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Hello,

I'm trying to get my head around the various properties of quaternions that all seem very similar, but I can't quite understand the underlying differences between them. I would like to know the differences between unit quaternions, purely imaginary quaternions, rotating a quaternion (does computing rotations using quaternions require them to be of unit magnitude?), and the exponential of a quaternion.

2. Relevant equations

One definition I have found is that "The exponential of a quaternion q = [a; v] is exp(q) =

exp(a)[cos(m);Lsin(m)], where m is a scalar and L a vector such that mL=v"

(taken from http://warwickmaths.org/files/Quaternions%20And%20Their%20Importance%2065.pdf [Broken] , page 6)

But another one says that the exponential of a purely imaginary q is "cos |q| + (q/|q|)sin|q|"Is this the same definition, but for the less general case of q having no scalar part?

If so, how are the two connected?

With regard to rotations, I know that we can think of the unit quaternion q = [a; v] , where v = tA, A being a unit vector (as q being a unit quaternion doesn't mean that v is a unit vector, correct?), as being represented as q = cos(θ/2) + A sin(θ/2) , where θ is the angle of rotation and A is the axis of rotation.

How does this relate to the exponential formula, as they look very similar indeed. What is the relation between |q| and θ?

3. The attempt at a solution

If someone can help me understanding these nuances with quaternions, I would be very grateful. I have already proven that the above representation of a rotation is true, but I would like to know the connection between that and the exponential.

Although, is it the case that the modulus of the quaternion is simply the angle of rotation? I think that is the connection between the two exponential cases. But how does that come about?

Many thanks.

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# Rotations with Quaternions and their exponential.

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