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
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