Geometry using Complex Numbers in C++

After going through previous post, we know what exactly are complex numbers and how we can use them to simulate points in a cartesian plane. Now, we will have an insight as to how to use the complex class from STL in C++.

To use the complex class from STL we use #include <complex>

Defining Point Class

We can define our point class by typedef complex<double> point; at the start of the program. The X and Y coordinates of the point are the real and imaginary part of the complex number respectively. To access our X- and Y-coordinates, we can macro the real() and imag() functions by using #define as follows:

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

Drawback: Since x and y have been used as macros, these can’t be used as variables. However, this drawback doesn’t stand in front of the many advantages this serves.

// CPP program to illustrate  
// the definition of point class 
#include <iostream> 
#include <complex> 
  
using namespace std; 
  
typedef complex<double> point; 
  
// X-coordinate is equivalent to the real part 
// Y-coordinate is equivalent to the imaginary part 
#define x real() 
#define y imag() 
  
int main() 
{ 
    point P(2.0, 3.0); 
    cout << "The X-coordinate of point P is: " << P.x << endl; 
    cout << "The Y-coordinate of point P is: " << P.y << endl; 
  
    return 0; 
} 

Output:

The X-coordinate of point P is: 2
The Y-coordinate of point P is: 3

Implementation of attributes with respect to P single point P in plane:

The X coordinate of P: P.x
The Y coordinate of P: P.y
The distance of P from origin (0, 0): abs(P)
The angle made by OP from the X-Axis where O is the origin: arg(z)

Rotation of P about origin: P * polar(r, θ)

// CPP program to illustrate 
// the implementation of single point attributes 
#include <iostream> 
#include <complex> 
  
using namespace std; 
  
typedef complex<double> point; 
#define x real() 
#define y imag() 
  
// The constant PI for providing angles in radians 
#define PI 3.1415926535897932384626 
  
// Function used to display X and Y coordiantes of a point 
void displayPoint(point P) 
{ 
    cout << "(" << P.x << ", " << P.y << ")" << endl; 
} 
  
int main() 
{ 
    point P(4.0, 3.0); 
  
    // X-Coordinate and Y-coordinate 
    cout << "The X-coordinate of point P is: " << P.x << endl; 
    cout << "The Y-coordinate of point P is: " << P.y << endl; 
  
    // Distances of P from origin 
    cout << "The distance of point P from orgin is: " << abs(P) <<endl; 
    cout << "The squared distance of point P from orgin is: " << norm(P) <<endl; 
  
    // Tangent Angle made by OP with the X-Axis 
    cout << "The angle made by OP with the X-Axis is: "
        << arg(P) << " radians" << endl; 
    cout << "The angle made by OP with the X-Axis is: "
        << arg(P)*(180/PI) << " degrees" << endl; 
  
  
    // Rotation of P about origin 
    // The angle of rotation = 90 degrees 
    point P_rotated = P * polar(1.0, PI/2); 
    cout<<"The point P on rotating 90 degrees anti-clockwise becomes: P_rotated";  
    displayPoint(P_rotated); 
  
    return 0; 
} 

Output:

The X-coordinate of point P is: 4
The Y-coordinate of point P is: 3
The distance of point P from orgin is: 5
The squared distance of point P from orgin is: 25
The angle made by OP with the X-Axis is: 0.643501 radians
The angle made by OP with the X-Axis is: 36.8699 degrees
The point P on rotating 90 degrees anti-clockwise becomes: P_rotated(-3, 4)

Let us consider points P (a, b) and Q (c, d) on the Euclidean Plane.
Implementation of attributes with respect to P and Q.

Vector Addition: P + Q
Vector Subtraction: P – Q
Euclidean Distance: abs(P – Q)
Slope of line PQ: tan(arg(Q – P))
```
point A = conj(P) * Q
```
Dot Product: A.x
Magnitude of Cross Product: abs(A.y)

// CPP program to illustrate  
// the implementation of two point attributes 
#include <iostream> 
#include <complex> 
  
using namespace std; 
  
typedef complex<double> point; 
#define x real() 
#define y imag() 
  
// Constant PI for providing angles in radians 
#define PI 3.1415926535897932384626 
  
// Function used to display X and Y coordiantes of a point 
void displayPoint(point P) 
{ 
    cout << "(" << P.x << ", " << P.y << ")" << endl; 
} 
  
int main() 
{ 
    point P(2.0, 3.0); 
    point Q(3.0, 4.0); 
  
    // Addition and Subtraction 
    cout << "Addition of P and Q is: P+Q"; displayPoint(P+Q); 
    cout << "Subtraction of P and Q is: P-Q"; displayPoint(P-Q); 
  
    // Distances between points P and Q 
    cout << "The distance between point P ans Q is: " << abs(P-Q) <<endl; 
    cout << "The squared distance between point P ans Q is: " << norm(P-Q) <<endl; 
  
    // Slope of line PQ 
    cout << "The angle of elevation for line PQ is: "
        << arg(Q-P)*(180/PI) << " degrees" << endl; 
    cout << "The slope of line PQ is: " << tan(arg(Q-P)) <<endl; 
  
    // Construction of point A 
    point A = conj(P)*Q; 
  
    // Dot Product and Cross Product 
    cout << "The dot product P.Q is: " << A.x << endl; 
    cout << "The magnitude of cross product PxQ is: " << abs(A.y) << endl; 
  
    return 0; 
} 

Output:

Addition of P and Q is: P+Q(5, 7)
Subtraction of P and Q is: P-Q(-1, -1)
The distance between point P ans Q is: 1.41421
The squared distance between point P ans Q is: 2
The angle of elevation for line PQ is: 45 degrees
The slope of line PQ is: 1
The dot product P.Q is: 18
The magnitude of cross product PxQ is: 1

This article is contributed by Aanya Jindal. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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My Personal Notes

Geometry using Complex Numbers in C++ | Set 2

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