It's always a hassle to define our 2D Geometry library during a contest. Is there a way to make our computational geometry lives easier in any way? Fortunately for us, there is, at least in C++, using complex numbers.
Complex numbers are of the form a + bi, where a is the real part and b imaginary. Thus, we can let a be the x-coordinate and b be the y-coordinate. Whelp, complex numbers can be represented as 2D vectors! Therefore, we can use complex numbers to define a point instead of defining the class ourselves. You can look at std::complex reference here.
Defining our point class
We can define our point class by typing typedef complex<double> point;
at the start of our program. To access our x- and y-coordinates, we can macro the real()
and imag()
functions by using #define
. Of course, don't forget to #include <complex>
before anything.
#include <iostream>
#include <complex>
using namespace std;
// define x, y as real(), imag()
typedef complex<double> point;
#define x real()
#define y imag()
// sample program
int main() {
point a = 2;
point b(3, 7);
cout << a << ' ' << b << endl; // (2, 0) (3, 7)
cout << a + b << endl; // (5, 7)
}
Oh goodie! We can use std:cout
for debugging! We can also add points as vectors without having to define operator+
. Nifty. And apparently, we can overall add points, subtract points, do scalar multiplication without defining any operator. Very nifty indeed.
Example
point a(3, 2), b(2, -7);
// vector addition and subtraction
cout << a + b << endl; // (5,-5)
cout << a - b << endl; // (1,9)
// scalar multiplication
cout << 3.0 * a << endl; // (9,6)
cout << a / 5.0 << endl; // (0.6,0.4)
Functions using std::complex
What else can we do with complex numbers? Well, there's a lot that is really easy to code.
Vector addition:
a + b
Scalar multiplication:
r * a
Dot product:
(a.conj() * b).x
Cross product:
(a.conj() * b).y
Squared distance:
norm(a - b)
orabs(a - b)
Euclidean distance:
sqrt(norm(a - b))
Angle of elevation:
arg(b - a)
Slope of line (a, b):
tan(arg(b - a))
or(b-a).y / (b-a).x
Polar to cartesian:
polar(r, theta)
Cartesian to polar:
point(sqrt(norm(cart)), arg(cart))
Rotation about the origin:
a * polar(1.0, theta)
Rotation about pivot p:
(a-p) * polar(1.0, theta) + p
notice: a.conj() * b == (ax*bx + ay*by) + i (ax*by - ay*bx) = dot(a,b) + i*cross(a,b)
Drawbacks
Using std::complex is very advantageous, but it has one disadvantage: you can't use std::cin
or scanf
. Also, if we macro x and y, we can't use them as variables. But that's rather minor, don't you think?