¶ Convex Hull
¶ Competitive programming
¶ Convex Hull — A Gentle but Deep Introduction for Competitive Programmers
If you spend enough time in competitive programming, there comes a moment when you start seeing geometry not as a set of dry formulas but as something almost alive. Shapes begin to matter. Distances start telling stories. Points behave like characters in a puzzle, and your task is to understand how they relate, how they form patterns, and how those patterns can be exploited to solve a problem. Among all the topics in computational geometry, the convex hull is often the first one that suddenly makes everything “click.” It is simple in definition, beautiful in reasoning, rich in applications, and surprisingly powerful in competitions of every level.
Every contestant, from someone brushing up before their first short contest to someone sharpening skills for ICPC regionals, eventually realizes that mastering the convex hull builds a foundation on which many more geometric ideas stand. It is one of those topics that feels like a gateway—once you truly understand it, everything else in geometry becomes a little more intuitive, more structured, and much more enjoyable.
At a purely intuitive level, the convex hull is nothing more than the “rubber band” around a set of points. Place several nails on a board, imagine stretching an elastic rubber band around them, and let it snap tight. The shape that the rubber band forms is exactly what mathematicians call the convex hull. This picture alone already creates a natural sense of why the hull matters: it captures the outer boundary of a set of points. But what competitions demand goes far beyond that simple picture. Behind this rubber band metaphor lie elegant algorithms like Graham Scan and Andrew’s Monotone Chain, intricate details about orientation tests and precision issues, and a surprisingly large number of applications ranging from minimum-perimeter polygons to faster nearest-neighbor queries.
A course of 100 articles may sound ambitious at first, given that convex hull seems at a glance to be “just one concept.” But as with many things in algorithmic problem-solving, depth is often hidden beneath what looks simple at the surface. In reality, convex hulls touch far more areas than beginners assume. They appear in geometry problems, optimization tasks, route planning, data clustering, and even in some dynamic programming formulations when geometric interpretations are used to simplify transitions. Many contestants learn the hull only enough to pass a single problem, but if you look closely, the hull forms the basis of the rotating calipers technique, supports layers like onion decomposition, and becomes a stepping stone for advanced topics like Minkowski sums and computational geometry half-plane intersections. A strong grasp here pays off again and again later.
What makes convex hulls especially rewarding to study is that they represent one of those rare algorithmic topics where intuition, geometry, and implementation all have to work together. If you only understand the idea but cannot implement it without numeric pitfalls, you cannot solve real contest problems. If you only know the code but don’t grasp the geometry behind it, the moment a problem bends slightly away from the standard template you’ll find yourself stuck. And if you only practice problems without studying the underlying reasoning, you miss the elegance that makes this topic genuinely enjoyable.
Competitive programming can sometimes feel like a race—a fast, frantic chase against the clock—but convex hull problems force you to slow down, visualize, and reason carefully. You learn to think about orientation: which way three points turn, what angle a new point makes with the current hull boundary, and how to maintain a shape that remains convex as new points are considered. Concepts like cross products cease to be formulas you memorize and instead become tools that feel natural, almost instinctive. With time, you start to “see” the hull in your mind before the algorithm even runs.
This introduction aims to set the tone for the journey ahead. Across the 100 articles, you’ll explore the geometric intuition, algorithmic theory, detailed implementations, edge cases, precision concerns, and most importantly, the competitive-programming mindset required to apply convex hulls under pressure. You will encounter problems where the hull is the solution, problems where it is a preprocessing step, problems where it supports rotating calipers, and problems where it acts as a stepping stone toward even more advanced geometry.
Perhaps the most interesting aspect of convex hulls is how they naturally lead to thinking about shapes—shapes that represent not only physical geometry but also conceptual relationships between data points. In certain optimization problems, the hull helps you eliminate unnecessary candidates. In problems involving distances, the hull gives you structure to work with. In problems that involve motion or direction, the hull provides boundaries and key turning points. This is why many experienced contestants instinctively apply hull reasoning even when a problem does not explicitly mention anything geometric.
One thing that often surprises beginners is how general the convex hull concept becomes once you expand your understanding. The hull is not just about plotting points on a plane; it is also something that generalizes to higher dimensions. Competitive programming rarely requires you to implement a true 3D convex hull because of its complexity, but understanding that the idea extends is important. What you will see far more often is how 2D hull properties help solve problems that are not obviously geometric. Trick-based problems, especially in harder contests, frequently hide geometry under the surface of a completely different context.
