Perimeter of Convex hull for a given set of points

Last Updated : 06 Apr, 2023

Given n 2-D points points[], the task is to find the perimeter of the convex hull for the set of points. A convex hull for a set of points is the smallest convex polygon that contains all the points. Examples:

Input: points[] = {{0, 3}, {2, 2}, {1, 1}, {2, 1}, {3, 0}, {0, 0}, {3, 3}} Output: 12 Input: points[] = {{0, 2}, {2, 1}, {3, 1}, {3, 7}} Output: 15.067

Approach: Monotone chain algorithm constructs the convex hull in O(n * log(n)) time. We have to sort the points first and then calculate the upper and lower hulls in O(n) time. The points will be sorted with respect to x-coordinates (with respect to y-coordinates in case of a tie in x-coordinates), we will then find the left most point and then try to rotate in clockwise direction and find the next point and then repeat the step until we reach the rightmost point and then again rotate in the clockwise direction and find the lower hull. We will then find the perimeter of the convex hull using the points on the convex hull which can be done in O(n) time as the points are already sorted in clockwise order. Below is the implementation of the above approach:

CPP

// C++ implementation of the approach #include <bits/stdc++.h> #define llu long long int using namespace std;  struct Point {      llu x, y;      bool operator<(Point p)     {         return x < p.x || (x == p.x && y < p.y);     } };  // Cross product of two vectors OA and OB // returns positive for counter clockwise // turn and negative for clockwise turn llu cross_product(Point O, Point A, Point B) {     return (A.x - O.x) * (B.y - O.y)         - (A.y - O.y) * (B.x - O.x); }  // Returns a list of points on the convex hull // in counter-clockwise order vector<Point> convex_hull(vector<Point> A) {     int n = A.size(), k = 0;      if (n <= 3)         return A;      vector<Point> ans(2 * n);      // Sort points lexicographically     sort(A.begin(), A.end());      // Build lower hull     for (int i = 0; i < n; ++i) {          // If the point at K-1 position is not a part         // of hull as vector from ans[k-2] to ans[k-1]         // and ans[k-2] to A[i] has a clockwise turn         while (k >= 2             && cross_product(ans[k - 2],                              ans[k - 1], A[i]) <= 0)             k--;         ans[k++] = A[i];     }      // Build upper hull     for (size_t i = n - 1, t = k + 1; i > 0; --i) {          // If the point at K-1 position is not a part         // of hull as vector from ans[k-2] to ans[k-1]         // and ans[k-2] to A[i] has a clockwise turn         while (k >= t             && cross_product(ans[k - 2],                          ans[k - 1], A[i - 1]) <= 0)             k--;         ans[k++] = A[i - 1];     }      // Resize the array to desired size     ans.resize(k - 1);      return ans; }  // Function to return the distance between two points double dist(Point a, Point b) {     return sqrt((a.x - b.x) * (a.x - b.x)                 + (a.y - b.y) * (a.y - b.y)); }  // Function to return the perimeter of the convex hull double perimeter(vector<Point> ans) {     double perimeter = 0.0;      // Find the distance between adjacent points     for (int i = 0; i < ans.size() - 1; i++) {         perimeter += dist(ans[i], ans[i + 1]);     }      // Add the distance between first and last point     perimeter += dist(ans[0], ans[ans.size() - 1]);      return perimeter; }  // Driver code int main() {     vector<Point> points;      // Add points     points.push_back({ 0, 3 });     points.push_back({ 2, 2 });     points.push_back({ 1, 1 });     points.push_back({ 2, 1 });     points.push_back({ 3, 0 });     points.push_back({ 0, 0 });     points.push_back({ 3, 3 });      // Find the convex hull     vector<Point> ans = convex_hull(points);      // Find the perimeter of convex polygon     cout << perimeter(ans);      return 0; }

