In non-Euclidean geometries, such as spherical or hyperbolic spaces, Voronoi cells may exhibit more complex topological structures. For instance, on a sphere, the Voronoi cells are not strictly convex but rather take the form of spherical polygons. Similarly, in hyperbolic space, the cells may become highly non-convex and may exhibit features like irregular shapes or “holes.” The topology of the manifold thus directly influences the topology of the Voronoi cells.

In some cases, Voronoi cells may be disconnected, especially in higher-dimensional manifolds or in spaces with complex curvature. The topological feature of connectedness is vital when analyzing the structure of a Voronoi diagram, as disconnected cells can indicate unusual geometric features or singularities in the space.

9.1.2. Boundary Structure

The boundaries of Voronoi cells are also subject to topological considerations. In Euclidean spaces, the boundaries are typically straight lines or curves, and the cells meet at shared edges or vertices. In curved spaces, the boundaries of Voronoi cells may be more intricate and may exhibit curvature depending on the underlying manifold. For example, in spherical space, the boundaries are segments of great circles, and in hyperbolic space, they may be segments of hyperbolic lines.

The topological type of the boundary can affect how the cells are connected and how they relate to one another. In general, the boundaries in non-Euclidean geometries can be more complex and may require more advanced tools from algebraic topology, such as homology and cohomology, to fully understand their structure.

9.1.3. Euler Characteristic

The Euler characteristic $\chi \left(M\right)$ is a topological invariant that provides a measure of the “shape” of a space. For a manifold M, the Euler characteristic is related to the number of vertices, edges, and faces of a polyhedral decomposition of the space. In the context of Voronoi diagrams, the Euler characteristic plays a role in determining the relationship between the number of Voronoi cells and their topological properties.

For instance, on a sphere, the Euler characteristic is 2, while for a torus, it is 0. The Euler characteristic of a manifold can impact the structure of the Voronoi diagram, especially in higher-dimensional spaces or complex manifolds. It is also used to analyze the global connectivity and the presence of singularities or other topological features in the diagram.

9.2. Geometric Properties

Geometric properties concern the distances, angles, and curvature of the manifold. These properties are crucial for understanding the shape and size of the Voronoi cells, as well as their relationship with each other.

9.2.1. Curvature of the Manifold

The curvature of the underlying manifold significantly affects the geometry of Voronoi cells. In Euclidean space, the curvature is zero, and Voronoi cells are typically convex and equidistant from their generating points. However, in curved spaces, the curvature introduces deviations from this simple structure.

In positively curved spaces, such as the surface of a sphere, Voronoi cells tend to be smaller near the poles and larger near the equator. The cells may also exhibit more irregular shapes as the curvature of the space distorts the standard Voronoi diagram structure. In negatively curved spaces, such as hyperbolic space, the cells grow exponentially as one moves away from the origin, and their boundaries become more distorted.

The curvature K of the manifold is typically quantified using tools from differential geometry, such as the Riemann curvature tensor or the Gaussian curvature for two-dimensional surfaces. Understanding the curvature is essential for predicting how the Voronoi diagram will behave in different regions of the space.

9.2.2. Length of Edges and Angles

In general, the edges of Voronoi cells in a curved space may not be straight lines but rather curves whose lengths and angles depend on the geometry of the manifold. The angles between the edges of adjacent cells can vary depending on the underlying curvature. In Euclidean space, these angles are typically 90° for orthogonal Voronoi diagrams, but in curved spaces, these angles can be much smaller or larger, and may change continuously across the space.

The length of edges in a Voronoi diagram also varies depending on the curvature of the manifold. For instance, on a sphere, the length of edges between two points on the surface is determined by the arc length of the great circle connecting them. In negatively curved spaces, such as hyperbolic space, the edges can become much longer as the distance between two points increases.

9.2.3. Volume of Voronoi Cells

The volume (or area in the case of two-dimensional manifolds) of a Voronoi cell is another geometric property influenced by the curvature of the manifold. In Euclidean space, the volume of each cell is relatively uniform, assuming uniform distribution of the generators. However, in curved spaces, the volume of each Voronoi cell can vary significantly based on the curvature.

For example, on the surface of a sphere, the area of each Voronoi cell is affected by its position relative to the poles and the equator. Cells near the poles tend to be smaller, while cells near the equator are larger. In hyperbolic space, the volume of Voronoi cells grows exponentially with distance from the origin, and the cells become increasingly irregular in shape.

