2.1 Convexity and convex sets

Convexity is a fundamental concept in computational geometry, simplifying complex problems and enabling efficient algorithms. It's all about shapes without dents or caves, like circles and triangles, which have special properties that make calculations easier.

Convex sets contain all line segments between any two points inside them. This simple idea leads to powerful mathematical tools and algorithms for solving geometric problems, from finding the smallest shape enclosing a set of points to detecting collisions in video games.

Definition of convexity

• Convexity plays a crucial role in computational geometry by simplifying complex geometric problems and enabling efficient algorithms
• Convex shapes and sets exhibit unique properties that allow for optimized computational approaches in various geometric calculations and analyses

Convex sets vs non-convex sets

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• Convex sets contain all line segments connecting any two points within the set
• Non-convex sets have at least one line segment between two points that lies partially outside the set
• Convex sets form a single, unbroken region without any indentations or holes
• Visualize convex sets as shapes without any "dents" or "caves" (circles, triangles, rectangles)

Mathematical formulation of convexity

• Defined using the concept of line segments between any two points in the set
• Formally expressed as $\forall x,y \in S, \forall t \in [0,1] : tx + (1-t)y \in S$
• Utilizes the notion of convex combinations to describe all points within the set
• Applies to both finite-dimensional vector spaces and infinite-dimensional function spaces

Properties of convex sets

• Convex sets exhibit fundamental characteristics that make them valuable in computational geometry
• Understanding these properties enables efficient algorithms for various geometric problems and optimizations

Intersection of convex sets

• Always results in another convex set or an empty set
• Preserves convexity, allowing for simplified intersection computations
• Enables efficient algorithms for finding common regions between multiple convex shapes
• Applies to both 2D and higher-dimensional spaces (planes intersecting in 3D space)

Union of convex sets

• Generally does not preserve convexity
• May result in non-convex shapes with complex geometries
• Requires additional processing to maintain convexity (convex hull computation)
• Presents challenges in computational geometry when dealing with multiple objects

Convex combinations

• Linear combinations of points in a convex set with non-negative coefficients summing to 1
• Expressed mathematically as $\sum_{i=1}^n \alpha_i x_i$ where $\sum_{i=1}^n \alpha_i = 1$ and $\alpha_i \geq 0$
• Forms the basis for defining convexity and understanding convex set properties
• Enables the representation of any point within a convex set as a combination of its vertices

Types of convex sets

• Convex sets encompass various geometric shapes and structures in computational geometry
• Understanding different types of convex sets aids in selecting appropriate algorithms and representations

Convex polygons

• Closed 2D shapes with straight sides and no internal angles greater than 180 degrees
• Characterized by vertices always pointing outwards
• Efficiently represented by an ordered list of vertices
• Include regular polygons (equilateral triangles, squares, regular hexagons)

Convex polyhedra

• 3D objects with flat faces, straight edges, and vertices pointing outwards
• Generalization of convex polygons to three-dimensional space
• Represented by sets of vertices, edges, and faces
• Encompass familiar shapes (cubes, tetrahedra, octahedra)

Convex hulls

• Smallest convex set containing a given set of points
• Computed using algorithms (Graham scan, Jarvis march, QuickHull)
• Serves as a fundamental operation in many computational geometry problems
• Finds applications in shape analysis, collision detection, and clustering

Algorithms for convex sets

• Computational geometry employs various algorithms to manipulate and analyze convex sets
• These algorithms form the foundation for solving complex geometric problems efficiently

Convex hull algorithms

• Graham scan algorithm operates in O(n log n) time complexity
  • Sorts points based on polar angle and builds the hull incrementally
• Jarvis march (gift wrapping) algorithm has O(nh) time complexity
  • Iteratively selects the next point with the smallest right-hand turn
• QuickHull algorithm uses divide-and-conquer approach
  • Recursively partitions points and constructs the hull

Point-in-polygon tests

• Ray casting algorithm counts intersections of a ray with polygon edges
• Winding number algorithm computes the number of times a polygon winds around a point
• Utilizes convexity properties for optimized testing in convex polygons
• Applies to both convex and non-convex polygons with different efficiency levels

