I would like to find the angle $\theta$ between two vectors:
$v_1$ from $i$ to $i+1$ : $v_1 = (x_{i1} - x_i , y_{i1} - y_1)$
and
$v_2$ from $i+1$ to $i+2$ : $v_2=(x_{i2}- x_{i1}, y_{i2} - y_{i1})$,
which are shown as in the figure (but $v_1$ and $v_2$ can be arbitrarily placed so $\theta$ may arbitrarily change as well).
Coordinates of the vector's starting and ending points are given in Cartesian coordinates.
Then this angle $\theta$ will be divided into two to find the bisector angle between two vectors. I would like to avoid calculation of the angles of vectors as much as possible and try to do this as vector operations.
I tried to subtract second vector from first one and find the angle by arctangent operation. However, that did not help for all possible cases (i.e. when the angles of the two vectors are not close to each other, etc.)
Another idea is to find the angle by $\arctan( \frac{ v_1 \cdot v_2}{ Det(v_1,v_2) })$, which basically corresponds to $\arctan( \frac{ \text{sin} }{ \text{cos} } )$.
What would be a reasonable approach to find this directed angle from first vector to the second one?
