求通过3点的平面方程的程序

给定三个点 (x1, y1, z1), (x2, y2, z2), (x3, y3, z3)。任务是找到通过这 3 个点的平面的方程。

例子：

Input: x1 = -1 y1 = w z1 = 1 
x2 = 0 y2 = -3 z2 = 2 
x3 = 1 y3 = 1 z3 = -4 
Output: equation of plane is 26 x + 7 y + 9 z + 3 = 0.
Input: x1 = 2, y1 = 1, z1 = -1, 1 
x2 = 0, y2 = -2, z2 = 0 
x3 = 1, y3 = -1, z3 = 2 
Output: equation of plane is -7 x + 5 y + 1 z + 10 = 0.

方法：设P、Q、R分别为坐标为(x1,y1,z1),(x2,y2,z2),(x3,y3,z3)的三个点。那么平面的方程是a * (x – x0) + b * (y – y0) + c * (z – z0) = 0 ，其中 a, b, c 是平面的法线方向比和 (x0, y0, z0) 是通过平面的任何点（即 P、Q 或 R）的坐标。求平面法线的方向比，取平面内任意两个向量，设向量PQ，向量PR。

=> Vector PQ = (x2 - x1, y2 - y1, z2 - z1) = (a1, b1, c1).
=> Vector PR = (x3 - x1, y3 - y1, z3 - z1) = (a2, b2, c2).

该平面的法向量将是向量 PQ x 向量 PR。

=> PQ X PR = (b1 * c2 - b2 * c1) i 
              + (a2 * c1 - a1 * c2) j 
              + (a1 * b2 - b1 *a2) k = ai + bj + ck.

法向量的方向比为 a、b、c。从 P、Q 或 R 中取任意一点，令其坐标为 (x0, y0, z0)。那么通过点(x0, y0, z0)并具有方向比a,b,c的平面方程为

=> a * (x - x0) + b * (y - y0) + c * (z - z0) = 0.
=> a * x - a * x0 + b * y - b * y0 + c * z - c * z0 = 0.
=> a * x + b * y + c * z + (- a * x0 - b * y0 - c * z0) = 0.

下面是上述方法的实现：

C++

// C++ program to find equation of a plane
// passing through given 3 points.
#include 
#include
#include 
#include 
 
using namespace std;
 
// Function to find equation of plane.
void equation_plane(float x1, float y1,
                    float z1, float x2,
                    float y2, float z2,
                    float x3, float y3, float z3)
{
    float a1 = x2 - x1;
    float b1 = y2 - y1;
    float c1 = z2 - z1;
    float a2 = x3 - x1;
    float b2 = y3 - y1;
    float c2 = z3 - z1;
    float a = b1 * c2 - b2 * c1;
    float b = a2 * c1 - a1 * c2;
    float c = a1 * b2 - b1 * a2;
    float d = (- a * x1 - b * y1 - c * z1);
    std::cout << std::fixed;
    std::cout << std::setprecision(2);
    cout << "equation of plane is " << a << " x + " << b
        << " y + " << c << " z + " << d << " = 0.";
}
 
// Driver Code
int main()
{
     
    float x1 =-1;
    float y1 = 2;
    float z1 = 1;
    float x2 = 0;
    float y2 =-3;
    float z2 = 2;
    float x3 = 1;
    float y3 = 1;
    float z3 =-4;
    equation_plane(x1, y1, z1, x2, y2, z2, x3, y3, z3);
    return 0;
}
// This code is contributed
// by Amber_Saxena.

C

// C program to find equation of a plane
// passing through given 3 points.
 
#include
 
// Function to find equation of plane.
void equation_plane(float x1, float y1,
                    float z1, float x2,
                    float y2, float z2,
                    float x3, float y3, float z3)
{
    float a1 = x2 - x1;
    float b1 = y2 - y1;
    float c1 = z2 - z1;
    float a2 = x3 - x1;
    float b2 = y3 - y1;
    float c2 = z3 - z1;
    float a = b1 * c2 - b2 * c1;
    float b = a2 * c1 - a1 * c2;
    float c = a1 * b2 - b1 * a2;
    float d = (- a * x1 - b * y1 - c * z1);
    printf("equation of plane is %.2f x + %.2f"
        " y + %.2f z + %.2f = 0.",a,b,c,d);
    return;
}
 
// Driver Code
int main()
{
    float x1 =-1;
    float y1 = 2;
    float z1 = 1;
    float x2 = 0;
    float y2 =-3;
    float z2 = 2;
    float x3 = 1;
    float y3 = 1;
    float z3 =-4;
    equation_plane(x1, y1, z1, x2, y2, z2, x3, y3, z3);
    return 0;
}
// This code is contributed
// by Amber_Saxena.

