Vrftsjtryk

I continue developing a 2D Collision Detection System in a programming language (Javascript) and one of the last things I need to sharpen it is to know a formula to find this angle:

NOTE: X and Y increase their value FROM LEFT TO RIGHT AND TOP TO BOTTOM

As you can see the angle is relative to the 0° degree or north pole of the 2D space.

Knowing the coordinates of the two points, how can I know that angle?
I might have an idea of finding the bearing to rectangle vertices and stuff like that (I just used them for the system) but I want to know if there is already a simple formula for this.

Thank you beforehand!

Define the bearing angle $theta$ from a point $A(a_1,a_2)$ to a point $B(b_1,b_2)$ as the angle measured in the clockwise direction from the north line with $A$ as the origin to the line segment $AB$.

Then,

$$
(b_1,b_2) = (a_1 + rsintheta, a_2 + rcostheta),
$$

where $r$ is the length of the line segment $AB$. It follows that $theta$ satisfies the equation

$$
tantheta = frac{b_1 - a_1}{b_2 - a_2}
$$

As suggested by @rogerl we can use the $mathrm{atan2}$ function to compute $theta$. Let

$$
hat{theta} =
mathrm{atan2}(b_1 - a_1, b_2 - a_2) in (-pi,pi]
$$

Then the bearing angle $thetain[0,2pi)$ is given by

$$
theta = left{
begin{array}{ll}
hat{theta}, & hat{theta} geq 0\
2pi + hat{theta}, & hat{theta} < 0
end{array}right.
$$

Note that the equations are given in terms of Cartesian coordinates, so it is necessary to transform to screen coordinates. I believe the formula for $hat{theta}$ in terms of screen coordinates $(a_1,a_2)$ and $(b_1,b_2)$ is $hat{theta} = mathrm{atan2}(b_1 - a_1,a_2 - b_2)$.

You could code this function in C++ as follows.

#include <cmath>



// Computes the bearing in degrees from the point A(a1,a2) to

// the point B(b1,b2). Note that A and B are given in terms of

// screen coordinates.

double bearing(double a1, double a2, double b1, double b2) {

    static const double TWOPI = 6.2831853071795865;

    static const double RAD2DEG = 57.2957795130823209;

    // if (a1 = b1 and a2 = b2) throw an error 

    double theta = atan2(b1 - a1, a2 - b2);

    if (theta < 0.0)

        theta += TWOPI;

    return RAD2DEG * theta;

}