# 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., & 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.

• They mean the unit sphere in four dimensions, which is what you're calling the hypersphere. In modern mathematical terminology it is rarely useful to restrict the term "sphere" to refer only to the one in three-dimensional Euclidean space.
– user106
Nov 20 '19 at 8:06
• Okay it might help to think that the quaternionion is a weirdly encoded vector representation where we encode the system so that it makes sense only if the vector length is one. Offcourse the encoding is weird indeed and has properties that the vector representation does not have. Nov 20 '19 at 16:14
• Thank you all. 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 $\mathbb{S}^2$?
– Ali
Nov 20 '19 at 22:06
• You will likely get much better answers for this sort of question here: math.stackexchange.com While quaternions are used in computer graphics, most graphics engineers don't put a significant amount of effort into understanding the theoretical mathematics any more deeply than what you've already stated in your question.
– Dan
Dec 21 '19 at 18:43
• A beautiful visual description here. Apr 27 '20 at 15:01

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.

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.

• Thank you for your help. But I still don't get why we can plot the unit quaternion as a point on a sphere. I have summarized my question as the second question, I wonder if you may have some suggestions?
– Ali
Nov 20 '19 at 22:05

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.