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Mirror of a point through a 3 D plane
  • Last Updated : 21 Nov, 2018

Given a point(x, y, z) in 3-D and coefficients of the equation of plane, the task is to find the mirror image of that point through the given plane.

Examples:

Input: a = 1, b = -2, c = 0, d = 0, x = -1, y = 3, z = 4
Output: x3 = 1.7999999999999998, y3 = -2.5999999999999996, z3 = 4.0

Input: a = 2, b = -1, c = 1, d = 3, x = 1, y = 3, z = 4
Output: x3 = -3.0, y3 = 5.0, z3 = 2.0



Approach: Equation of plane is as ax + by + cz + d = 0. Therefore, direction ratios of the normal to the plane are (a, b, c). Let N be the foot of perpendicular from given point to the given plane so, line PN has directed ratios (a, b, c) and it passes through P(x1, y1, z1).

The equation of line PN will be as:-

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

Hence any point on line PN can be written as:-

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

since N lies in both line and plane so will satisfy(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).

Now, the coordinates of Point N in terms of k will be:-

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

Since, Point N(x2, y2, z2) is midpoint of point P(x1, y1, z1) and point Q(x3, y3, z3), coordinates of Point Q are:-

=> 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 <bits/stdc++.h> 
#include<math.h>
#include <iostream>
#include <iomanip>
  
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<stdio.h>
      
// 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




<?php 
// PHP program to find Mirror of
// a point through a 3 D plane
  
// Function to mirror image
function mirror_point($a, $b, $c, $d
                      $x1, $y1, $z1)
    $k = (-$a * $x1 - $b
           $y1 - $c * $z1 - $d) / 
          ($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;
    echo sprintf("x3 = %.1f ", $x3); 
    echo sprintf("y3 = %.1f ", $y3); 
    echo sprintf("z3 = %.1f ", $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);
  
// This code is contributed 
// by Amber_Saxena.
?> 
Output:
x3 = 1.8 y3 = -2.6 z3 = 4.0

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