求平面通过三点方程的程序-yiteyi-C++库

给出三个点（x1，y1，z1），（x2，y2，z2），（x3，y3，z3）。任务是找到通过这三个点的平面方程。

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图片[1]-求平面通过三点方程的程序-yiteyi-C++库

例如：

输入： x1=-1 y1=w z1=1 x2=0 y2=-3 z2=2 x3=1 y3=1 z3=-4 输出： 平面方程为26x+7y+9z+3=0。 输入： x1=2，y1=1，z1=-1，1 x2=0，y2=-2，z2=0 x3=1，y3=-1，z3=2 输出： 平面方程为-7x+5y+1z+10=0。

推荐：请尝试你的方法 {IDE} 首先，在进入解决方案之前。

方法： 设P、Q和R分别为坐标为（x1、y1、z1）、（x2、y2、z2）、（x3、y3、z3）的三个点。那么平面方程是 a*（x-x0）+b*（y-y0）+c*（z-z0）=0 式中，a、b、c是法向与平面的方向比，（x0、y0、z0）是通过平面的任何点（即P、Q或R）的坐标。为了求法线与平面的方向比，取平面上的任意两个向量，设为向量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).

该平面的法向量为向量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取任意一点，其坐标为（x0、y0、z0）。然后，通过点（x0，y0，z0）并具有方向比a，b，c的平面方程

=> 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.

以下是上述方法的实施情况：

C++

                         // C++ program to find equation of a plane                       
                         // passing through given 3 points.                       
                         #include <bits/stdc++.h>                       
                         #include<math.h>                       
                         #include <iostream>                       
                         #include <iomanip>                       
           
                         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<stdio.h>                       
           
                         // 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                       

PHP

                         <?php                       
                         // PHP 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                                     )                       
                         {                       
                                                 $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                                     );                       
                                                 echo                                     sprintf(                                     "equation of the plane is %.2fx"                                     .                       
                                                 " + %.2fy + %.2fz + %.2f = 0"                                     ,                       
                                                 $a                                     ,                                     $b                                     ,                                     $c                                     ,                                     $d                                     );                       
                         }                       
           
                         // 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                                     );                       
           
                         // This code is contributed                       
                         // by Amber_Saxena.                       
                         ?>                       

Javascript

                         <script>                       
                         // 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                       
                         </script>                       

输出：

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

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