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, $\textbf{because they lie on the surface of the unit sphere.}$
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?
Any help is appreciated. Thank you.
Later in the paragraph the author uses the phrase "four-dimensional unit sphere" to mean the unit sphere in four dimensions. In math, we call this the "3-sphere" since this sphere is itself three-dimensional, even though it is situated in 4D Euclidean space. Outside of math, people may call it "hypersphere." The author is simply omitting the word "hyper," which is common. Since the hypersphere is three-dimensional, it makes sense to say only three dimensions are needed to visualize the unit (three-)sphere.
The tricky part is that the unit 3-sphere, which we denote $S^3$, is topologically different from 3D Euclidean space, which we denote $\mathbb{R}^3$. They seem to be interpolating a curve that exists within a 2D cross-section of $S^3$, just as how $S^2$ has circular cross-sections when intersecting it with (possibly skew) planes. These cross-sections will topologically be 2-spheres (what people usually think of when they hear the word "sphere"), which they show figures of.