9.3. Impact of Metric on Voronoi Structure

The choice of metric plays a crucial role in determining the geometric and topological properties of the Voronoi diagram. Different metrics, such as Euclidean, Manhattan, or geodesic distance on a curved manifold, will yield different shapes for the Voronoi cells. In particular, the curvature of the manifold, as encoded in the metric, affects the overall shape and size of the cells, as well as their connectivity and boundary structure.

The impact of the metric can be seen in the way that the Voronoi diagram adapts to the underlying geometry of the space. For example, on a flat plane with Euclidean distance, Voronoi cells are convex polygons. However, on a sphere with geodesic distance, the cells are spherical polygons, and on hyperbolic space, the cells may be highly distorted and non-convex.

9.4. Summary

The topological and geometric properties of a space play a significant role in determining the structure of Voronoi partitions. The connectivity, boundary structure, and Euler characteristic of the space govern the topological properties of the Voronoi cells, while the curvature, edge lengths, angles, and volumes govern the geometric properties. Understanding these properties is essential for analyzing and predicting the behavior of Voronoi diagrams in different settings, particularly in non-Euclidean geometries where the interaction between topology and geometry is more complex. The choice of metric, whether Euclidean, geodesic, or anisotropic, profoundly influences the shape and structure of the Voronoi diagram, and a deeper understanding of these properties is key to solving problems in fields such as computational geometry, image analysis, and machine learning.

10. Computational Considerations

Computing Voronoi diagrams on manifolds, particularly in non-Euclidean geometries, involves several computational challenges. These challenges arise from the complex nature of the underlying manifolds, the curvature of the space, and the choice of distance metrics. In this section, we discuss the key computational aspects involved in generating Voronoi diagrams on manifolds, including the complexity of different algorithms, numerical methods, and optimization strategies used to handle the geometry of the space efficiently.

10.1. Algorithmic Complexity

The computational complexity of Voronoi diagram algorithms can vary significantly depending on the nature of the manifold and the metric used. In Euclidean space, the problem is relatively straightforward, with several well-established algorithms, such as Fortune’s algorithm, that run in $O(nlogn)$ time, where n is the number of points (generators) in the diagram. However, for manifolds with non-Euclidean geometry or for spaces with complex topological structures, the complexity of Voronoi diagram generation increases, as it requires handling curved spaces and potentially non-convex cells.

The fundamental challenge lies in computing the distance function efficiently. In Euclidean space, distance is computed using simple formulas, but in curved spaces, computing the distance between points involves more intricate operations, such as solving geodesic equations or using numerical optimization techniques to approximate shortest paths. The computational complexity of computing geodesic distances often depends on the underlying manifold’s curvature and dimensionality, making the problem significantly harder for higher-dimensional or highly curved spaces.

For instance, in spherical space, the distance between two points involves arc-length calculations, and in hyperbolic space, distance computation requires hyperbolic trigonometry. Algorithms such as the fast marching method or geodesic shooting methods are often employed to compute these distances, and their complexity can be higher, typically in the range of $O\left(n^2\right)$ for dense graphs or $O(nlogn)$ for sparsely connected networks.

10.2. Numerical Methods for Geodesic Computation

As geodesic distances are fundamental to computing Voronoi diagrams on manifolds, choosing efficient methods for geodesic computation is crucial. Several numerical methods are employed to approximate the shortest path between two points on a manifold:

• Geodesic Shooting Method [51]: This method involves solving the geodesic equation by “shooting” geodesics from a starting point and adjusting the trajectory until the geodesic reaches the target point. This approach requires solving a system of differential equations and is computationally expensive, especially for non-trivial manifolds.
• Riemannian Gradient Descent [52]: This approach approximates geodesic paths by iteratively moving along the gradient direction on the manifold. It is particularly useful in high-dimensional spaces but can converge slowly for manifolds with complex geometry.
• Fast Marching Method [53]: The fast marching method is an efficient algorithm for computing geodesic distances in Euclidean and Riemannian spaces. It operates by solving the eikonal equation, which provides a way of approximating distances by propagating the solution from an initial point, updating the distance values in a fast and efficient manner.
• Dijkstra’s Algorithm (on Manifolds) [54]: For discretized manifolds (e.g., meshes or graphs), Dijkstra’s algorithm can be adapted to compute geodesic distances by considering the edge weights as the distance between adjacent points. This algorithm runs efficiently on graphs with a time complexity of $O(nlogn)$, assuming that a priority queue is used.

