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Mirror of a point through a 3 D plane

  • Last Updated : 28 Apr, 2021

Given a point(x, y, z) in 3-D and coefficients of the equation of a 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 a 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.
?>

Javascript




<script>
 
      // JavaScript 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)
      {
        var k = parseFloat(
          (-a * x1 - b * y1 -
          c * z1 - d) / parseFloat(a * a + b * b +
          c * c)
        );
        var x2 = parseFloat(a * k + x1);
        var y2 = parseFloat(b * k + y1);
        var z2 = parseFloat(c * k + z1);
        var x3 = parseFloat(2 * x2 - x1).toFixed(1);
        var y3 = parseFloat(2 * y2 - y1).toFixed(1);
        var z3 = parseFloat(2 * z2 - z1).toFixed(1);
 
        document.write(" x3 = " + x3);
        document.write(" y3 = " + y3);
        document.write(" z3 = " + z3);
      }
 
      // Driver Code
      var a = 1;
      var b = -2;
      var c = 0;
      var d = 0;
      var x1 = -1;
      var y1 = 3;
      var z1 = 4;
 
      // function call
      mirror_point(a, b, c, d, x1, y1, z1);
       
</script>
Output: 
x3 = 1.8 y3 = -2.6 z3 = 4.0

 

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