**Indian Technical Authorship Contest**starts on 1st July 2023. Stay tuned.

In this article, we will learn how to find the mirror image position of a point in a 2D plane along with C++ implementation of the approach.

## Table of contents

- Problem Statement
- Mirror Equation
- Implementation
- Code

# Problem Statement

Given a point P(x1,y1) in 2-D plane, find the image of the point P(x1,y1) formed due to the mirror.

**Sample input :**

(5,1)

**Sample output :**

Image -> (1,5)

# Mirror Equation

In order to get mirror image we must have a mirror first.

You can observe in this image that mirror image of point P(x,y) is Q(x,-y) when we take **x-axis** as mirror.

Mirror image is always equidistant from the mirror, in this case the distance from x-axis(mirror) to point P and distance from x-axis to point Q is same that is |y|.

Formula for line equation (mirror equation) -> **a x+by+c=0**

where x and y is the middle point of mirror image and actual point.

In this image we can observe that the mirror used here is y-axis and both the points are equidistant from the mirror.

Now, for simplicity we will take our mirror in the first quadrant where x and y are both positive. Imagine a line from the origin at 45 degrees from both x-axis and y-axis.

For this line mirror equation will be -> **(-1)x + (1)y + 0 = 0**

Now, when we have a mirror equation, we can use this formula below to get mirror image of a actual point with respect to the mirror (-1)x +(1)y + 0 = 0

Formula to get mirror image :-

(x-x1)/a = (y-y1)/b = -2(ax1+by1+c)/(a^{2}+b^{2})

# Implementation

- Make a function which applies the above formula and returns the value of x and y cordinate of mirror image in a pair of double integer (x,y).
- Initialize coefficient of mirror equation.
- Take input from user , the actual point (x1,y1)
- Run the function and output the mirror image position.

->Note that we can use any line as mirror, but in this we will use this mirror equation because it is symmetric through both the axis.

# Code :

```
#include <iostream>
using namespace std;
pair<double, double> mirrorimage(double x1, double y1)
{
//coefficients of mirror equation
double a = -1.0;
double b = 1.0;
double c = 0.0;
//Formula used for mirror image
double temp = -2 * (a * x1 + b * y1 + c) / (a * a + b * b);
//mirror image of (x1,y1)
//kept x one side and sent a and -x1 to other side to get this equation , same for y.
double x = temp * a + x1;
double y = temp * b + y1;
return make_pair(x, y);
}
int main()
{
double x1,y1;
cout<<"Enter the point x1 : ";
cin>>x1;
cout<<"Enter the point y1 : ";
cin>>y1;
pair<double,double> image;
image = mirrorimage(x1,y1);
cout<<"Mirror image of ("<<x1<<","<<y1<<") is : ("<<image.first<<","<<image.second<<")";
return 0;
}
```

## Sample output

```
Enter the point x1 : 10
Enter the point y1 : 9
Mirror image of (10,9) is : (9,10)
```

With this article at OpenGenus, you must have the complete idea of Finding mirror image of point in 2D plane.