Intersection detection

• Separating Axis Theorem (SAT) efficiently detects intersections between convex shapes
• Binary Space Partitioning (BSP) trees organize space for faster intersection queries
• Bounding Volume Hierarchies (BVH) use nested bounding volumes to accelerate intersection tests
• Exploits convexity properties to simplify and speed up collision detection algorithms

Applications of convexity

• Convexity concepts find widespread use in various computational geometry applications
• Understanding these applications highlights the practical importance of convex sets

Optimization problems

• Linear programming relies on convex feasible regions for efficient solutions
• Convex optimization problems guarantee global optima, simplifying solution methods
• Gradient descent algorithms converge to global minima in convex functions
• Applies to resource allocation, network flow, and machine learning problems

Collision detection

• Separating Axis Theorem (SAT) efficiently detects collisions between convex objects
• Bounding volume hierarchies use convex shapes (spheres, axis-aligned bounding boxes) for broad-phase collision detection
• GJK (Gilbert-Johnson-Keerthi) algorithm computes distance between convex shapes
• Enables real-time physics simulations in video games and robotics

Computer graphics

• Convex decomposition simplifies complex 3D models for efficient rendering
• Occlusion culling uses convex hulls to determine object visibility
• Shadow volume algorithms utilize convex polyhedra for real-time shadow rendering
• Convex hull computation aids in level-of-detail generation for 3D models

Convexity in higher dimensions

• Convexity concepts extend beyond three dimensions in computational geometry
• Higher-dimensional convex sets present unique challenges and properties

Hyperplanes and halfspaces

• Hyperplanes generalize the concept of planes to higher dimensions
• Defined by linear equations in n-dimensional space
• Halfspaces represent all points on one side of a hyperplane
• Form building blocks for describing convex sets in higher dimensions

Convex polytopes

• Higher-dimensional analogs of convex polygons and polyhedra
• Bounded by hyperplanes in n-dimensional space
• Vertex representation becomes inefficient as dimensionality increases
• Facet representation often preferred for high-dimensional polytopes

Computational complexity

• Analyzing the efficiency of convex set algorithms informs algorithm selection and problem-solving approaches
• Complexity considerations become crucial as problem sizes and dimensions increase

Time complexity of convex algorithms

• Convex hull algorithms vary in complexity (O(n log n) for Graham scan, O(n^2) for Jarvis march)
• Linear programming algorithms (Simplex, Interior Point) have different worst-case and average-case complexities
• Intersection detection algorithms often have O(n log n) complexity for preprocessing and O(log n) for queries
• Dimensionality significantly impacts algorithm performance in higher-dimensional problems

Space complexity considerations

• Convex hull storage requires O(n) space for output in 2D and 3D cases
• Higher-dimensional convex sets may require exponential space to represent explicitly
• Trade-offs between time and space complexity often arise in convex set algorithms
• Implicit representations and approximation techniques help manage space complexity in high dimensions

Duality and convexity

• Duality concepts in computational geometry often involve convex sets
• Understanding duality provides alternative perspectives on geometric problems

Polar duality

• Maps points to hyperplanes and vice versa in projective geometry
• Preserves incidence relationships and convexity properties
• Transforms convex polytopes into their polar duals
• Enables solving certain geometric problems by working in the dual space

Voronoi diagrams

• Partition space based on proximity to a set of points or objects
• Dual to Delaunay triangulations, which maximize the minimum angle of triangles
• Voronoi cells for convex sites are always convex
• Find applications in nearest neighbor searches and computational biology

Convexity in machine learning

• Convex optimization plays a crucial role in many machine learning algorithms
• Convexity properties enable efficient training and guarantee global optima

Support vector machines

• Utilize convex optimization to find the optimal separating hyperplane
• Kernel trick extends SVM to non-linearly separable data while maintaining convexity
• Soft-margin SVM formulation results in a convex quadratic programming problem
• Convexity ensures efficient training and convergence to global optima

Convex optimization techniques

• Gradient descent converges to global minima for convex functions
• Stochastic gradient descent (SGD) efficiently handles large-scale machine learning problems
• Interior point methods solve convex optimization problems with inequality constraints
• Convex relaxation techniques approximate non-convex problems as convex ones for tractable solutions