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# Angle between 2 quaternions

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Patrícia Falcão on 23 Aug 2018
Commented: James Tursa on 6 May 2021
How do we calculate angle between 2 quaternions? For example, what is the angle between x = ( 0.968, 0.008, -0.008, 0.252) and y = (0.382, 0.605, 0.413, 0.563)? How can I do it in matlab?
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Rik on 23 Aug 2018
What is the angle between two points? You need to define that first, or explain the mathematical definition, as I couldn't find the definition with a quick search.

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### Answers (2)

James Tursa on 23 Aug 2018
Edited: James Tursa on 23 Aug 2018
Assuming these represent attitude rotations from one coordinate frame to another, if you are simply asking what is the minimum rotation to take you from one quaternion to the other, you simply multiply one quaternion by the conjugate of the other and then pick off the rotation angle of the resulting quaternion.
But we really need to know what these quaternions represent, and what angle you are trying to recover, before we know what you want.
E.g., suppose x and y represent ECI->BODY rotation quaternions, and you want to know the minimum rotation angle that would take you from the x BODY position to the y BODY position. Then you could do this:
>> x = [ 0.968, 0.008, -0.008, 0.252]; x = x/norm(x); % ECI->BODY1
>> y = [ 0.382, 0.605, 0.413, 0.563]; y = y/norm(y); % ECI->BODY2
>> z = quatmultiply(quatconj(x),y) % BODY1->BODY2
z =
0.5132 0.6911 0.2549 0.4405
>> a = 2*acosd(z(4)) % min angle rotation from BODY1 to BODY2
a =
127.7227
But, again, these calculations are dependent on how I have the quaternions defined. Your specific case may be different.
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James Tursa on 6 May 2021
@Brent Raiteri Yes, correct. Looking back at this post I see I was using my own scalar-last functions for the reply. Which in retrospect is of course confusing given that the MATLAB functions are the same name. Good catch.

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Erik Blake on 13 May 2020
Just as with vectors, the cosine of the rotation angle between two quaternions can be calculated as the dot product of the two quaternions divided by the 2-norm of the both quaternions. Normalization by the 2-norms is not required if the quaternions are unit quaternions (as is often the case when describing rotations).
As with vectors, the dot product is calculated by summing the products of the four elements of the quaternion.
Note that this calculation yields the full rotation angle, not the half-angle as when converting from quaternions to rotation vectors.
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James Tursa on 13 May 2020
What you are describing is just the math for the scalar part of the quaternion multiply I described. This still only recovers the half angle of the rotation since it is the same calculation. I.e., the scalar part of the quatmultiply(quatconj(x),y) is this:
xs*ys - dot(-xv,yv) = xs*ys + dot(xv,yv)
And this is the same as the dot(x,y) operation you describe.
So if the quaternions represent two coordinate system transformations (my assumption), this result contains cos(half rotation angle), so acos( ) of this will recover the half rotation angle, not the full rotation angle. Or maybe I am misunderstanding you?
To be more robust for numerical issues, I could have done this:
z = quatmultiply(quatconj(x),y)
angle = 2 * atan2(norm(zv),zs)
where zv is the vector part of z and zs is the scalar part of z (either first or last element depending on convention).

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