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Triangle with no point inside

Given N points in 2-dimensional space, we need to find three points such that triangle made by choosing these points should not contain any other points inside. All given points will not lie on the same line so solution will always exist.
Examples:



In above diagram possible triangle with no point 
inside can be formed by choosing these triplets,
[(0, 0), (2, 0), (1, 1)]
[(0, 0), (1, 1), (0, 2)]
[(1, 1), (2, 0), (2, 2)]
[(1, 1), (0, 2), (2, 2)]

So any of the above triplets can be the final answer.

The solution is based on the fact that if there exist triangle(s) with no points inside, then we can form a triangle with any random point among all points.
We can solve this problem by searching all three points one by one. The first point can be chosen randomly. After choosing the first point, we need two points such that their slope should be different and no point should lie inside the triangle of these three points. We can do this by choosing the second point as nearest point to the first and third point as second nearest point with the different slope. For doing this, we first iterate over all points and choose the point which is nearest to the first one and designate that as second point of the required triangle. Then we iterate one more time to find point which has different slope and has the smallest distance, which will be the third point of our triangle.

C++

// C/C++ program to find triangle with no point inside
#include <bits/stdc++.h>
using namespace std;
  
// method to get square of distance between
// (x1, y1) and (x2, y2)
int getDistance(int x1, int y1, int x2, int y2)
{
    return (x2 - x1)*(x2 - x1) +
           (y2 - y1)*(y2 - y1);
}
  
// Method prints points which make triangle with no
// point inside
void triangleWithNoPointInside(int points[][2], int N)
{
    //    any point can be chosen as first point of triangle
    int first = 0;
    int second, third;
    int minD = INT_MAX;
  
    // choose nearest point as second point of triangle
    for (int i = 0; i < N; i++)
    {
        if (i == first)
            continue;
  
        // Get distance from first point and choose
        // nearest one
        int d = getDistance(points[i][0], points[i][1],
                    points[first][0], points[first][1]);
        if (minD > d)
        {
            minD = d;
            second = i;
        }
    }
  
    // Pick third point by finding the second closest
    // point with different slope.
    minD = INT_MAX;
    for (int i = 0; i < N; i++)
    {
        // if already chosen point then skip them
        if (i == first || i == second)
            continue;
  
        // get distance from first point
        int d = getDistance(points[i][0], points[i][1],
                     points[first][0], points[first][1]);
  
        /*  the slope of the third point with the first
            point should not be equal to the slope of
            second point with first point (otherwise
            they'll be collinear)     and among all such
            points, we choose point with the smallest
            distance  */
        // here cross multiplication is compared instead
        // of division comparison
        if ((points[i][0] - points[first][0]) *
            (points[second][1] - points[first][1]) !=
            (points[second][0] - points[first][0]) *
            (points[i][1] - points[first][1]) &&
            minD > d)
        {
            minD = d;
            third = i;
        }
    }
  
    cout << points[first][0] << ", "
         << points[first][1] << endl;
    cout << points[second][0] << ", "
         << points[second][1] << endl;
    cout << points[third][0] << ", "
         << points[third][1] << endl;
}
  
// Driver code to test above methods
int main()
{
    int points[][2] = {{0, 0}, {0, 2}, {2, 0},
                       {2, 2}, {1, 1}};
    int N = sizeof(points) / sizeof(points[0]);
    triangleWithNoPointInside(points, N);
    return 0;
}

Java

// Java program to find triangle
// with no point inside
import java.io.*;
  
class GFG 
{
    // method to get square of distance between
    // (x1, y1) and (x2, y2)
    static int getDistance(int x1, int y1, int x2, int y2)
    {
        return (x2 - x1)*(x2 - x1) +
                  (y2 - y1)*(y2 - y1);
    }
      
    // Method prints points which make triangle with no
    // point inside
    static void triangleWithNoPointInside(int points[][], int N)
    {
        // any point can be chosen as first point of triangle
        int first = 0;
        int second =0;
        int third =0;
        int minD = Integer.MAX_VALUE;
      
        // choose nearest point as second point of triangle
        for (int i = 0; i < N; i++)
        {
            if (i == first)
                continue;
      