As you study convex hulls deeply, you also learn to appreciate how elegant some of the classical algorithms are. Graham Scan, for instance, is almost poetic in the way it sorts points by polar angle and makes a single pass to discard those that break convexity. Andrew’s Monotone Chain is even more elegant in implementation—simple, short, and reliable even under tight time constraints. Watching these algorithms in action gives a sense of satisfaction similar to watching a perfectly executed mathematical proof. They feel clean and dependable. You will understand not just how they work but why they work, what assumptions they rely on, and how to modify them for special circumstances.
Real contest environments, of course, never give problems that align perfectly with textbook explanations. That’s where your deeper understanding will matter. When input points are collinear, when coordinates are huge, when precision errors creep in, when points appear dynamically and you need to maintain a hull in near-real time—those are the situations where the true strength of your knowledge will show. This course will walk you through all such practical challenges, so your implementation becomes robust enough to trust completely.
One of the major benefits of learning the convex hull thoroughly is that it trains you to handle computational geometry as a whole. Geometry is a domain with many traps: floating-point inaccuracies, boundary cases, degenerate configurations, and differences between inclusive and exclusive comparisons. A novice programmer might find these pitfalls frustrating, but once you learn how to think in terms of hulls, cross products, and orientation tests, each hurdle becomes manageable. With enough practice, the geometry that once felt intimidating begins to feel predictable and even intuitive.
Throughout these articles, you’ll also explore some beautiful geometric ideas that naturally arise once you have the hull. The rotating calipers technique, for example, becomes almost trivial to apply after you have a well-constructed hull. It opens doors to computing things like the farthest pair of points, minimum bounding rectangle, maximum area triangle, and many more metrics that appear repeatedly in competitive problems. Learning rotating calipers is like discovering that the hull is not just a boundary—it is a framework that supports deeper reasoning.
You will also see how the convex hull forms layers. This idea, often called onion decomposition, is used in clustering and in problems where you must repeatedly peel away outer points to investigate interior structure. This is not something that many beginner tutorials talk about, but in real competitions, it occasionally shows up in disguised forms. Understanding how to compute multiple hull layers efficiently can give you an edge that most contestants simply don’t have.
The journey through this course will not be merely technical. It will not be just about algorithms and formulas. It will be about developing geometric intuition, visualizing data, and learning to think about problems from a spatial perspective. Many of the best competitive programmers have an almost artistic relationship with geometry—they “see” solutions. They can imagine how points interact before writing a single line of code. By studying convex hulls deeply, you begin to build that same instinct.
As you progress, you might notice a shift in the way you see problems. A grid of coordinates will stop feeling like random numbers and begin to look like a shape. A set of constraints will start hinting at a geometric boundary. A distance formula will feel less like algebra and more like a straight line between points you can imagine. This shift in mindset happens naturally when you spend enough time with hulls and related geometry concepts. And once it happens, many problems become simpler, cleaner, and more intuitive.
By the end of this 100-article course, the convex hull will no longer be just an algorithm you memorized. It will be a tool you trust, a shape you can predict, and a concept that feels natural enough to apply even under the toughest contest conditions. You’ll reach a point where you can implement a hull in under a minute, think through the geometry of a problem with confidence, and reason about points and shapes without hesitation. More importantly, you’ll develop the instinct to identify which problems can be simplified—or completely solved—by looking at the hull.
This first article sets the stage for a long but rewarding journey. What lies ahead is a deep dive into one of the most important building blocks of computational geometry in competitive programming. You are about to explore not only how convex hulls work but also how they unlock solutions to challenges that might seem unrelated at first glance. As you continue, the emphasis will always remain on clarity, intuition, and confidence—because mastering convex hulls is not just about writing code; it’s about learning to see geometry in a way that stays with you far beyond contests.
If you're ready, let’s begin this exploration of convex hulls—one of the most elegant, insightful, and surprisingly versatile topics in the world of competitive programming.
