检查 3-D 平面中的 4 个点是否共面的程序

给定 4 个点 (x1, y1, z1), (x2, y2, z2), (x3, y3, z3), (x4, y4, z4)。任务是编写一个程序来检查这 4 个点是否共面。
注意：如果 3-D 平面中的 4 个点位于同一平面内，则称它们共面。

例子：

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

方法：

1. 要检查 4 个点是否共面，首先要找到通过给定点中的任意三个点的平面的方程。
   求通过3点的平面方程的方法。
2. 然后，检查第4个点是否满足步骤1中得到的方程。即，将第4个点的值代入得到的方程中。如果它满足方程，则 4 个点共面，否则不共面。

下面是上述想法的实现：

C++

// C++ program to check if 4 points
// in a 3-D plane are Coplanar
 
#include
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
}

Java

//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) ;                           
    }
}

Python3

# 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)

C#

// 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

Javascript

//javascript 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)
    {
        var a1 = x2 - x1;
        var b1 = y2 - y1;
        var c1 = z2 - z1;
        var a2 = x3 - x1;
        var b2 = y3 - y1;
        var c2 = z3 - z1;
        var a = b1 * c2 - b2 * c1;
        var b = a2 * c1 - a1 * c2;
        var c = a1 * b2 - b1 * a2;
        var d = (-a * x1 - b * y1 - c * z1);
 
        // equation of plane is: a*x + b*y + c*z = 0 #
 
        // checking if the 4th povar satisfies
        // the above equation
        if (a * x + b * y + c * z + d == 0)
            document.write("Coplanar");
        else
            document.write("Not Coplanar");
 
    }
 
    // Driver Code
        var x1 = 3;
        var y1 = 2;
        var z1 = -5;
        var x2 = -1;
        var y2 = 4;
        var z2 = -3;
        var x3 = -3;
        var y3 = 8;
        var z3 = -5;
        var x4 = -3;
        var y4 = 2;
        var z4 = 1;
 
        // function calling
        equation_plane(x1, y1, z1, x2, y2, z2,
        x3, y3, z3, x4, y4, z4);
 
 
// This code is contributed by Rajput-Ji

输出：

Coplanar

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