# Geometry using Complex Numbers in C++

In this section, we will see how to make point class using complex class from STL in C++. And apply them on some geometry related problems. The complex number is present inside the complex class from STL (#include <complex>)

## Defining Point Class

To make complex to point, we will change the name of the complex<double> as point, then change x to real() of complex class and y to imag() of complex class. Thus, we can simulate the point class.

# include <complex>
typedef complex<double> point;
# define x real()
# define y imag()

We have to keep in mind that the x and y have been applied as macros, these can’t be applied as variables.

## Example

Let us see the following implementation to get better understanding −

Live Demo

#include <iostream>
#include <complex>
using namespace std;
typedef complex<double> point;
#define x real()
#define y imag()
int main() {
point my_pt(4.0, 5.0);
cout << "The point is :" << "(" << my_pt.x << ", " << my_pt.y << ")";
}

## Output

The point is :(4, 5)

To apply geometry, we can find that the distance of P from origin (0, 0), this is denoted as −abs(P). The angle made by OP from axis X where O is the origin: arg(z). Rotation of P about origin is P * polar(r, θ).

## Example

Let us see the following implementation to get better understanding −

Live Demo

#include <iostream>
#include <complex>
#define PI 3.1415
using namespace std;
typedef complex<double> point;
#define x real()
#define y imag()
void print_point(point my_pt){
cout << "(" << my_pt.x << ", " << my_pt.y << ")";
}
int main() {
point my_pt(6.0, 7.0);
cout << "The point is:" ;
print_point(my_pt);
cout << endl;
cout << "Distance of the point from origin:" << abs(my_pt) << endl;
cout << "Tangent angle made by OP with X-axis: (" << arg(my_pt) << ") rad = (" << arg(my_pt)*(180/PI) << ")" << endl;
point rot_point = my_pt * polar(1.0, PI/2);
cout << "Point after rotating 90 degrees counter-clockwise, will be: ";
print_point(rot_point);
}

## Output

The point is:(6, 7)
Distance of the point from origin:9.21954
6.00032) 