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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

## Recommended: Please try your approach on {IDE} first, before moving on to the solution. 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  ``#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

 ` `
Output:
```x3 = 1.8 y3 = -2.6 z3 = 4.0
```

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