Program to check whether 4 points in a 3-D plane are Coplanar

Given 4 points (x1, y1, z1), (x2, y2, z2), (x3, y3, z3), (x4, y4, z4). The task is to write a program to check whether these 4 points are coplanar or not.

Note: 4 points in a 3-D plane are said to be coplanar if they lies in the same plane.

Examples:

Input:
x1 = 3, y1 = 2, z1 = -5
x2 = -1, y2 = 4, z2 = -3
x3 = -3, y3 = 8, z3 = -5
x4 = -3, y4 = 2, z4 = 1
Output: Coplanar

Input:
x1 = 0, y1 = -1, z1 = -1
x2 = 4, y2 = 5, z2 = 1
x3 = 3, y3 = 9, z3 = 4
x4 = -4, y4 = 4, z4 = 3
Output: Not Coplanar

Approach:

  1. To check whether 4 points are coplanar or not, first of all, find the equation of the plane passing through any three of the given points.
    Approach to find equation of a plane passing through 3 points.
  2. Then, check whether the 4th point satisfies the equation obtained in step 1. That is, putting the value of 4th point in the equation obtained. If it satisfies the equation then the 4 points are Coplanar otherwise not.

Below is the implementation of the above idea:

C++

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// C++ program to check if 4 points 
// in a 3-D plane are Coplanar 
  
#include<bits/stdc++.h>
using namespace std ;
  
// Function to find equation of plane.
void equation_plane(int x1,int y1,int z1,int x2,int y2,int z2,
             int x3, int y3, int z3, int x, int y, int z)
    {
    int a1 = x2 - x1 ;
    int b1 = y2 - y1 ;
    int c1 = z2 - z1 ;
    int a2 = x3 - x1 ;
    int b2 = y3 - y1 ;
    int c2 = z3 - z1 ;
    int a = b1 * c2 - b2 * c1 ;
    int b = a2 * c1 - a1 * c2 ;
    int c = a1 * b2 - b1 * a2 ;
    int d = (- a * x1 - b * y1 - c * z1) ;
        
    // equation of plane is: a*x + b*y + c*z = 0 # 
        
    // checking if the 4th point satisfies 
    // the above equation 
    if(a * x + b * y + c * z + d == 0) 
        cout << "Coplanar" << endl; 
    else
        cout << "Not Coplanar" << endl; 
                   
    }
      
// Driver Code 
int main()
{
       
int x1 = 3 ;
int y1 = 2 ;
int z1 = -5 ;
int x2 = -1 ;
int y2 = 4 ;
int z2 = -3 ;
int x3 = -3 ;
int y3 = 8 ;
int z3 = -5 ;
int x4 = -3 ;
int y4 = 2 ;
int z4 = 1 ;
  
// function calling
equation_plane(x1, y1, z1, x2, y2, z2, x3,  
                            y3, z3, x4, y4, z4) ;                            
return 0;
  
// This code is contributed by ANKITRAI1
}

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Java

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//Java program to check if 4 points 
//in a 3-D plane are Coplanar 
  
public class GFG {
  
    //Function to find equation of plane.
    static void equation_plane(int x1,int y1,int z1,int x2,int y2,int z2,
              int x3, int y3, int z3, int x, int y, int z)
     {
     int a1 = x2 - x1 ;
     int b1 = y2 - y1 ;
     int c1 = z2 - z1 ;
     int a2 = x3 - x1 ;
     int b2 = y3 - y1 ;
     int c2 = z3 - z1 ;
     int a = b1 * c2 - b2 * c1 ;
     int b = a2 * c1 - a1 * c2 ;
     int c = a1 * b2 - b1 * a2 ;
     int d = (- a * x1 - b * y1 - c * z1) ;
          
     // equation of plane is: a*x + b*y + c*z = 0 # 
          
     // checking if the 4th point satisfies 
     // the above equation 
     if(a * x + b * y + c * z + d == 0
         System.out.println("Coplanar"); 
     else
         System.out.println("Not Coplanar"); 
                     
     }
        
    //Driver Code 
    public static void main(String[] args) {
          
        int x1 = 3 ;
        int y1 = 2 ;
        int z1 = -5 ;
        int x2 = -1 ;
        int y2 = 4 ;
        int z2 = -3 ;
        int x3 = -3 ;
        int y3 = 8 ;
        int z3 = -5 ;
        int x4 = -3 ;
        int y4 = 2 ;
        int z4 = 1 ;
  
        //function calling
        equation_plane(x1, y1, z1, x2, y2, z2, x3,  
                                 y3, z3, x4, y4, z4) ;                            
    }
}