Java

// Java program to find equation
// of a plane passing through
// given 3 points.
import java .io.*;
 
class GFG
{
     
// Function to find equation of plane.
static void equation_plane(float x1, float y1,
                           float z1, float x2,
                           float y2, float z2,
                           float x3, float y3,
                           float z3)
{
    float a1 = x2 - x1;
    float b1 = y2 - y1;
    float c1 = z2 - z1;
    float a2 = x3 - x1;
    float b2 = y3 - y1;
    float c2 = z3 - z1;
    float a = b1 * c2 - b2 * c1;
    float b = a2 * c1 - a1 * c2;
    float c = a1 * b2 - b1 * a2;
    float d = (- a * x1 - b * y1 - c * z1);
    System.out.println("equation of plane is " + a +
                       " x + " + b + " y + " + c +
                       " z + " + d + " = 0.");
}
 
// Driver code
public static void main(String[] args)
{
    float x1 =-1;
    float y1 = 2;
    float z1 = 1;
    float x2 = 0;
    float y2 =-3;
    float z2 = 2;
    float x3 = 1;
    float y3 = 1;
    float z3 =-4;
    equation_plane(x1, y1, z1, x2,
                   y2, z2, x3, y3, z3);
}
}
 
// This code is contributed
// by Amber_Saxena.

Python

# Python program to find equation of a plane
# passing through given 3 points.
 
# Function to find equation of plane.
def equation_plane(x1, y1, z1, x2, y2, z2, x3, y3, z3):
     
    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)
    print "equation of plane is ",
    print a, "x +",
    print b, "y +",
    print c, "z +",
    print d, "= 0."
 
# Driver Code
x1 =-1
y1 = 2
z1 = 1
x2 = 0
y2 =-3
z2 = 2
x3 = 1
y3 = 1
z3 =-4
equation_plane(x1, y1, z1, x2, y2, z2, x3, y3, z3)

C#

// C# program to find equation
// of a plane passing through
// given 3 points.
using System;
 
class GFG
{
     
// Function to find equation of plane.
static void equation_plane(float x1, float y1,
                           float z1, float x2,
                           float y2, float z2,
                           float x3, float y3,
                           float z3)
{
    float a1 = x2 - x1;
    float b1 = y2 - y1;
    float c1 = z2 - z1;
    float a2 = x3 - x1;
    float b2 = y3 - y1;
    float c2 = z3 - z1;
    float a = b1 * c2 - b2 * c1;
    float b = a2 * c1 - a1 * c2;
    float c = a1 * b2 - b1 * a2;
    float d = (- a * x1 - b * y1 - c * z1);
    Console.Write("equation of plane is " + a +
                      "x + " + b + "y + " + c +
                          "z + " + d + " = 0");
}
 
// Driver code
public static void Main()
{
    float x1 =-1;
    float y1 = 2;
    float z1 = 1;
    float x2 = 0;
    float y2 =-3;
    float z2 = 2;
    float x3 = 1;
    float y3 = 1;
    float z3 =-4;
    equation_plane(x1, y1, z1,
                   x2, y2, z2,
                   x3, y3, z3);
}
}
 
// This code is contributed
// by ChitraNayal

Javascript

// javascript program to find equation
// of a plane passing through
// given 3 points.
 
    // Function to find equation of plane.
    function equation_plane(x1 , y1 , z1 , x2 ,
    y2 , z2 , x3 , y3, z3) {
        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);
        document.write("equation of plane is " + a + " x + "
        + b + " y + " + c + " z + " + d + " = 0.");
    }
 
    // Driver code
     
        var x1 = -1;
        var y1 = 2;
        var z1 = 1;
        var x2 = 0;
        var y2 = -3;
        var z2 = 2;
        var x3 = 1;
        var y3 = 1;
        var z3 = -4;
        equation_plane(x1, y1, z1, x2, y2, z2, x3, y3, z3);
 
// This code is contributed by Rajput-Ji

输出：

equation of plane is  26 x + 7 y + 9 z + 3 = 0.