# convex hull algorithm

> algorithm for computing convex hull

**Wikidata**: [Q5166520](https://www.wikidata.org/wiki/Q5166520)  
**Wikipedia**: [English](https://en.wikipedia.org/wiki/Convex_hull_algorithms)  
**Source**: https://4ort.xyz/entity/convex-hull-algorithm

## Summary
A convex hull algorithm is a computational method used to determine the smallest convex shape that encloses a given set of points in a plane or higher-dimensional space. These algorithms are fundamental in computational geometry, with applications in computer graphics, robotics, and pattern recognition.

## Key Facts
- **Classification**: A type of geometric algorithm, which is a subclass of combinatorial algorithms.
- **Purpose**: Computes the convex hull of a set of points, which is the smallest convex polygon containing all the points.
- **Related Algorithms**:
  - **Monotone Chain**: An algorithm for finding the convex hull of planar points.
  - **Chan's Algorithm**: Another method for computing the convex hull in the plane.
  - **Gift Wrapping Algorithm**: Traces the boundary of the hull to compute it.
  - **Kirkpatrick–Seidel Algorithm**: Computes the convex hull in 𝒪(𝑛 log ℎ) time, where 𝑛 is the number of input points and ℎ is the number of hull points.
  - **Graham Scan**: A widely used algorithm for finding the convex hull of planar points.
- **Wikidata Description**: "Algorithm for computing convex hull."
- **Wikipedia Coverage**: Available in multiple languages, including English, French, Japanese, and Russian.
- **Microsoft Academic ID (Discontinued)**: 2777841862.

## FAQs
### Q: What is the purpose of a convex hull algorithm?
A: It computes the smallest convex shape (polygon or polytope) that encloses a given set of points, useful in geometry, graphics, and optimization.

### Q: What are some well-known convex hull algorithms?
A: Notable examples include Graham Scan, Gift Wrapping, Chan's Algorithm, and the Kirkpatrick–Seidel Algorithm.

### Q: How is the convex hull used in real-world applications?
A: It is applied in collision detection, pattern recognition, computer graphics, and robotics for shape analysis and boundary determination.

### Q: What is the time complexity of convex hull algorithms?
A: Some algorithms, like Kirkpatrick–Seidel, achieve 𝒪(𝑛 log ℎ) time, where 𝑛 is the number of points and ℎ is the hull size.

### Q: Is the convex hull algorithm limited to 2D points?
A: While many algorithms focus on planar (2D) points, extensions exist for higher-dimensional spaces.

## Why It Matters
Convex hull algorithms are foundational in computational geometry, enabling efficient solutions to problems requiring boundary determination or shape analysis. They play a critical role in fields like computer graphics (e.g., collision detection, shape rendering), robotics (path planning), and data analysis (outlier detection). By providing a minimal enclosing shape, these algorithms optimize computations in spatial queries, pattern recognition, and geometric modeling. Their efficiency and versatility make them indispensable in both theoretical research and practical applications, from game development to scientific simulations.

## Notable For
- **Fundamental Role**: A core tool in computational geometry for shape analysis.
- **Diverse Algorithms**: Multiple approaches (e.g., Graham Scan, Gift Wrapping) tailored for different use cases.
- **Efficiency**: Some algorithms achieve near-linear time complexity (e.g., Kirkpatrick–Seidel’s 𝒪(𝑛 log ℎ)).
- **Broad Applications**: Used in computer graphics, robotics, and data science.
- **Theoretical Impact**: Influences research in geometric algorithms and optimization.

## Body
### Definition and Scope
A convex hull algorithm computes the convex hull of a set of points, which is the smallest convex polygon (in 2D) or polytope (in higher dimensions) that contains all the points. The hull is defined as the intersection of all convex sets containing the points.

### Types of Algorithms
Several algorithms exist for computing convex hulls, each with distinct approaches:
- **Graham Scan**: Sorts points by polar angle and constructs the hull incrementally.
- **Gift Wrapping (Jarvis March)**: Iteratively selects the next hull point by checking all candidates.
- **Chan’s Algorithm**: A randomized algorithm with optimal expected time complexity.
- **Kirkpatrick–Seidel Algorithm**: Achieves 𝒪(𝑛 log ℎ) time by leveraging geometric properties.

### Applications
- **Computer Graphics**: Collision detection, shape rendering, and mesh generation.
- **Robotics**: Path planning and obstacle avoidance.
- **Data Science**: Outlier detection and clustering.
- **Geometric Modeling**: Simplifying complex shapes for analysis.

### Technical Details
- **Input**: A set of points in a plane or higher-dimensional space.
- **Output**: The vertices of the convex hull in order.
- **Complexity**: Ranges from 𝒪(𝑛 log 𝑛) for Graham Scan to 𝒪(𝑛 log ℎ) for Kirkpatrick–Seidel.

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