Java

import java.util.*; import java.lang.*; import java.io.*;  class Point {     int x, y;     Point(int x, int y) {         this.x = x;         this.y = y;     } }  class Main {     static int cross_product(Point o, Point a, Point b) {         return (a.x - o.x) * (b.y - o.y) - (a.y - o.y) * (b.x - o.x);     }      static List<Point> convex_hull(List<Point> points) {         int n = points.size();         if (n <= 3)             return points;          Collections.sort(points, new Comparator<Point>() {             @Override             public int compare(Point p1, Point p2) {                 if (p1.x == p2.x)                     return p1.y - p2.y;                 return p1.x - p2.x;             }         });          Point[] ans = new Point[2 * n];         int k = 0;          // Build lower hull         for (int i = 0; i < n; i++) {             while (k >= 2 && cross_product(ans[k-2], ans[k-1], points.get(i)) <= 0)                 k--;             ans[k++] = points.get(i);         }          // Build upper hull         int t = k + 1;         for (int i = n-2; i >= 0; i--) {             while (k >= t && cross_product(ans[k-2], ans[k-1], points.get(i)) <= 0)                 k--;             ans[k++] = points.get(i);         }          List<Point> result = new ArrayList<>();         for (int i = 0; i < k-1; i++)             result.add(ans[i]);         return result;     }      static double dist(Point a, Point b) {         return Math.sqrt(Math.pow(a.x - b.x, 2) + Math.pow(a.y - b.y, 2));     }      static double perimeter(List<Point> points) {         int n = points.size();         double perimeter = 0.0;         for (int i = 0; i < n-1; i++)             perimeter += dist(points.get(i), points.get(i+1));         perimeter += dist(points.get(0), points.get(n-1));         return perimeter;     }      public static void main(String[] args) throws java.lang.Exception {         List<Point> points = new ArrayList<>();         points.add(new Point(0, 3));         points.add(new Point(2, 2));         points.add(new Point(1, 1));         points.add(new Point(2, 1));         points.add(new Point(3, 0));         points.add(new Point(0, 0));         points.add(new Point(3, 3));          List<Point> convex_points = convex_hull(points);         System.out.println(perimeter(convex_points));     } }

Python3

from typing import List from math import sqrt  class Point:     def __init__(self, x: int, y: int):         self.x = x         self.y = y  def cross_product(o: Point, a: Point, b: Point) -> int:     return (a.x - o.x) * (b.y - o.y) - (a.y - o.y) * (b.x - o.x)  def convex_hull(points: List[Point]) -> List[Point]:     n = len(points)     if n <= 3:         return points      points.sort(key=lambda p: (p.x, p.y))     ans = [None] * (2 * n)      # Build lower hull     k = 0     for i in range(n):         while k >= 2 and cross_product(ans[k-2], ans[k-1], points[i]) <= 0:             k -= 1         ans[k] = points[i]         k += 1      # Build upper hull     t = k + 1     for i in range(n-2, -1, -1):         while k >= t and cross_product(ans[k-2], ans[k-1], points[i]) <= 0:             k -= 1         ans[k] = points[i]         k += 1      ans = ans[:k-1]     return ans  def dist(a: Point, b: Point) -> float:     return sqrt((a.x - b.x) ** 2 + (a.y - b.y) ** 2)  def perimeter(points: List[Point]) -> float:     n = len(points)     perimeter = 0.0     for i in range(n-1):         perimeter += dist(points[i], points[i+1])     perimeter += dist(points[0], points[n-1])     return perimeter  # Driver code points = [Point(0, 3), Point(2, 2), Point(1, 1), Point(2, 1), Point(3, 0), Point(0, 0), Point(3, 3)] convex_points = convex_hull(points) print(perimeter(convex_points))