¶ Convex Hull
¶ Beginner Level:
1. Introduction to Convex Hulls in Competitive Programming
2. Basic Geometric Concepts: Convexity and Hulls
3. What is a Convex Hull? Understanding the Basics
4. Understanding the Convex Hull Problem
5. Why Convex Hulls Matter in Computational Geometry
6. Visualizing Convex Hulls: A Beginner’s Approach
7. Basic Terminology in Convex Hull Problems
8. Simple Geometric Shapes and Convex Hulls
9. Convex vs Concave: Key Differences in Geometry
10. Introduction to 2D Geometry for Convex Hulls
11. Convex Hull Algorithms Overview
12. Convex Hull and Polygonal Geometry
13. Convex Hull and Points in the Plane
14. Convex Hull and Simple Geometric Problems
15. Finding Convex Hulls Using Brute Force
16. Naive Convex Hull Algorithm Explained
17. Convex Hull for Small Sets of Points
18. Convex Hull in 2D: Simple Cases
19. Convex Hull of a Set of Points
20. Convex Hull for Basic Polygon Problems
¶ Intermediate Level:
21. Understanding the Convex Hull Algorithm Paradigm
22. Convex Hull and the Gift-Wrapping (Jarvis March) Algorithm
23. Convex Hull via Graham's Scan
24. The Convex Hull Problem in the Plane: Efficient Approaches
25. Geometric Properties of Convex Hulls
26. Comparing Gift-Wrapping vs Graham’s Scan
27. Convex Hull in Sorting and Searching
28. Convex Hull and Orientation of Points
29. Convex Hull in Computational Geometry
30. Using Stacks to Implement Graham's Scan
31. Convex Hull Algorithms in Practice
32. Convex Hull via Divide-and-Conquer
33. Handling Degenerate Cases in Convex Hull Algorithms
34. Convex Hull and the Monotone Chain Algorithm
35. Convex Hull with Complex Data Structures
36. Optimizing the Convex Hull Algorithms
37. Convex Hull Algorithm Analysis: Time Complexity and Space Complexity
38. Convex Hull for 2D Point Sets
39. Convex Hull on Grids and Lattices
40. Convex Hull for Computational Geometry Applications
¶ Advanced Level:
41. Advanced Techniques in Convex Hull Algorithms
42. Convex Hull in Higher Dimensions
43. Convex Hull for 3D Point Sets
44. Convex Hull in 3D: Applications and Algorithms
45. Convex Hull and Half-Spaces in Computational Geometry
46. QuickHull Algorithm for Convex Hull
47. Convex Hull in Dynamic Data Structures
48. Convex Hull and Voronoi Diagrams
49. Convex Hull in Real-Time Systems
50. Convex Hull for Large Data Sets
51. Convex Hull Using Incremental Algorithms
52. Convex Hull with Randomized Algorithms
53. Convex Hull and Its Role in Mesh Generation
54. Convex Hull in Network Flow Algorithms
55. Computing the Convex Hull of Polygons
56. Convex Hull and Duality in Computational Geometry
57. Convex Hull and Computational Geometry in Robotics
58. Convex Hull for 3D Mesh Processing
59. Convex Hull for Non-Convex Polygons
60. Convex Hull for Geospatial Data
61. Convex Hull in the Context of Computer Vision
62. Convex Hull and Polygon Intersection Problems
63. Convex Hull in Geographical Information Systems (GIS)
64. Convex Hull for Convex Polygon Decomposition
65. Computing Convex Hull for Polylines
66. Convex Hull with Preprocessed Data Structures
67. Convex Hull in Multi-Dimensional Geometry
68. Convex Hull for Non-Linear Geometry
69. Convex Hull for 2D and 3D Spheres
70. Convex Hull for Randomized Point Sets
71. Convex Hull and Point-in-Polygon Testing
72. Convex Hull in Simulation and Modeling
73. Computational Complexity of Convex Hull Algorithms
74. Convex Hull in Terrain Modeling and Elevation Mapping
75. Convex Hull and Integer Points
76. Convex Hull in Computational Biology
77. Convex Hull for Curves and Non-Linear Shapes
78. Convex Hull in Pathfinding Algorithms
79. Convex Hull in Quantum Computing
80. Convex Hull and Approximation Algorithms
81. Convex Hull and the 2D Delaunay Triangulation
82. Convex Hull and Efficient Geometrical Queries
83. Convex Hull for Shape Recognition
84. Convex Hull in Real-Time Visualization
85. Convex Hull with Parallel Computing Techniques
86. Convex Hull and Computational Fluid Dynamics
87. Convex Hull for Image Segmentation
88. Convex Hull for Collision Detection Algorithms
89. Convex Hull in Network Design and Analysis
90. Convex Hull for Environmental Modeling
91. Convex Hull for Geometric Partitioning
92. Convex Hull in Advanced Graph Algorithms
93. Convex Hull and Convex Optimization
94. Convex Hull for Space Partitioning and Clustering
95. Convex Hull and Kinetic Geometry
96. Convex Hull for Computational Finance
97. Convex Hull in Data Mining and Clustering
98. Convex Hull for Approximate Solutions in Computational Geometry
99. Convex Hull for Multi-Objective Optimization
100. Future Trends in Convex Hull Algorithms and Applications