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I have begun reading about quaternions after a long time. However I should answer this soon to other members of my work group. My understanding of quaternions is composed of four values w,x,y and z and that they can be used to represent rotations.

I have received data that supposedly represent quaternions

What are the conditions that have to be met for these data to be valid?

I have been told one condition of the quaternions is they have to have unit length so I am checking this condition on the data

I was told that a second one is that the w part(the scalar part) has to be non-negative to represent a proper rotation. However watching videos about quaternions make me doubt this.

Are there other conditions for data to be valid as quaternions?

KansaiRobot
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  • If you have one value it represents a valid real value. If you have two values it represents a valid complex value. If you have 4 values, they represent a valid quaternion. If you're asking about quaternions that represent subgroups of operations in affine 3D space, that is an entirely different subject. – R. J. Mathar Mar 21 '24 at 12:00
  • I have data with four values w,x,y,z and they represent rotation. I have to verify if that data is valid and therefore asking for conditions of validity. I checked and the data is of unit length. – KansaiRobot Mar 21 '24 at 12:26

1 Answers1

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Every unit quaternion represents a rotation, i.e., if you have $(w, z, y, z)$ and $$ w^2 + x^2 + y^2 + z^2 = 1 $$ then it represents a rotation. Unfortunately, the rotation represented by $(w, x, y, z)$ and the one represented by $(-w, -x, -y, -z)$ are one and the same, so folks often say "Only use non-negative values of $w$" as a way to make things unique. Unfortunately, that doesn't really work, because $(0, 1, 0, 0)$ and $(0, -1, 0, 0)$ still represent the same rotation. But you can't just say "OK, make $w, x, y,$ and $z$ all be nonnegative", because that ends up ruling out some rotations that you want, like $(0, s, -s, 0)$, where $s = \sqrt{2}/2$.

So when you want to check whether your data is valid, the sum-of-squares is the only sure thing; non-negative $w$ is something you might expect to see ... a quick look at the data in a spreadsheet would tell you a lot, as would an email to the person who sent you the data ... and beyond that, there's probably no other fixed pattern in the data that you can check for.

John Hughes
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