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 ↓


Geometric Algorithms


Geometric algorithms are at the core of computational geometry, a field that combines computer science and mathematics. These algorithms deal with the study and manipulation of geometric objects, such as points, lines, polygons, and polyhedrons. They are important in a variety of applications, including computer graphics, robotics, geographic information systems (GIS), and computer-aided design (CAD).

Overview of geometrical objects and operations

Before delving into specific algorithms, let's understand the main geometric objects we work with:

  • Point: The simplest geometric object, defined by coordinates in a space such as 2D or 3D.
  • Lines: An infinite series of points extending in both directions, usually defined by a linear equation.
  • Line segment: The portion of a line bounded by two endpoints.
  • Polygons: Closed, multi-sided shapes in the plane. Triangles, squares, and pentagons are examples.
  • Polyhedrons: The 3D equivalents of polygons, such as cubes and pyramids.

Geometric operations often involve checking intersections, calculating distances, finding convex covers, or performing transformations such as translation, rotation, and scaling. Here is a simple example of calculating the distance between two points:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Basic geometric algorithms

Convex hull

The convex cover of a set of points is the smallest convex polygon that encloses all the points. It is often compared to stretching a rubber band around the outermost points. The most common algorithms for finding the convex cover are the Graham scan and the Jarvis march (also known as the gift wrapping algorithm).

Here's an example of how points are enclosed in a convex cover:

Line segment intersection

Detecting whether two line segments intersect is fundamental in computational geometry, with applications ranging from computer graphics to path planning in robotics. The Bentley–Ottman algorithm is a known solution for efficiently reporting all intersections in a set of line segments.

Consider two line segments AB and CD. If they intersect, the intersection can be found by solving these linear equations:

A = (x1, y1), B = (x2, y2) 
C = (x3, y3), D = (x4, y4) 
AB: y = mx + c 
CD: y = nx + k 
Intersection occurs when: mx + c = nx + k

Voronoi diagram

Voronoi diagrams divide a plane into regions based on the distance to a specified set of points. Each region corresponds to a point and contains all locations closer to that point than to any other point. These diagrams are useful in a variety of fields, including meteorology and urban planning.

Simple illustration of a Voronoi diagram with 4 points:

Advanced topics in geometric algorithms

Triangulation

Triangulation is the division of a polygon into non-overlapping triangles. This process is important for finite element analysis in graphics rendering and engineering simulations. The resulting triangles form a mesh that is easy to manipulate and analyze.

Keeping the polygon convex while triangulating reduces the complexity:

Quadtrees and octrees

Quadtrees (in 2D) and octrees (in 3D) are tree data structures used to partition spaces to support efficient spatial querying. These structures can improve the performance of algorithms dealing with large sets of points by rapidly removing large regions of space when looking for points.

Ray tracing and computation in graphics

Ray tracing is a rendering technique that simulates the way light interacts with objects to create realistic images. The algorithms calculate the path of the rays as they intersect with surfaces, tracing them backwards from the eye to the light source. This process involves multiple intersection tests and shows the beauty of geometric algorithms in practical applications.

Applications of geometric algorithms

Geometric algorithms are important in many fields. Here's how they contribute to various applications:

Computer graphics

In graphics, geometric algorithms are used to model, transform, and render images. Triangulation and rasterization are commonly used to convert vector graphics to pixel-based displays.

Robotics

Robotics uses geometric algorithms for path planning and obstacle avoidance, ensuring that robots can navigate an environment efficiently. Algorithms for collision detection are particularly important.

Geographic Information Systems (GIS)

GIS benefits greatly from geometric algorithms for managing spatial data - analyzing and visualizing geographic information. From mapping political boundaries using Voronoi diagrams to calculating shortest paths with Dijkstra's algorithm, these systems rely heavily on computational geometry.

In conclusion, geometric algorithms provide powerful tools for solving complex spatial problems in a variety of domains. Their development remains a rich area of research within mathematics and computer science, pushing the boundaries of what is possible with digital computations.


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