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

Last Updated : 17 Sep, 2024

Given coordinates of the N points of a Convex Polygon. The task is to check if the given point (X, Y) lies inside the polygon. Examples:

Input: N = 7, Points: {(1, 1), (2, 1), (3, 1), (4, 1), (4, 2), (4, 3), (4, 4)}, Query: X = 3, Y = 2 Below is the image of plotting of the given points: Output: YES Input: N = 7, Points: {(1, 1), (2, 1), (3, 1), (4, 1), (4, 2), (4, 3), (4, 4)}, Query: X = 3, Y = 9 Output: NO

Approach: The idea is to use Graham Scan Algorithm to find if the given point lies inside the Convex Polygon or not. Below are some of the observations:

Suppose the point (X, Y) is a point in the set of points of the convex polygon. If the Graham Scan Algorithm is used on this set of points, another set of points would be obtained, which makes up the Convex Hull.
If the point (X, Y) lies inside the polygon, it won’t lie on the Convex Hull and hence won’t be present in the newly generated set of points of the Convex Hull.
If the point (X, Y) lies outside the polygon, it will then lie on the Convex Hull formed and hence would be present in the newly generated set of points of the Convex Hull.

Below are the steps to solve the problem:

Sort the given points along with the query point in the increasing order of their abscissa values. If the abscissa values(x-coordinates) of any two points are the same, then sort them on the basis of their ordinate value.
Set the bottom-left point as the start point and top-right point as the end point of the convex hull.
Iterate over all the points and find out the points, forming the convex polygon, that lies in between the start and endpoints in the clockwise direction. Store these points in a vector.
Iterate over all the points and find out the points, forming the convex polygon, that lies in between the start and endpoints in the counter-clockwise direction. Store these points in the vector.
Check if the query point exists in the vector then the point lies outside the convex hull. So return “No”.
If the point doesn’t exist in the vector, then the point lies inside the convex hull print “Yes”.

Below is the implementation based on the above approach:

CPP

// C++ program for the above approach #include <bits/stdc++.h> using namespace std;    // Sorting Function to sort points bool cmp(pair<int, int>& a,          pair<int, int>& b) {        if (a.first == b.first)         return a.second < b.second;     return a.first < b.first; }    // Function To Check Clockwise // Orientation int cw(pair<int, int>& a,        pair<int, int>& b,        pair<int, int>& c) {        int p = a.first * (b.second - c.second)             + b.first * (c.second - a.second)             + c.first * (a.second - b.second);        return p < 0ll; }    // Function To Check Counter // Clockwise Orientation int ccw(pair<int, int>& a,         pair<int, int>& b,         pair<int, int>& c) {        int p = a.first * (b.second - c.second)             + b.first * (c.second - a.second)             + c.first * (a.second - b.second);        return p > 0ll; }    // Graham Scan algorithm to find Convex // Hull from given points vector<pair<int, int> > convexHull(     vector<pair<int, int> >& v) {     // Sort the points     sort(v.begin(),          v.end(), cmp);        int n = v.size();     if (n <= 3)         return v;        // Set starting and ending points as     // left bottom and top right     pair<int, int> p1 = v[0];     pair<int, int> p2 = v[n - 1];        // Vector to store points in     // upper half and lower half     vector<pair<int, int> > up, down;        // Insert StartingEnding Points     up.push_back(p1);     down.push_back(p1);        // Iterate over points     for (int i = 1; i < n; i++) {            if (i == n - 1 || !ccw(p1, v[i], p2)) {                while (up.size() > 1                    && ccw(up[up.size() - 2],                           up[up.size() - 1],                           v[i])) {                    // Exclude this point                 // if we can form better                    up.pop_back();             }                up.push_back(v[i]);         }            if (i == n - 1 || !cw(p1, v[i], p2)) {                while (down.size() > 1                    && cw(down[down.size() - 2],                          down[down.size() - 1],                          v[i])) {                    // Exclude this point                 // if we can form better                 down.pop_back();             }             down.push_back(v[i]);         }     }        // Combine upper and  lower half     for (int i = down.size() - 2;          i > 0; i--)         up.push_back(down[i]);        // Remove duplicate points     up.resize(unique(up.begin(),                      up.end())               - up.begin());        // Return the points on Convex Hull     return up; }    // Function to find if point lies inside // a convex polygon bool isInside(vector<pair<int, int> > points,               pair<int, int> query) {     // Include the query point in the     // polygon points     points.push_back(query);        // Form a convex hull from the points     points = convexHull(points);        // Iterate over the points     // of convex hull     for (auto x : points) {            // If the query point lies         // on the convex hull         // then it wasn't inside         if (x == query)             return false;     }        // Otherwise it was Inside     return true; }    // Driver Code int main() {        // Points of the polygon     // given in any order     int n = 7;     vector<pair<int, int> > points;        points = { { 1, 1 }, { 2, 1 }, { 3, 1 },                { 4, 1 }, { 4, 2 }, { 4, 3 },                { 4, 4 } };        // Query Points     pair<int, int> query = { 3, 2 };        // Check if its inside     if (isInside(points, query)) {         cout << "YES" << endl;     }     else {         cout << "NO" << endl;     }        return 0; }

