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.

Solution 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)

Solution 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 );

Comments

[Derek Ledbetter]

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.

[Cerin]

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

[rraallvv]

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

[Derek Ledbetter]

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

[rraallvv]

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

[Gustavo Maciel]

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

[user2083364]

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}.

[bright-star]

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

[Helin Wang]

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.

[Lennart Rolland]

And how would that look like in 3d?

[Tara]

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

[Roi Danton]

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

[Tara]

@RoiDanton Thank you!

[SaPropper]

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