Cfrgtkky

I have 3 points in 3D and I want to calculate $ABC$ angle, which could be reflex angle. I also know the normal of the plane they are in. Is there a way to calculate the angle?

EDIT:

Take a look at this picture. The two vectors create two angles one would be $a$ and the other $360 - a$. Now I also know the direction of normal between the two vectors. With that information how can I determine if they construct angle $a$ or $360-a$.

You can use scalar product
$$
langle mathbf{u},mathbf{v}rangle = lVert mathbf{u}rVert lVert mathbf{v}rVert cosalpha
$$
where $alpha$ is the angle between $mathbf{u}$ and $mathbf{v}$.

Hence
$$
alpha=arccos frac{langle mathbf{u},mathbf{v}rangle}{lVert mathbf{u}rVert lVert mathbf{v}rVert}
$$

In your case
begin{align}
mathbf{u}=A-B\
mathbf{v}=C-B
end{align}

Notice that you'll get $alphale180°$.

I finally found a solution for this case in here.

The part that is relevant for this question:

When your vectors are not placed arbitrarily, but lie within a plane with a known normal vector $n$. Then the axis of rotation will be in direction $n$ as well, and the orientation of $n$ will fix an orientation for that axis. In this case you can calculate the angle via:

$$dot = x_1x_2 + y_1y_2 + z_1z_2$$
$$det = x_1y_2z_n + x_2y_nz_1 + x_ny_1z_2 - z_1y_2*x_n - z_2y_nx_1 - z_ny_1*x_2$$
$$angle = arctan(det, dot)$$

One condition for this to work is that the normal vector n has unit length. If not, you'll have to normalize it.