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It’s hard enough to visualize a quaternion, geometrically speaking. A complex number is simple: it’s a point in a plane.

Suppose we had a number like this:

a + bi + cj

I supose you can visualize this as a point in a 3-dimensional space, where 1 of the dimensions is real, and two of them are imaginary.

Quaternions, on the other hand, have 4 dimensions, 1 real and 3 imaginary dimensions. So they are points in a very non-intuitive space. And unit quaternions are points on the “surface” of a hypersphere (whatever such mythical beast calls it its surface). How can I visualize linearly interpolating 2 such things as quaternions? I understand you often use something different for quaternion interpolation, called spheric interpolation. But before I try to understand the spheric interpolation, I would like to be able to root my knowledge on the familiar notion of linear interpolation.

And, if it's not to much to ask, how would I write the interpolation equation?

FinnTheHuman
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  • Have you seen this question (and the answer): http://math.stackexchange.com/q/30716/11323 – Kimball Jun 25 '16 at 11:19
  • Yes. That's more like a "how to do it" question. I want to be able to have some kind of visual grasp of what it means to linearly interpolate two quaternions. I'm not even thinking of their application as a means to describe a rotation. I'm thinking about them purely as numbers. – FinnTheHuman Jun 25 '16 at 13:31
  • The quaternions are a 4-d vector space. I visualize higher-dimensional vector spaces the same way I visualize $\mathbb R^3$. You just have two points in a linear space and your tracing a line between them. – Kimball Jun 25 '16 at 13:41
  • Still hard for me... I tried the analogy of linearly interpolating two points in space-time (because time, as a fourth dimension, is different from the other 3, which is kind of similar to the real part of the quaternion being different from the other 3 imaginary parts). But I could not visualize how to linearly interpolate 2 different points in space and time, so I still got nothing. – FinnTheHuman Jun 25 '16 at 14:14
  • Hey @Kimball turns out my space and time analogy works. I'm able to visualize quaternion linear interpolation using it. Our conversation ended up leading to some sort of answer. I would still love a visual response though, like a video or gif or something. – FinnTheHuman Jun 25 '16 at 17:28
  • I've seen videos of 4-d visualizations, but I don't know of any links off the top of my head. Try google? By the way, this may be of interest: http://mathoverflow.net/q/48448/6518 – Kimball Jun 27 '16 at 10:14
  • @Kimball Apparently I'm like that Hamilton guy, and need to ground my mathematical knowledge somewhere, or it just doesn't stick. Now whenever I'm confused about Quaternions I take a step back and just think of them as a point in euclidean space and an instant in time. Of course, I have yet to think about quaternion multiplication under this lens, because space has no imaginary component. the logarithmic and exponential functions are easier, just forget about time and think geometrically (1 scalar, 1 vector = 1 quaternion). But the multiplication is still beyond me. – FinnTheHuman Jun 29 '16 at 15:12

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