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.

score 64 · Answer 1 · answered Jan 27 '10 at 21:44

64

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 );

answered Jan 27 '10 at 21:44

Derek Ledbetter

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do you know whether the second equation always return angles less than 180º? – rraallvv Mar 02 '13 at 16:32
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The angle will be between -pi and pi radians, inclusive. – Derek Ledbetter Mar 06 '13 at 02:30
great that solves the issue when a = (-1,1) and b = (-1,-1) pointed above – rraallvv Mar 06 '13 at 03:20
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Man, this was all that I needed! Works flawlessy in 2d, thanks! – Gustavo Maciel Oct 15 '13 at 00:09
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All other duplicate questions should link to this question and this answer; this is so sparsely documented (doesn't even have a wikipedia article) – bright-star Nov 23 '13 at 04:23
And how would that look like in 3d? – Lennart Rolland Sep 01 '14 at 18:59
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Can you elaborate how the 2nd version works? Specifically the calculations you pass into atan2. – Tara Apr 27 '16 at 05:08
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@Tara The first parameter is the determinant, the second one the dot product. See also [this answer](https://stackoverflow.com/a/16544330/4566599). – Roi Danton Mar 19 '18 at 17:57
@RoiDanton Thank you! – Tara Mar 20 '18 at 07:20
didn't produce the correct angle between -Pi and Pi for me, but [this answer](https://stackoverflow.com/a/21484228/6077426) did – SaPropper Dec 13 '20 at 19:43

score 37 · Accepted Answer · answered Jan 27 '10 at 20:44

37

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

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

answered Jan 27 '10 at 20:44

Sparr

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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. – Derek Ledbetter Jan 27 '10 at 21:52
@Derek Good catch. I actually discovered this myself while implementing the solution. – Cerin Jan 28 '10 at 13:06
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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}. – user2083364 Oct 17 '13 at 08:26
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. – Helin Wang Jan 10 '14 at 02:47

Finding Signed Angle Between Vectors

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