Each of these methods has trade-offs in terms of computational cost, accuracy, and applicability to different types of manifolds. For example, the geodesic shooting method provides high accuracy but is computationally expensive, while the fast marching method is more efficient for large-scale computations.

10.3. Discretization of Manifolds

For computational purposes, continuous manifolds are often discretized into simpler structures, such as meshes or graphs. Discretization allows for the use of graph-based algorithms, such as Dijkstra’s or A* search, to approximate Voronoi diagrams. Meshes are commonly used to represent manifolds, with vertices corresponding to points on the manifold and edges representing the connections between them. The key challenge here is to approximate geodesic distances accurately on these discrete structures.

When discretizing a manifold, it is important to maintain the geometric properties of the space. For example, when discretizing a sphere, the vertices should be placed on the surface of the sphere, and edges should represent geodesic paths (arcs of great circles). Similarly, in hyperbolic space, the discretization should account for the exponential growth of distances and the curvature of the space.

In some cases, the manifold may not have a natural discretization, and advanced techniques such as mesh generation or simplicial complexes may be required to ensure that the discretization reflects the underlying geometry and topology of the manifold.

10.4. Optimization Techniques

Computing Voronoi diagrams on manifolds is not just about distance calculations but also involves solving optimization problems. In particular, Voronoi partitioning often requires finding the region corresponding to each generator, which involves solving a set of optimization problems to minimize the distance to the generators subject to the constraints imposed by the manifold.

One popular optimization approach is the use of variational principles, where the partition is computed by minimizing a cost functional representing the distance to the generators. This can be solved using numerical optimization techniques, such as gradient descent, Newton’s method, or constrained optimization algorithms. These methods are particularly useful for computing weighted Voronoi diagrams or handling anisotropic metrics, where the distance between two points is not simply a linear function but depends on more complex geometric structures.

For example, in the case of weighted Voronoi diagrams, where the influence of each generator is modified by a weight, optimization techniques can be used to solve the weighted distance function. The complexity of this optimization process depends on the metric and topology of the manifold and may require the use of advanced algorithms such as sequential quadratic programming or interior-point methods.

10.5. Parallel and Distributed Computing

Given the computational complexity of Voronoi diagram algorithms, parallel and distributed computing techniques are increasingly being employed to speed up the process. Since the computation of Voronoi cells is often independent for each generator, the problem is naturally parallelizable. In parallel algorithms, the geodesic distance computations, as well as the identification of Voronoi cells, can be distributed across multiple processors or machines.

Graphics processing units (GPUs) have been particularly useful in accelerating these computations, as they are well-suited for handling the large number of geometric and distance computations required in Voronoi diagram generation. GPUs allow for massive parallelization of the calculations, enabling the processing of large datasets or high-dimensional spaces that would otherwise be computationally prohibitive.

Distributed computing frameworks, such as Apache Spark or CUDA, allow for the computation of Voronoi diagrams on very large scales, such as for high-dimensional or complex manifolds, by distributing the work across multiple nodes or machines. These frameworks also help in managing large datasets and performing computations more efficiently.

10.6. Approximation Methods

In some cases, exact computation of Voronoi diagrams on complex manifolds may not be feasible due to the high computational cost. In such cases, approximation methods are employed to estimate the Voronoi diagram efficiently. One common approach is to sample the manifold at a set of discrete points and compute a Voronoi diagram based on these sample points. This approximation becomes more accurate as the density of the sampling points increases, though it comes at the cost of increased computation.

Other methods, such as Monte Carlo simulations or random sampling algorithms, can be used to approximate the Voronoi diagram by randomly selecting points and estimating the corresponding regions. These methods are particularly useful when dealing with large-scale problems where exact solutions are too computationally expensive.

10.7. Summary

Computing Voronoi diagrams on manifolds is a challenging task that requires handling the complexities of curved spaces and non-Euclidean geometry. The computational considerations involved include the algorithmic complexity, numerical methods for geodesic computation, discretization techniques, optimization strategies, and the use of parallel and distributed computing. Advanced techniques such as variational optimization and mesh generation play crucial roles in ensuring that Voronoi diagrams are accurately computed on complex manifolds. While the problem can be computationally expensive, the use of approximation methods and modern parallel computing frameworks has made it possible to handle large-scale problems efficiently. As computational tools and algorithms continue to evolve, the ability to compute Voronoi diagrams on more complex manifolds will continue to expand, enabling a wide range of applications in fields such as computational geometry, machine learning, and computer graphics.