Java

import java.util.ArrayList; import java.util.Arrays; import java.util.List;  public class GFG {     // Function To Check Clockwise Orientation     static boolean cw(int[] a, int[] b, int[] c)     {         int p = a[0] * (b[1] - c[1]) + b[0] * (c[1] - a[1])                 + c[0] * (a[1] - b[1]);          return p < 0;     }      // Function To Check Counter Clockwise Orientation     static boolean ccw(int[] a, int[] b, int[] c)     {         int p = a[0] * (b[1] - c[1]) + b[0] * (c[1] - a[1])                 + c[0] * (a[1] - b[1]);          return p > 0;     }      // Graham Scan Algorithm To Find Convex Hull From Given     // Points     static int[][] convexHull(int[][] v)     {         // Sort The Points         Arrays.sort(v, (a, b) -> a[0] - b[0]);          int n = v.length;         if (n <= 3) {             return v;         }          // Set Starting And Ending Points As Left Bottom And         // Top Right         int[] p1 = v[0];         int[] p2 = v[n - 1];          // Vector To Store Points In Upper Half And Lower         // Half         List<int[]> up = new ArrayList<int[]>();         List<int[]> down = new ArrayList<int[]>();          // Insert Starting/Ending Points         up.add(p1);         down.add(p1);          // Iterate Over Points         for (int i = 1; i < n; i++) {             if (i == n - 1 || !ccw(p1, v[i], p2)) {                 while (up.size() > 1                        && ccw(up.get(up.size() - 2),                               up.get(up.size() - 1),                               v[i])) {                     // Exclude This Point If We Can Form                     // Better                     up.remove(up.size() - 1);                 }                  up.add(v[i]);             }              if (i == n - 1 || !cw(p1, v[i], p2)) {                 while (down.size() > 1                        && cw(down.get(down.size() - 2),                              down.get(down.size() - 1),                              v[i])) {                     // Exclude This Point If We Can Form                     // Better                     down.remove(down.size() - 1);                 }                  down.add(v[i]);             }         }          // Combine Upper And Lower Half         for (int i = down.size() - 2; i >= 0; i--) {             up.add(down.get(i));         }          // Return The Points On Convex Hull         return up.toArray(new int[0][]);     }      // Function To Find If Point Lies Inside A convex     // polygon     static boolean isInside(int[][] points, int[] query)     {         // Include the query point in the         // polygon points         int[][] points1 = new int[points.length + 1][];         for (int i = 0; i < points.length; i++)             points1[i] = points[i];         points1[points.length] = query;          // Form a convex hull from the points         points = convexHull(points);          // Iterate over the points         // of convex hull         for (int[] x : points) {              // If the query point lies             // on the convex hull             // then it wasn't inside             if (Arrays.equals(x, query))                 return false;         }          // Otherwise it was Inside         return true;     }      // Driver Code      public static void main(String[] args)     {         // Points of the polygon         // given in any order         int n = 7;         int[][] points             = { new int[] { 1, 1 }, new int[] { 2, 1 },                 new int[] { 3, 1 }, new int[] { 4, 1 },                 new int[] { 4, 2 }, new int[] { 4, 3 },                 new int[] { 4, 4 } };          // Query Points         int[] query = { 3, 2 };          // Check if its inside         if (isInside(points, query)) {             System.out.println("YES");         }         else {             System.out.println("NO");         }     } }

