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quaternion rotation

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Let q be a unit quaternion that rotates angle t about axis n. I know q and -q represent the same rotation, but -q rotates 2pi - t about axis -n. In other words, -q rotates the long way.

So when slerping I choose either slerp(a, b, t) or slerp(a, -b, t) based on the b or -b that gives an acute angle with a.

So my question is:

Choosing the shorter arc corresponds to interpolating the orientation path in the shortest way and choosing the larger arc corresponds to interpolating the orientation path the longest way (like 2pi - t)?

Most books just state this but don't prove that the shortest arc corresponds to the most direct orientation path (although I guess it makes sense it doesn't seem obvious).

So is it like one hemisphere of the 4D unit sphere representats rotations (n, theta) and the other half represents rotations (-n, 2pi - theta)?

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Choosing the shorter arc corresponds to interpolating the orientation path in the shortest way and choosing the larger arc corresponds to interpolating the orientation path the longest way (like 2pi - t)?


Yes.

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