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Python3

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# Python program to check if 4 points
# in a 3-D plane are Coplanar
  
# Function to find equation of plane.
def equation_plane(x1, y1, z1, x2, y2, z2, x3, 
                                y3, z3, x, y, z): 
      
    a1 = x2 - x1
    b1 = y2 - y1
    c1 = z2 - z1
    a2 = x3 - x1
    b2 = y3 - y1
    c2 = z3 - z1
    a = b1 * c2 - b2 * c1
    b = a2 * c1 - a1 * c2
    c = a1 * b2 - b1 * a2
    d = (- a * x1 - b * y1 - c * z1)
      
    # equation of plane is: a*x + b*y + c*z = 0 #
      
    # checking if the 4th point satisfies
    # the above equation
    if(a * x + b * y + c * z + d == 0):
        print("Coplanar")
    else:
        print("Not Coplanar")
      
      
# Driver Code 
x1 = 3
y1 = 2
z1 = -5
x2 = -1
y2 = 4
z2 = -3
x3 = -3
y3 = 8
z3 = -5
x4 = -3
y4 = 2
z4 = 1
equation_plane(x1, y1, z1, x2, y2, z2, x3, 
                            y3, z3, x4, y4, z4)

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C#

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// C# program to check if 4 points 
// in a 3-D plane are Coplanar
using System;
  
class GFG
{
  
// Function to find equation of plane. 
static void equation_plane(int x1, int y1, int z1,
                           int x2, int y2, int z2, 
                           int x3, int y3, int z3,
                           int x, int y, int z) 
    int a1 = x2 - x1 ; 
    int b1 = y2 - y1 ; 
    int c1 = z2 - z1 ; 
    int a2 = x3 - x1 ; 
    int b2 = y3 - y1 ; 
    int c2 = z3 - z1 ; 
    int a = b1 * c2 - b2 * c1 ; 
    int b = a2 * c1 - a1 * c2 ; 
    int c = a1 * b2 - b1 * a2 ; 
    int d = (- a * x1 - b * y1 - c * z1) ; 
          
    // equation of plane is: a*x + b*y + c*z = 0 # 
          
    // checking if the 4th point satisfies 
    // the above equation 
    if(a * x + b * y + c * z + d == 0) 
        Console.WriteLine("Coplanar"); 
    else
        Console.WriteLine("Not Coplanar"); 
                      
      
// Driver Code 
static public void Main ()
{
    int x1 = 3 ; 
    int y1 = 2 ; 
    int z1 = -5 ; 
    int x2 = -1 ; 
    int y2 = 4 ; 
    int z2 = -3 ; 
    int x3 = -3 ; 
    int y3 = 8 ; 
    int z3 = -5 ; 
    int x4 = -3 ; 
    int y4 = 2 ; 
    int z4 = 1 ; 
  
    //function calling 
    equation_plane(x1, y1, z1, x2, y2, z2, 
                   x3, y3, z3, x4, y4, z4);                         
  
// This code is contributed by jit_t

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PHP

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<?php
// PHP program to check if 4 points 
// in a 3-D plane are Coplanar 
  
// Function to find equation of plane.
function equation_plane($x1, $y1, $z1, $x2
                        $y2, $z2, $x3, $y3
                        $z3, $x, $y, $z)
{
    $a1 = $x2 - $x1;
    $b1 = $y2 - $y1;
    $c1 = $z2 - $z1;
    $a2 = $x3 - $x1;
    $b2 = $y3 - $y1;
    $c2 = $z3 - $z1;
    $a = $b1 * $c2 - $b2 * $c1;
    $b = $a2 * $c1 - $a1 * $c2;
    $c = $a1 * $b2 - $b1 * $a2;
    $d = (- $a * $x1 - $b
            $y1 - $c * $z1);
      
    // equation of plane is:
    // a*x + b*y + c*z = 0 # 
      
    // checking if the 4th point  
    // satisfies the above equation 
    if($a * $x + $b * $y
       $c * $z + $d == 0) 
        echo ("Coplanar"); 
    else
        echo ("Not Coplanar"); 
}
      
// Driver Code 
$x1 = 3; $y1 = 2; $z1 = -5;
$x2 = -1; $y2 = 4; $z2 = -3;
$x3 = -3; $y3 = 8; $z3 = -5;
$x4 = -3; $y4 = 2; $z4 = 1;
  
// function calling
equation_plane($x1, $y1, $z1
               $x2, $y2, $z2
               $x3, $y3, $z3
               $x4, $y4, $z4);                     
  
// This code is contributed 
// by Shivi_Aggarwal
?>

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Output:

Coplanar


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