Mirror of a point through a 3 D plane
Last Updated :
20 Aug, 2022
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++
#include <bits/stdc++.h>
#include<math.h>
#include <iostream>
#include <iomanip>
using namespace std;
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;
}
int main()
{
float a = 1;
float b = -2;
float c = 0;
float d = 0;
float x1 = -1;
float y1 = 3;
float z1 = 4;
mirror_point(a, b, c, d,
x1, y1, z1);
return 0;
}
|
C
#include<stdio.h>
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);
}
int main()
{
float a = 1;
float b = -2;
float c = 0;
float d = 0;
float x1 = -1;
float y1 = 3;
float z1 = 4;
mirror_point(a, b, c, d,
x1, y1, z1);
}
|
Java
import java.io.*;
class GFG
{
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 + " ");
}
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;
mirror_point(a, b, c, d,
x1, y1, z1) ;
}
}
|
Python
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,
a = 1
b = -2
c = 0
d = 0
x1 = -1
y1 = 3
z1 = 4
mirror_point(a, b, c, d, x1, y1, z1)
|
C#
using System;
class GFG
{
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 + " ");
}
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;
mirror_point(a, b, c, d,
x1, y1, z1);
}
}
|
PHP
<?php
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);
}
$a = 1;
$b = -2;
$c = 0;
$d = 0;
$x1 = -1;
$y1 = 3;
$z1 = 4;
mirror_point($a, $b, $c, $d,
$x1, $y1, $z1);
?>
|
Javascript
<script>
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);
}
var a = 1;
var b = -2;
var c = 0;
var d = 0;
var x1 = -1;
var y1 = 3;
var z1 = 4;
mirror_point(a, b, c, d, x1, y1, z1);
</script>
|
Output:
x3 = 1.8 y3 = -2.6 z3 = 4.0
Time Complexity: O(1)
Auxiliary Space: O(1)
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