11. Applications and Implications

Voronoi diagrams, due to their ability to partition a space based on proximity to a set of points, have a wide range of applications across various fields. The flexibility of Voronoi diagrams to adapt to different distance metrics and their ability to model spatial relationships in non-Euclidean geometries make them particularly useful for solving complex problems in computational geometry, machine learning, robotics, and other scientific domains. In this section, we explore the significant applications and implications of Voronoi diagrams on manifolds, especially in contexts that require non-Euclidean distance metrics and in domains with complex geometries.

11.1. Geographic and Environmental Mapping

Voronoi diagrams are commonly used in geographic information systems (GISs) and environmental sciences to model regions of influence for geographical features such as cities, service stations, or weather patterns. The traditional Euclidean Voronoi diagram is used extensively in urban planning, where it helps to determine optimal locations for infrastructure based on proximity to existing facilities or population centers. However, when dealing with spherical Earth models, irregular terrains, or geodesic distances, Voronoi diagrams on manifolds become crucial.

For example, in mapping ecological zones, where the underlying surface is not flat, Voronoi diagrams based on geodesic distances can accurately reflect the true influence regions of environmental factors, such as rainfall, temperature, or vegetation coverage, especially when topographical features like mountains or valleys are present. Similarly, in urban planning on spherical surfaces (e.g., modeling satellite coverage or airport reach), geodesic Voronoi diagrams become essential for modeling true proximity relationships that are influenced by the Earth’s curvature.

11.2. Robotics and Multi-Agent Systems

In robotics, Voronoi diagrams are a fundamental tool in path planning, motion coordination, and sensor network optimization. For multi-robot systems, Voronoi partitioning can be used to allocate distinct regions of the environment to different robots, ensuring efficient coverage and coordination. In non-Euclidean settings such as spherical or hyperbolic spaces, Voronoi diagrams are useful for robots operating on curved surfaces, where Euclidean distances are not appropriate.

For example, a team of autonomous robots performing coverage tasks on a curved surface, such as a spherical planet or an irregularly shaped environment, can utilize Voronoi diagrams with geodesic or anisotropic metrics to divide the workspace into regions of influence. This allows each robot to operate within its allocated region, optimizing coverage and avoiding collisions. In scenarios with obstacles or varying terrain, Voronoi-based algorithms can adapt to ensure efficient exploration and task completion.

11.3. Machine Learning and Data Clustering

Voronoi diagrams are an important tool in machine learning, particularly in clustering algorithms and classification tasks. The ability to partition data points based on proximity to centroids or other reference points makes Voronoi diagrams valuable in k-nearest neighbor (k-NN) classification, support vector machines (SVMs), and other algorithms. By using alternative metrics such as Minkowski or weighted distances, Voronoi diagrams can be tailored to address specific types of data, such as those with varying scales or anisotropic properties.

In high-dimensional spaces, Voronoi diagrams can also be used to approximate data partitioning in non-linear manifolds. This is particularly useful in cases where the data does not naturally lie on a flat Euclidean space but instead resides on a curved or complex manifold. Dimensionality reduction techniques, such as t-SNE or PCA, can be used in combination with Voronoi diagrams to improve clustering performance and provide insights into the underlying structure of the data.

11.4. Wireless Sensor Networks and Communication Systems

Voronoi diagrams are essential in optimizing the layout and communication strategies of wireless sensor networks (WSNs) and communication systems. The primary goal in these networks is to ensure efficient coverage and communication between sensors or transmitters while minimizing energy consumption. In WSNs, Voronoi-based approaches are employed to optimize sensor placement and improve data transmission, especially in environments with irregular topologies.

For non-Euclidean spaces, such as sensor networks deployed on curved surfaces (e.g., satellites orbiting Earth, underwater systems, or even within buildings), Voronoi diagrams using geodesic or anisotropic metrics can be employed to improve communication efficiency. The resulting partitioning ensures that sensor nodes are optimally placed to minimize signal loss and interference, while also accounting for curvature and other physical constraints.

