Visualize the interpolated unit quaternion on the surface of the unit sphere 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., &amp; 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.
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.