Visualize the interpolated unit quaternion on the surface of the unit sphere

Question

My major is mechanical, please forgive me for asking questions that may seem trivial to you.

First, I'm reading the following paper:

Dam, Erik B., Martin Koch, and Martin Lillholm. Quaternions, interpolation and animation. Vol. 2. Copenhagen: Datalogisk Institut, Københavns Universitet, 1998.

On page 35, it says:

Since quaternion space is four-dimensional, we cannot visualise the interpolated curves directly. We will always interpolate between unit quaternions, and the interpolated quaternions will always (with a few exceptions in chapter 6 on page 38 and 69) be unit quaternions. This means that we only need three dimensions to visualize the interpolation curves, because they lie on the surface of the unit sphere.

I don't quite get the last sentence: why only three dimensions are needed to visualize the unit quaternion or why unit quaternion lies on the surface of the unit sphere? I thought they lie on the unit hyper-sphere. Does the author assume that we are viewing the quaternions from the south pole of the hyper-sphere?

Second, I was also following this paper:

Ramamoorthi, R., & Barr, A. H. (1997). Fast construction of accurate quaternion splines.

I found they are also treating (plotting) a unit quaternion as a point on a unit sphere $\mathbb{S}^{2}$. I wonder if you could recommend some resources in why we can view a unit quaternion $q = (a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k})$ as a point on a unit sphere $\mathbb{S}^{2}$?

Any help is appreciated. Thank you.

Answer 1

I think Fig. 1 in the referenced paper is only a 2D-sketch, as the unit quaternions lie on the 3-dimensional sphere embedded in four dimensions.

Quaternions are used to represent orientations / rotations and these can be associated with the axis-angle representation (determining the angle from the first quaternion coefficient and the axis from the last three coefficients.) However, I could think of a visualization in 3D, where a point in space is associated to a unit quaternion / or axis-angle: Place the point in direction of the axis "angle"-units away from the origin. This would be a sphere of radius two pi, where also internal points represent rotations. I dont know of a direct visualization in 3D of the quaternion coefficients.

Answer 2

Because given 3 values of a unit quaterion we can derive the fourth value.

Which means there exists a bijective mapping from unit quaternion to 3D space.

Though for visualization you want the mapping that makes what is happening the most easily discernible. However a usable visualization doesn't need to be fully bijective.

Answer 3

Though quaternions cannot be visualized as points on the surface of a unit sphere in 3 dimensions, they can be readily visualized as unit tangents attached to the surface of the unit sphere. Here is an example visualization.

One could construct a similar visualization by drawing an arrow, starting at $q * \hat{x}$ and ending at $q * \hat{y}$, where $\hat{x}$ is the unit x vector, $\hat{y}$ is the unit y vector, and $q$ is the unit quaternion we are trying to visualize. This allows the drawing of nice smooth curves, while remaining bijective.