JavaScript

class Point {   constructor(x, y) {     this.x = x;     this.y = y;   } }  function crossProduct(o, a, b) {   return (a.x - o.x) * (b.y - o.y) - (a.y - o.y) * (b.x - o.x); }  function convexHull(points) {   let n = points.length;   if (n <= 3) {     return points;   }    points.sort((a, b) => a.x - b.x || a.y - b.y);   let ans = new Array(2 * n).fill(null);    // Build lower hull   let k = 0;   for (let i = 0; i < n; i++) {     while (k >= 2 && crossProduct(ans[k-2], ans[k-1], points[i]) <= 0) {       k--;     }     ans[k] = points[i];     k++;   }    // Build upper hull   let t = k + 1;   for (let i = n - 2; i >= 0; i--) {     while (k >= t && crossProduct(ans[k-2], ans[k-1], points[i]) <= 0) {       k--;     }     ans[k] = points[i];     k++;   }    ans = ans.slice(0, k-1);   return ans; }  function dist(a, b) {   return Math.sqrt((a.x - b.x) ** 2 + (a.y - b.y) ** 2); }  function perimeter(points) {   let n = points.length;   let perimeter = 0.0;   for (let i = 0; i < n - 1; i++) {     perimeter += dist(points[i], points[i+1]);   }   perimeter += dist(points[0], points[n-1]);   return perimeter; }  // Driver code let points = [new Point(0, 3), new Point(2, 2), new Point(1, 1), new Point(2, 1), new Point(3, 0), new Point(0, 0), new Point(3, 3)]; let convex_points = convexHull(points); console.log(perimeter(convex_points));

using System; using System.Collections.Generic; using System.Linq;  class Point {     public int x, y;     public Point(int x, int y) {         this.x = x;         this.y = y;     } }  class MainClass {          // Cross product of two vectors OA and OB     // returns positive for counter clockwise     // turn and negative for clockwise turn     static int cross_product(Point o, Point a, Point b) {         return (a.x - o.x) * (b.y - o.y) - (a.y - o.y) * (b.x - o.x);     }        // Returns a list of points on the convex hull     // in counter-clockwise order     static List<Point> convex_hull(List<Point> points) {         int n = points.Count;         if (n <= 3)             return points;          points.Sort((p1, p2) => {             if (p1.x == p2.x)                 return p1.y - p2.y;             return p1.x - p2.x;         });          Point[] ans = new Point[2 * n];         int k = 0;          // Build lower hull         for (int i = 0; i < n; i++) {                          // If the point at K-1 position is not a part             // of hull as vector from ans[k-2] to ans[k-1]             // and ans[k-2] to A[i] has a clockwise turn             while (k >= 2 && cross_product(ans[k-2], ans[k-1], points[i]) <= 0)                 k--;             ans[k++] = points[i];         }          // Build upper hull         int t = k + 1;         for (int i = n-2; i >= 0; i--) {                          // If the point at K-1 position is not a part             // of hull as vector from ans[k-2] to ans[k-1]             // and ans[k-2] to A[i] has a clockwise turn             while (k >= t && cross_product(ans[k-2], ans[k-1], points[i]) <= 0)                 k--;             ans[k++] = points[i];         }          List<Point> result = new List<Point>();         for (int i = 0; i < k-1; i++)             result.Add(ans[i]);         return result;     }        // Function to return the distance between two points     static double dist(Point a, Point b) {         return Math.Sqrt(Math.Pow(a.x - b.x, 2) + Math.Pow(a.y - b.y, 2));     }        // Function to return the perimeter of the convex hull     static double perimeter(List<Point> points) {         int n = points.Count;         double perimeter = 0.0;         for (int i = 0; i < n-1; i++)             perimeter += dist(points[i], points[i+1]);         perimeter += dist(points[0], points[n-1]);         return perimeter;     }        // Driver code     public static void Main(string[] args) {         List<Point> points = new List<Point>();                  // Add points         points.Add(new Point(0, 3));         points.Add(new Point(2, 2));         points.Add(new Point(1, 1));         points.Add(new Point(2, 1));         points.Add(new Point(3, 0));         points.Add(new Point(0, 0));         points.Add(new Point(3, 3));         // Find the convex hull         List<Point> convex_points = convex_hull(points);                  // Find the perimeter of convex polygon         Console.WriteLine(perimeter(convex_points));     } }

Output:

Time Complexity : O(nlogn), where n is the number of points in the input vector.

Auxiliary Space Complexity : O(n), where n is the number of points in the input vector.

Check if the given point lies inside given N points of a Convex Polygon

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Perimeter of Convex hull for a given set of points

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