11.5. Computer Graphics and Mesh Generation

In computer graphics, Voronoi diagrams play an essential role in mesh generation, texture mapping, and the simulation of natural phenomena. For instance, Voronoi diagrams are used in the generation of terrains, where each Voronoi cell represents a distinct region of the terrain, such as a mountain or a valley. These techniques are also applied in texture mapping, where textures are assigned to regions based on their Voronoi cells.

In 3D modeling and mesh generation, Voronoi diagrams are useful for defining the structure of complex models, especially when working with non-Euclidean surfaces. When modeling objects with irregular geometries, such as human organs or geological formations, Voronoi-based methods help in discretizing the surface into manageable cells, ensuring that meshes conform to the curvature of the surface.

11.6. Astronomy and Astrophysics

Voronoi diagrams are also applied in astronomy and astrophysics for modeling the distribution of celestial bodies and other astrophysical phenomena. The shape of galaxies, star clusters, and the large-scale structure of the universe can be analyzed using Voronoi diagrams. In the study of the cosmic web, Voronoi diagrams can help one to understand the distribution of galaxies and clusters in the universe, with the distances between them often governed by cosmological metrics that are not Euclidean, such as the expanding universe model.

Additionally, Voronoi diagrams are employed in the optimization of telescope arrays for astronomical observation. By using non-Euclidean metrics, the placement of telescopes in spherical or hyperbolic spaces can be optimized to achieve the best coverage of the sky or to maximize the sensitivity of the array to cosmic phenomena.

11.7. Biological and Medical Sciences

In biology and medicine, Voronoi diagrams have found applications in cellular biology, tissue growth modeling, and anatomical studies. For example, Voronoi diagrams can be used to model the growth patterns of cells or tissues, where each cell occupies a region determined by the nearest neighboring cells. This model is particularly useful in studying phenomena such as tumor growth, wound healing, and the spread of diseases.

In medical imaging, Voronoi diagrams are used for the segmentation and analysis of 3D images, such as MRI scans, where the tissue regions are partitioned based on proximity to key anatomical landmarks. These applications often rely on non-Euclidean metrics, particularly when working with data from curved surfaces, like the brain or organs, with complex topologies.

11.8. Implications for Future Research

The study of Voronoi diagrams on manifolds has profound implications for both theoretical and applied mathematics. As computational power increases and new numerical methods emerge, the potential for solving previously intractable problems in geometry, physics, and engineering grows significantly. The implications of this research extend across various domains, from the design of more efficient algorithms for spatial partitioning to the application of Voronoi diagrams in highly curved or complex spaces.

Moreover, the exploration of anisotropic metrics, weighted Voronoi diagrams, and geodesic distances on manifolds opens new avenues for optimization and modeling in non-Euclidean spaces. Future research may focus on further improving the efficiency of Voronoi diagram generation, extending the applications to more complex manifolds, and integrating machine learning algorithms with Voronoi-based methods to tackle large-scale data analysis in diverse fields.

12. Conclusions

Voronoi diagrams have proven to be a powerful tool in computational geometry, enabling efficient spatial partitioning based on proximity to a set of points. Their application extends across diverse fields, including robotics, machine learning, environmental science, and biological modeling, with the ability to adapt to different distance metrics, such as Minkowski, Manhattan, and geodesic distances. This paper explored the versatility of Voronoi diagrams, particularly in non-Euclidean spaces and on manifolds, highlighting their significance in modeling complex geometries. By incorporating weighted, anisotropic, and geodesic metrics, Voronoi diagrams offer a flexible framework for solving spatial optimization problems, allowing for more accurate modeling in fields like multi-robot systems, geographic mapping, and biological research. Despite the progress made, computational challenges remain in efficiently constructing Voronoi diagrams in non-Euclidean spaces, particularly with high-dimensional or dynamically changing environments.

Future work in this area could focus on improving the computational efficiency of algorithms for non-Euclidean Voronoi diagrams, with particular attention to high-dimensional or complex manifolds. Integration with machine learning techniques could further enhance clustering and classification tasks, while parallel and distributed computing methods can handle larger datasets in real-time applications. Additionally, exploring custom, domain-specific metrics for specialized applications such as sensor networks, traffic management, and robotics could lead to significant advances in those fields. The continued development of Voronoi diagrams in real-world systems—such as multi-agent robotics or environmental modeling—will provide deeper insights into their practical viability. As computational methods advance, Voronoi diagrams will undoubtedly remain a cornerstone tool in both theoretical and applied computational mathematics.