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📜  通过3D平面的点的镜像

📅  最后修改于: 2021-05-04 21:53:29             🧑  作者: Mango

给定3-D中的点(x，y，z)和平面方程的系数，任务是通过给定平面找到该点的镜像。

例子：

方法：平面的等式为ax + x + cz + d =0。因此，法线与平面的方向比为(a，b，c) 。令N为从给定点到给定平面的垂直脚，因此，线PN的有向比(a，b，c)穿过P(x1，y1，z1)。

PN线的等式为：- 

(x - x1) / a = (y - y1) / b = (z - z1) / c = k

因此，PN线上的任何点都可以写成：- 

x = a*k + x1
y = b*k + y1
z = c*k + z1

由于N位于直线和平面中，因此将满足(ax + by + cz + d = 0)。 

=>a * (a * k + x1) + b * (b * k + y1) + c * (c * k + z1) + d = 0.
=>a * a * k + a * x1 + b * b * k + b * y1 + c * c * k + c * z1 + d = 0.
=>(a * a + b * b + c * c)k = -a * x1 - b * y1 - c * z1 - d.
=>k = (-a * x1 - b * y1 - c * z1 - d) / (a * a + b * b + c * c).

现在，以k表示的点N的坐标将是： 

x2 = a * k + x1
y2 = b * k + y1
z2 = c * k + z1

因为点N(x2，y2，z2)是点P(x1，y1，z1)和点Q(x3，y3，z3)的中点，所以点Q的坐标为：- 

=> x3 = 2 * x2 - x1
=> y3 = 2 * y2 - y1
=> z3 = 2 * z2 - z1
C++
// C++ program to find 
// Mirror of a point  
// through a 3 D plane
#include  
#include
#include 
#include 
  
using namespace std;
  
// Function to mirror image 
void mirror_point(float a, float b,  
                  float c, float d,  
                  float x1, float y1, 
                  float z1)
{
    float k = (-a * x1 - b *  
                y1 - c * z1 - d) /  
        (float)(a * a + b * b + c * c); 
    float x2 = a * k + x1; 
    float y2 = b * k + y1; 
    float z2 = c * k + z1; 
    float x3 = 2 * x2 - x1; 
    float y3 = 2 * y2 - y1; 
    float z3 = 2 * z2 - z1; 
        
    std::cout << std::fixed;
    std::cout << std::setprecision(1);
    cout << " x3 = " << x3;  
    cout << " y3 = " << y3;  
    cout << " z3 = " << z3;
}
  
// Driver Code 
int main()
{
    float a = 1; 
    float b = -2; 
    float c = 0; 
    float d = 0; 
    float x1 = -1; 
    float y1 = 3; 
    float z1 = 4; 
    
    // function call 
    mirror_point(a, b, c, d,  
                 x1, y1, z1);
    return 0;
}
// This code is contributed 
// by Amber_Saxena.

C
// C program to find
// Mirror of a point 
// through a 3 D plane
#include
      
// Function to mirror image
void mirror_point(float a, float b, 
                  float c, float d, 
                  float x1, float y1,
                  float z1)
{
    float k = (-a * x1 - b * 
                y1 - c * z1 - d) / 
        (float)(a * a + b * b + c * c);
    float x2 = a * k + x1;
    float y2 = b * k + y1;
    float z2 = c * k + z1;
    float x3 = 2 * x2 - x1;
    float y3 = 2 * y2 - y1;
    float z3 = 2 * z2 - z1;
      
    printf("x3 = %.1f ", x3); 
    printf("y3 = %.1f ", y3); 
    printf("z3 = %.1f ", z3);
}
  
// Driver Code 
int main()
{
    float a = 1;
    float b = -2;
    float c = 0;
    float d = 0;
    float x1 = -1;
    float y1 = 3;
    float z1 = 4;
  
    // function call
    mirror_point(a, b, c, d, 
                 x1, y1, z1);
}
  
// This code is contributed 
// by Amber_Saxena.

Java
// Java program to find
// Mirror of a point 
// through a 3 D plane
import java.io.*;
  
class GFG 
{
      
// Function to mirror image
static void mirror_point(int a, int b,  
                         int c, int d, 
                         int x1, int y1,
                         int z1)
{
    float k = (-a * x1 - b * y1 - c * z1 - d) / 
        (float)(a * a + b * b + c * c);
    float x2 = a * k + x1;
    float y2 = b * k + y1;
    float z2 = c * k + z1;
    float x3 = 2 * x2 - x1;
    float y3 = 2 * y2 - y1;
    float z3 = 2 * z2 - z1;
      
    System.out.print("x3 = " + x3 + " "); 
    System.out.print("y3 = " + y3 + " "); 
    System.out.print("z3 = " + z3 + " ");
}
  
// Driver Code 
public static void main(String[] args)
{
    int a = 1;
    int b = -2;
    int c = 0;
    int d = 0;
    int x1 = -1;
    int y1 = 3;
    int z1 = 4;
  
    // function call
    mirror_point(a, b, c, d, 
                 x1, y1, z1) ;
}
}
  
// This code is contributed
// by inder_verma

Python
# Function to mirror image
def mirror_point(a, b, c, d, x1, y1, z1): 
       
    k =(-a * x1-b * y1-c * z1-d)/float((a * a + b * b + c * c))
    x2 = a * k + x1
    y2 = b * k + y1
    z2 = c * k + z1
    x3 = 2 * x2-x1
    y3 = 2 * y2-y1
    z3 = 2 * z2-z1
    print "x3 =", x3, 
    print "y3 =", y3, 
    print "z3 =", z3,
  
  
# Driver Code 
a = 1
b = -2
c = 0
d = 0
x1 = -1
y1 = 3
z1 = 4
  
# function call
mirror_point(a, b, c, d, x1, y1, z1)

C#
// C# program to find Mirror of 
// a point through a 3 D plane 
using System;
  
class GFG
{
          
// Function to mirror image 
static void mirror_point(int a, int b, 
                         int c, int d, 
                         int x1, int y1, 
                         int z1) 
{ 
    float k = (-a * x1 - b * y1 - c * z1 - d) / 
              (float)(a * a + b * b + c * c); 
    float x2 = a * k + x1; 
    float y2 = b * k + y1; 
    float z2 = c * k + z1; 
    float x3 = 2 * x2 - x1; 
    float y3 = 2 * y2 - y1; 
    float z3 = 2 * z2 - z1; 
      
    Console.Write("x3 = " + x3 + " "); 
    Console.Write("y3 = " + y3 + " "); 
    Console.Write("z3 = " + z3 + " "); 
} 
  
// Driver Code 
static public void Main ()
{
    int a = 1; 
    int b = -2; 
    int c = 0; 
    int d = 0; 
    int x1 = -1; 
    int y1 = 3; 
    int z1 = 4; 
      
    // function call 
    mirror_point(a, b, c, d, 
                 x1, y1, z1); 
}
}
  
// This code is contributed by jit_t

PHP

输出： 
x3 = 1.8 y3 = -2.6 z3 = 4.0