Finding Signed Angle Between Vectors

Question

How would you find the signed angle theta from vector a to b?

And yes, I know that theta = arccos((a.b)/(|a||b|)).

However, this does not contain a sign (i.e. it doesn't distinguish between a clockwise or counterclockwise rotation).

I need something that can tell me the minimum angle to rotate from a to b. A positive sign indicates a rotation from +x-axis towards +y-axis. Conversely, a negative sign indicates a rotation from +x-axis towards -y-axis.

assert angle((1,0),(0,1)) == pi/2.
assert angle((0,1),(1,0)) == -pi/2.

Crossways

Answer 1

If you have an atan2() function in your math library of choice:

signed_angle = atan2(b.y,b.x) - atan2(a.y,a.x)

Digit

What about a = (-1,1) and b = (-1,-1), where the answer should be pi/2? You should check if the absolute value is bigger than pi, and then add or subtract 2*pi if it is. – Offhand

@Derek Good catch. I actually discovered this myself while implementing the solution. – Crossways

it is not very appropriate for example for computer graphics because it confuse -pi and pi if I have a = {-1, 0} and b = {0, 1}. – Garlic

In degree, the result of this answer should be [-180, 180), but some time I discover result like: 358.5. Derek Ledbetter's answer works fine. – Atul

Answer 2

What you want to use is often called the “perp dot product”, that is, find the vector perpendicular to one of the vectors, and then find the dot product with the other vector.

if(a.x*b.y - a.y*b.x < 0)
    angle = -angle;

You can also do this:

angle = atan2( a.x*b.y - a.y*b.x, a.x*b.x + a.y*b.y );

Offhand

do you know whether the second equation always return angles less than 180º? – Main

The angle will be between -pi and pi radians, inclusive. – Offhand

great that solves the issue when a = (-1,1) and b = (-1,-1) pointed above – Main

Man, this was all that I needed! Works flawlessy in 2d, thanks! – Anastomosis

All other duplicate questions should link to this question and this answer; this is so sparsely documented (doesn't even have a wikipedia article) – Arachne

And how would that look like in 3d? – Effy

Can you elaborate how the 2nd version works? Specifically the calculations you pass into atan2. – Ileanaileane

@Ileanaileane The first parameter is the determinant, the second one the dot product. See also this answer. – Tyner

@RoiDanton Thank you! – Ileanaileane

didn't produce the correct angle between -Pi and Pi for me, but this answer did – Lx