Python

# Python3 program for the above approach  # Function To Check Clockwise # Orientation def cw(a, b, c):     p = a[0] * (b[1] - c[1]) + b[0] * (c[1] - a[1]) + c[0] * (a[1] - b[1]);        return p < 0;  # Function To Check Counter # Clockwise Orientation def ccw(a, b, c):     p = a[0] * (b[1] - c[1]) + b[0] * (c[1] - a[1]) + c[0] * (a[1] - b[1]);        return p > 0;  # Graham Scan algorithm to find Convex # Hull from given points def convexHull(v):      # Sort the points     v.sort();             n = len(v);     if (n <= 3):         return v;        # Set starting and ending points as     # left bottom and top right     p1 = v[0];     p2 = v[n - 1];        # Vector to store points in     # upper half and lower half     up = []     down = [];        # Insert StartingEnding Points     up.append(tuple(p1));     down.append(p1);        # Iterate over points     for i in range(1, n):                  if i == n - 1 or (not ccw(p1, v[i], p2)):              while len(up) > 1 and ccw(up[len(up) - 2], up[len(up) - 1], v[i]):                   # Exclude this point                 # if we can form better                    up.pop();                             up.append(tuple(v[i]));                     if i == n - 1 and  (not cw(p1, v[i], p2)):                 while (len(down) > 1) and cw(down[len(down) - 2], down[len(down) - 1], v[i]):                    # Exclude this point                 # if we can form better                 down.pop();                          down.append(v[i]);              # Combine upper and  lower half     for i in range(len(down) - 2, -1, -1):         up.append(tuple(down[i]));        # Remove duplicate points     up = set(up)     up = list(up)        # Return the points on Convex Hull     return up;  # Function to find if point lies inside # a convex polygon def isInside( points, query):      # Include the query point in the     # polygon points     points.append(query);        # Form a convex hull from the points     points = convexHull(points);        # Iterate over the points     # of convex hull     for x in points:             # If the query point lies         # on the convex hull         # then it wasn't inside         if x == query:             return False;          # Otherwise it was Inside     return True;  # Driver Code  # Points of the polygon # given in any order n = 7; points = [[1, 1 ], [2, 1 ], [ 3, 1 ], [ 4, 1 ], [ 4, 2 ], [4, 3 ], [ 4, 4 ]];    # Query Points query = [ 3, 2 ];    # Check if its inside if (isInside(points, query)) :     print("YES");      else :     print("NO");        # This code is contributed by phasing17.

using System; using System.Linq; using System.Collections.Generic;  class GFG {     // Function To Check Clockwise Orientation     static bool cw(int[] a, int[] b, int[] c)     {         int p = a[0] * (b[1] - c[1]) + b[0] * (c[1] - a[1])                 + c[0] * (a[1] - b[1]);          return p < 0;     }      // Function To Check Counter Clockwise Orientation     static bool ccw(int[] a, int[] b, int[] c)     {         int p = a[0] * (b[1] - c[1]) + b[0] * (c[1] - a[1])                 + c[0] * (a[1] - b[1]);          return p > 0;     }      // Graham Scan Algorithm To Find Convex Hull From Given     // Points     static int[][] convexHull(int[][] v)     {         // Sort The Points         Array.Sort(v, (a, b) => a[0] - b[0]);          int n = v.Length;         if (n <= 3) {             return v;         }          // Set Starting And Ending Points As Left Bottom And         // Top Right         int[] p1 = v[0];         int[] p2 = v[n - 1];          // Vector To Store Points In Upper Half And Lower         // Half         List<int[]> up = new List<int[]>();         List<int[]> down = new List<int[]>();          // Insert Starting/Ending Points         up.Add(p1);         down.Add(p1);          // Iterate Over Points         for (int i = 1; i < n; i++) {             if (i == n - 1 || !ccw(p1, v[i], p2)) {                 while (up.Count > 1                        && ccw(up[up.Count - 2],                               up[up.Count - 1], v[i])) {                     // Exclude This Point If We Can Form                     // Better                     up.RemoveAt(up.Count - 1);                 }                  up.Add(v[i]);             }              if (i == n - 1 || !cw(p1, v[i], p2)) {                 while (down.Count > 1                        && cw(down[down.Count - 2],                              down[down.Count - 1], v[i])) {                     // Exclude This Point If We Can Form                     // Better                     down.RemoveAt(down.Count - 1);                 }                  down.Add(v[i]);             }         }          // Combine Upper And Lower Half         for (int i = down.Count - 2; i >= 0; i--) {             up.Add(down[i]);         }          // Remove Duplicate Points         up = new List<int[]>(new HashSet<int[]>(up));          // Return The Points On Convex Hull         return up.ToArray();     }      // Function To Find If Point Lies Inside A     //  convex polygon     static bool isInside(int[][] points, int[] query)     {         // Include the query point in the         // polygon points         int[][] points1 = new int[points.Length + 1][];         for (int i = 0; i < points.Length; i++)             points1[i] = points[i];         points1[points.Length] = query;          // Form a convex hull from the points         points = convexHull(points);          // Iterate over the points         // of convex hull         foreach(var x in points)         {              // If the query point lies             // on the convex hull             // then it wasn't inside             if (Enumerable.SequenceEqual(x, query))                 return false;         }          // Otherwise it was Inside         return true;     }      // Driver Code      public static void Main(string[] args)     {         // Points of the polygon         // given in any order         int n = 7;         int[][] points             = { new int[] { 1, 1 }, new int[] { 2, 1 },                 new int[] { 3, 1 }, new int[] { 4, 1 },                 new int[] { 4, 2 }, new int[] { 4, 3 },                 new int[] { 4, 4 } };          // Query Points         int[] query = { 3, 2 };          // Check if its inside         if (isInside(points, query)) {             Console.WriteLine("YES");         }         else {             Console.WriteLine("NO");         }     } }

