Not Frequent
0/8

# Convex Hull

Authors: Benjamin Qi, Neo Wang

### Prerequisites

Smallest convex polygon containing a set of points on a grid.

## Introduction

The Convex Hull is the subset of points that forms the smallest convex polygon which encloses all points in the set. To visualize this, imagine that each point is a pole. Then, imagine what happens if you were to wrap a rope around the outside of all the poles, and then pull infinitely hard, such that the connections between any two points that lie on the edge of the rope are lines. The set of points that touch the rope is the convex hull.

Convex Hull Visualization

Focus Problem – read through this problem before continuing!

## With Graham Scan

### Solution

C++

The Graham Scan algorithm works in 3 steps. First, it sorts all of the $n$ points by their counterclockwise angle around a pivot $P_0$, breaking ties by distance. This algorithm uses the leftmost (and bottommost if there is a tie), point as $P_0$.

We maintain a stack containing the points such that the following invariant holds: every three consecutive points $a,b,c$ of the stack form a counterclockwise turn. In other words, $c$ lies to the left of the line from $a$ to $b$. This condition implies that the points of the stack form the vertices of a convex polygon.

To start the creation of the convex hull, we choose 2 points. The pivot (first point), and the second point based on our initial sorting. After that, we attempt to add each point in $\texttt{cand}$ to the stack.

Denote our stack as $\texttt{hull}$, the top element of $\texttt{hull}$ as $\texttt{hull}[i]$ and $\texttt{cand}[j]$ as the $j$-th sorted candidate point. Before adding $\texttt{cand}[j]$ to the stack, we check whether $\texttt{hull}[i-1] \rightarrow \texttt{hull}[i] \rightarrow \texttt{cand}[j]$ forms a counterclockwise turn.

• If so, then we add $\texttt{cand}[j]$ to the stack and the invariant holds. Continue with $\texttt{cand}[j+1]$.
• Otherwise, $\texttt{hull}[i]$ lies within the convex hull of the other points in the stack along with $\texttt{cand}[j]$, so we pop $\texttt{hull}[i]$ from the stack and continue with $\texttt{cand}[j]$.

Illustration

Worked Example

#include <bits/stdc++.h>
using namespace std;
#define FOR(i, a, b) for (int i = (a); i < (b); i++)#define FORE(i, a, b) for (int i = (a); i <= (b); i++)#define F0R(i, a) for (int i = 0; i < (a); i++)#define trav(a, x) for (auto &a : x)
#define f first

## With Monotone Chain

### This section is not complete.

Any help would be appreciated! Just submit a Pull Request on Github.

## Rotating Caliphers

Focus Problem – read through this problem before continuing!

### This section is not complete.

Any help would be appreciated! Just submit a Pull Request on Github.

## Problems

StatusSourceProblem NameDifficultyTags
CFEasy
Show TagsConvex
PlatNormal
Show TagsConvex
CFNormal
Show TagsConvex, PURS
Old GoldNormal
Show TagsConvex
KattisHard
Show TagsConvex
ACVery Hard
Show TagsConvex

### Join the USACO Forum!

Stuck on a problem, or don't understand a module? Join the USACO Forum and get help from other competitive programmers!