Counter-clockwise angle between edges

Question

I have a couple of connected lines (or rather edges) in the form of coordinates, that is for each edge a starting point $(x_s,y_s)$ and end point $(x_e,y_e)$. and I want to know a specific angle between them. Here is a drawing to illustrate which angles I want: The blue arrows are (in this case 4) edges. The orange-like pieces of a circle are supposed to show the desired angles. That is, I want the counter-clockwise angle of the respective second edge to the respective first edge. How do I do that?

Answer 1

Using the start and end point, you can determine the related vector like this $(x_e-x_s, y_e-y_s)$. Then, you have the vectors, and you can compute the angles using the interior product. Assume a and b are two vectors, then the angle between them can be computed using the following.

$$\cos(\theta)=\frac{\mathbf {a \cdot b}}{\mathbf {|a||b|}}$$

Please note that this angle is not always the one you are looking for, and you may need to consider $\pi-\theta$ as the one which you need to find.