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++ 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 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 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 |
# 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# 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 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. ?> |
<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> |
x3 = 1.8 y3 = -2.6 z3 = 4.0
Time Complexity: O(1)
Auxiliary Space: O(1)