JavaScript

// JS program for the above approach     // Sorting Function to sort points function cmp(a, b) {        if (a[0] == b[0])         return a[1] < b[1];     return a[0] < b[0]; }    // Function To Check Clockwise // Orientation function cw(a, b, c) {        let p = a[0] * (b[1] - c[1])             + b[0] * (c[1] - a[1])             + c[0] * (a[1] - b[1]);        return p < 0; }    // Function To Check Counter // Clockwise Orientation function ccw(a, b, c) {        let p = a[0] * (b[1] - c[1])             + b[0] * (c[1] - a[1])             + c[0] * (a[1] - b[1]);        return p > 0; }    // Graham Scan algorithm to find Convex // Hull from given points function convexHull(v) {     // Sort the points     v.sort(cmp);             let n = v.length;     if (n <= 3)         return v;        // Set starting and ending points as     // left bottom and top right     let p1 = v[0];     let p2 = v[n - 1];        // Vector to store points in     // upper half and lower half     let  up = [], down = [];        // Insert StartingEnding Points     up.push(p1);     down.push(p1);        // Iterate over points     for (var i = 1; i < n; i++) {            if (i == n - 1 || !ccw(p1, v[i], p2)) {                while (up.length > 1                    && ccw(up[up.length - 2],                           up[up.length - 1],                           v[i])) {                    // Exclude this point                 // if we can form better                    up.pop();             }                up.push(v[i]);         }            if (i == n - 1 || !cw(p1, v[i], p2)) {                while (down.length > 1                    && cw(down[down.length - 2],                          down[down.length - 1],                          v[i])) {                    // Exclude this point                 // if we can form better                 down.pop();             }             down.push(v[i]);         }     }        // Combine upper and  lower half     for (var i = down.length - 2;          i > 0; i--)         up.push(down[i]);        // Remove duplicate points     up = new Set(up)     up = Array.from(up)        // Return the points on Convex Hull     return up; }    // Function to find if point lies inside // a convex polygon function isInside( points, query) {     // Include the query point in the     // polygon points     points.push(query);        // Form a convex hull from the points     points = convexHull(points);        // Iterate over the points     // of convex hull     for (let x of points) {            // If the query point lies         // on the convex hull         // then it wasn't inside         if (x == query)             return false;     }        // Otherwise it was Inside     return true; }    // Driver Code         // Points of the polygon     // given in any order let n = 7; let  points = [[1, 1 ], [2, 1 ], [ 3, 1 ],                [ 4, 1 ], [ 4, 2 ], [4, 3 ],                [ 4, 4 ]];        // Query Points     let query = [ 3, 2 ];        // Check if its inside     if (isInside(points, query)) {         console.log("YES");     }     else {        console.log("NO");     }

Output

YES

Time Complexity: O(N * log(N)) Auxiliary Space: O(N)

Tangents between two Convex Polygons

s5960r

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

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