Geometry using Complex Numbers in C++ | Set 2
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:
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# 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
// 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
// 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 coordinates of a pointvoid 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)
- 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
// 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 coordinates of a pointvoid 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
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