            // Get distance from first point and choose
            // nearest one
            int d = getDistance(points[i][0], points[i][1],
                        points[first][0], points[first][1]);
            if (minD > d)
            {
                minD = d;
                second = i;
            }
        }
      
        // Pick third point by finding the second closest
        // point with different slope.
        minD = Integer.MAX_VALUE;
        for (int i = 0; i < N; i++)
        {
            // if already chosen point then skip them
            if (i == first || i == second)
                continue;
      
            // get distance from first point
            int d = getDistance(points[i][0], points[i][1],
                        points[first][0], points[first][1]);
      
            /* the slope of the third point with the first
                point should not be equal to the slope of
                second point with first point (otherwise
                they'll be collinear) and among all such
                points, we choose point with the smallest
                distance */
            // here cross multiplication is compared instead
            // of division comparison
            if ((points[i][0] - points[first][0]) *
                (points[second][1] - points[first][1]) !=
                (points[second][0] - points[first][0]) *
                (points[i][1] - points[first][1]) &&
                minD > d)
            {
                minD = d;
                third = i;
            }
        }
      
        System.out.println(points[first][0] + ", "
            + points[first][1]);
        System.out.println(points[second][0]+ ", "
            + points[second][1]) ;
        System.out.println(points[third][0] +", "
            + points[third][1]);
    }
      
    // Driver code 
    public static void main (String[] args) 
    {
        int points[][] = {{0, 0}, {0, 2}, {2, 0},
                         {2, 2}, {1, 1}};
        int N = points.length;
        triangleWithNoPointInside(points, N);
    }
}
  
// This article is contributed by vt_m. 

C#

using System;
  
// C# program to find triangle 
// with no point inside 
  
public class GFG
{
    // method to get square of distance between 
    // (x1, y1) and (x2, y2) 
    public static int getDistance(int x1, int y1, int x2, int y2)
    {
        return (x2 - x1) * (x2 - x1) + (y2 - y1) * (y2 - y1);
    }
  
    // Method prints points which make triangle with no 
    // point inside 
    public static void triangleWithNoPointInside(int[][] points, int N)
    {
        // any point can be chosen as first point of triangle 
        int first = 0;
        int second = 0;
        int third = 0;
        int minD = int.MaxValue;
  
        // choose nearest point as second point of triangle 
        for (int i = 0; i < N; i++)
        {
            if (i == first)
            {
                continue;
            }
  
            // Get distance from first point and choose 
            // nearest one 
            int d = getDistance(points[i][0], points[i][1], points[first][0], points[first][1]);
            if (minD > d)
            {
                minD = d;
                second = i;
            }
        }
  
        // Pick third point by finding the second closest 
        // point with different slope. 
        minD = int.MaxValue;
        for (int i = 0; i < N; i++)
        {
            // if already chosen point then skip them 
            if (i == first || i == second)
            {
                continue;
            }
  
            // get distance from first point 
            int d = getDistance(points[i][0], points[i][1], points[first][0], points[first][1]);
  
            /* the slope of the third point with the first 
                point should not be equal to the slope of 
                second point with first point (otherwise 
                they'll be collinear) and among all such 
                points, we choose point with the smallest 
                distance */
            // here cross multiplication is compared instead 
            // of division comparison 
            if ((points[i][0] - points[first][0]) * (points[second][1] - points[first][1]) != (points[second][0] - points[first][0]) * (points[i][1] - points[first][1]) && minD > d)
            {
                minD = d;
                third = i;
            }
        }
  
        Console.WriteLine(points[first][0] + ", " + points[first][1]);
        Console.WriteLine(points[second][0] + ", " + points[second][1]);
        Console.WriteLine(points[third][0] + ", " + points[third][1]);
    }
  
    // Driver code  
    public static void Main(string[] args)
    {
        int[][] points = new int[][]
        {
            new int[] {0, 0},
            new int[] {0, 2},
            new int[] {2, 0},
            new int[] {2, 2},
            new int[] {1, 1}
        };
        int N = points.Length;
        triangleWithNoPointInside(points, N);
    }
}
  
  // This code is contributed by Shrikant13

/div>


Output:

0, 0
1, 1
0, 2

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.



This article is attributed to GeeksforGeeks.org

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