PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Geometry → Computational Geometry ↓


Triangulations


In computational geometry, the concept of triangulation plays an essential role in the study of planar subdivisions. Triangulation is essentially dividing a plane into triangles. More formally, given a set of points in the plane, triangulation is the set of vertices at these points. There is a collection of non-overlapping triangles with vertices such that every point is a vertex of some triangle, and the union of these triangles is the convex cover of the point set.

Triangulation has countless applications in various fields such as computer graphics, geographic information systems (GIS), finite element analysis, and many more. The study of triangulation includes the properties of different types of triangulations, algorithms for their construction, their application to specific applications, and more. This includes understanding optimization and their mathematical properties.

Basic definitions and properties

Consider a set of points P = {p1, p2, ..., pn} in a plane. The basic requirement for triangulation of these points is that no triangle overlaps with any other triangle, except possibly along edges and of vertices. If T is a triangulation of P, it is represented graphically as where the vertices are the points from P and the edges are the line segments connecting these points, forming a triangle.

Triangulation has several interesting properties:

  • Euler's formula: For a planar graph with V vertices, E edges, and F faces, the following relation applies: 
    V - E + F = 2
    In the case of triangulation, where every face (except the outer face) is a triangle, this formula is particularly helpful in proving properties related to the numbers of triangles and edges.
  • Number of edges and triangles: The triangulation of n points in the plane has 2n - 3 edges and n - 2 triangles, provided that the convex cover of the points is itself a triangle.

Algorithms for triangulation

Various algorithms have been proposed for computing triangulation, each with its own specific advantages and complexities.

Incremental algorithm

The incremental algorithm works by adding points one by one and updating the triangulation dynamically. When a new point is added, the algorithm finds the triangle containing the new vertex and then re-triangulates the faces affected by this addition Here, we describe a simple step-by-step incremental approach:

  1. Start with a basic triangle with all the points.
  2. For each point, add it to the triangulation.
  3. Identify the triangle containing the new point and divide it into three smaller triangles.
  4. Fix resulting edge-fan inconsistencies by "flipping" the edges to maintain valid triangulation.

Delaunay triangulation

A special type of triangulation is Delaunay triangulation, which maximizes the minimum angle of each triangle. This property avoids thin triangles, making it quite useful in many practical applications. Delaunay triangulation has a unique property: any Also the circumcircle of a triangle does not include any other points in its interior.

Divide and conquer algorithm

The divide and conquer algorithm first divides the set of points into two halves, recursively triangulates each half, and merges the two triangulations on the boundary to get the final triangulation. Although it is complicated, it is remarkably efficient. It is more efficient than and is often used to obtain Delaunay triangulations.

Optimal triangulation

In some contexts, it is necessary to optimize the triangulation to satisfy specific conditions, such as minimizing the total edge length, maximizing the smallest angle, or obtaining triangles with limited aspect ratios. One such important optimization The goal is to find a minimum weight triangulation (MWT) where the sum of the edge lengths in the triangulation is minimal.

Applications of triangulation

Triangulation has far-reaching applications such as:

  • Computer graphics: Triangulation is used in mesh generation and rendering in 3D computer graphics. By decomposing an area into triangles, graphics software processes complex scenes efficiently.
  • Finite element analysis (FEA): Engineers use triangulation to divide domains for simulations of physical phenomena such as heat transfer, fluid dynamics, and stress analysis.
  • Geographic information systems (GIS): Triangulation is important for modeling terrain and managing spatial information in real-world geographic data.

Challenges and open problems

Despite extensive research in the field of trigonometry, some questions still remain open:

  • Finding the most efficient algorithms that work effectively for a wide range of inputs, especially in high-dimensional spaces.
  • Developing algorithms that can deal with dynamic datasets where points are constantly added or deleted.

Conclusion

Triangulations serve as a fundamental structure in computational geometry. Understanding their properties, algorithms for their construction, and their diverse applications in various fields reveals their importance in both theoretical and practical aspects.

The study of triangulation remains a vibrant area of research, with significant impact on technological advancements as computational capabilities increase and our modeling needs become more sophisticated.


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Triangulations

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