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 ↓


Convex Hulls


In the study of computational geometry, the concept of a "convex cover" is a basic building block. Imagine a set of scattered points on a plane. The convex cover is like wrapping a rubber band around those points. The band will take the shape of the smallest convex polygon that can enclose all the scattered points.

Understanding convex shapes

Before diving into convex covers, it is important to understand what convexity means. A shape is convex if, for any two points inside the shape, the line segment connecting them lies entirely within the shape. Think of it like a balloon. If you poke two pins anywhere on the surface of the balloon and connect them with a piece of thread, the thread will not slip out of the balloon.

Definition of convex cover

The convex cover of a set of points is the smallest convex set that encloses all the points. Mathematically, if you have a set S of points, then the convex cover is the intersection of all convex sets that contain S Alternatively, it is a boundary that uses the smallest perimeter while enclosing all the points.

Mathematical representation

The convex cover of a set of points S can be represented as:

Convex hull(S) = { x : x is a convex combination of points in S }

Visualization of convex covers

Example 1: Basic convex cover

Consider a simple scenario where the set has only three points that form a triangle.

In this example, the convex cover is the triangle itself, since all three points are its vertices.

Example 2: A more complex set

Let's consider some more complex points:

Here, the points form a figure that includes an interior point. The convex cover is a triangle, excluding the point at (130, 80), which does not affect the perimeter of the cover.

Properties of convex cover

Convexity

The output shape is always convex, that is, every internal angle is less than or equal to 180 degrees.

Specification

For a given set of points, the convex cover is unique. This uniqueness arises because there is no difference between the definition of convexity and the nature of Euclidean space.

Minimalism

The convex cover has the smallest possible boundary enclosing all the points of the set.

Algorithms for computing the convex cover

Various algorithms can calculate the convex cover of a set of points, each with varying complexities and use cases. Some of the popular algorithms are as follows:

Gift wrapping algorithm

Also known as the Jarvis March algorithm, it is like wrapping a gift by moving around the edge of the shape.

1. Start at the leftmost point in the set (smallest x-coordinate).
2. Select the point that makes the smallest counterclockwise angle with the line formed by the starting point.
3. Repeat step 2 with the most recently added point until you return to the starting point.

The computational complexity of gift wrapping is typically O(nh), where n is the number of points in the input set, and h is the number of points on the solution.

Graham scan

This algorithm computes the convex cover by first ordering the vertices and then considering them one by one.

1. Find the point with the lowest y-coordinate, breaking away from the smallest x-coordinate.
2. Sort the points in increasing order of the angle made by them and the point chosen in step 1 with the x-axis.
3. Repeat the points and remove the points that cause clockwise rotation.

Its computational complexity is O(n log n), mainly due to the initial sort operation.

Quickhull algorithm

The Quickhull algorithm is a method based on the divide-and-conquer paradigm.

1. Find the points with minimum and maximum x-coordinates, which should be part of the convex cover.
2. Use a recursive divide-and-conquer procedure to find the set of points that form a convex cover on each side of the line defined by the two points.

The average complexity is O(n log n), while in the worst case, it can be O(n^2).

Incremental algorithm

In this algorithm, the points are added one by one and the convex cover is updated at each step.

1. Start with a small subset of the points that form the initial cover.
2. Add a new point and check if it lies inside the current hull.
3. If it lies outside, update the hull to include the new point.

Although not the most efficient in terms of Big-O notation, this algorithm can still be intuitive and useful in specific applications.

Applications of convex cover

Convex covers have many practical applications in computational geometry and related fields:

Computer graphics

Convex covers are used in computer graphics to determine the boundary of a set of objects and for collision detection and path finding.

Geographic analysis

In GIS systems, convex covers help define the boundary of geographic units, such as groupings of locations.

Pattern recognition

Used in pattern recognition to give shape to groups of points, aiding in classification and detection.

Robotics

In robotics, convex shells aid spatial navigation and motion planning, and provide safe paths around sets of obstacles.

Conclusion

The study of convex covers is a gateway to understanding more complex computational geometry problems. While their definition is simple, efficient computation and application require a deep understanding of geometry and algorithm design. As a fundamental aspect of geometry, convex covers remain an area of active research and play an essential role in a variety of technology applications.


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Convex Hulls

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