# Real usage of “pure” quaternions in stereometry?

There are two major categories of the "quaternions". It is well-known that a (nonzero) versor represents a three-dimensional rotation operator. A versor is a unit quaternion or a normalized quaternion: $$a+b{\bf i}+c{\bf j}+d{\bf k}\text{ with }a^2+b^2+c^2+d^2=1^2\text{, where }a,b,c,d\in\mathbb R\text.$$ It is also known that (the vector part of) a vector quaternion or a pure quaternion describes a spatial location, where a vector quaternion is a quaternion that has vanishing scalar component (see, e.g., this section and this section).

The former has proved useful in many places; nonetheless, I cannot find any application (other than rotation operations in 3 dimensions) for the latter (namely, a quaternion of the form $$0+b{\bf i}+c{\bf j}+d{\bf k}$$). Of note, the latter itself is a (special class of) quaternion rather that the location of a 3D point – just like although a complex number corresponds to a 2D point, a complex number itself is neither coordinates of that point nor a $${\rm2}$$-dimensional vector. Whilst I notice that complex numbers are useful in fielding certain intimidating problems (including, but not limited to, olympiad-level ones) in planar geometry (not complex geometry), I wonder if there exists some case where the so-called vector quaternions can be used to address non-trivial problems in solid geometry.
For instance, now that Ptolemy's classical theorem may be derived by elucidating the four points as complex numbers, and it has been generalized to $${\rm3}$$-dimensional Euclidean space (see below), can vector quaternions be utilized for showing the following 3D versions?

1. For any four (distinct) points $$A$$, $$B$$, $$C$$, and $$D$$ in $$\mathbb R^\color{red}3$$, they are either concyclic or collinear iff $$\begin{vmatrix}0&{\left\lvert AB\right\rvert}^2&{\left\lvert AC\right\rvert}^2&{\left\lvert AD\right\rvert}^2\\{\left\lvert BA\right\rvert}^2&0&{\left\lvert BC\right\rvert}^2&{\left\lvert BD\right\rvert}^2\\{\left\lvert CA\right\rvert}^2&{\left\lvert CB\right\rvert}^2&0&{\left\lvert CD\right\rvert}^2\\{\left\lvert DA\right\rvert}^2&{\left\lvert DB\right\rvert}^2&{\left\lvert DC\right\rvert}^2&0\end{vmatrix}=0\text.$$
2. For five arbitrary points $$A$$, $$B$$, $$C$$, $$D$$, and $$E$$ in $$\mathbb R^\color{red}3$$, they lie on a sphere of (in)finite radius iff $$\begin{vmatrix}0&{\left\lvert AB\right\rvert}^2&{\left\lvert AC\right\rvert}^2&{\left\lvert AD\right\rvert}^2&{\left\lvert AE\right\rvert}^2\\{\left\lvert BA\right\rvert}^2&0&{\left\lvert BC\right\rvert}^2&{\left\lvert BD\right\rvert}^2&{\left\lvert BE\right\rvert}^2\\{\left\lvert CA\right\rvert}^2&{\left\lvert CB\right\rvert}^2&0&{\left\lvert CD\right\rvert}^2&{\left\lvert CE\right\rvert}^2\\{\left\lvert DA\right\rvert}^2&{\left\lvert DB\right\rvert}^2&{\left\lvert DC\right\rvert}^2&0&{\left\lvert DE\right\rvert}^2\\{\left\lvert EA\right\rvert}^2&{\left\lvert EB\right\rvert}^2&{\left\lvert EC\right\rvert}^2&{\left\lvert ED\right\rvert}^2&0\end{vmatrix}=0\text{. }$$

A proof using (elementary) linear algebra for the first generalization can be found here, and the second one was proved by Arthur Cayley in 1841. Notwithstanding, neither of these proofs makes use of the alleged vector quaternions. Inasmuch as complex numbers are instrumental in the classic proof the original Ptolemy's theorem, is it possible to prove the two aforementioned three-dimensional analogues using vector quaternions?

Edit. Complex numbers can doubly serve as rotation operators as well as position vectors, and so can quaternions. What I am looking forward to is: In view of the fact that when we use complex numbers, the context does not need to be rotation operations, and we may simply use complex numbers to undertake geometrical analysis in a coordinate-free manner (for example, proving Ptolemy's theorem without "synthetic" axiomatic methods or explicit coordinate-based method), which often has nothing to do with the algebra of rotations for the plane, can't we do the same thing for “pure” quaternions in stereometry?

• I cannot find any application for the latter (namely, a quaternion of the form 0+𝑥𝐢+𝑦𝐣+𝑧𝐤). Huh? Unless you're going about composing rotations without ever actually applying them, you'd have to express the input position $x$ in that form to compute $qxq^{-1}$ for a unit quaternion $q$ to see what the output position is. Commented Mar 27 at 14:39
• You seem to be drawing a false analogy. If complex numbers a points in 2D space, then analogously general quaternions are points in 4D space. Why would you expect vector quaternions (pure imaginary) to have a similar role to complex numbers (real+imaginary)? Commented Mar 27 at 15:55
• (2) can almost certainly be expressed using conformal geometric algebra. (1) probably can as well but isn't as clear to me. I don't think you can do anything using only vector quaternions. Commented Mar 27 at 16:08
• @rschwieb I think that you may misunderstand my words: "The former" means the unit quaternion, and "the latter" means the vector/pure quaternion. If I understand right, what we use in 3D rotation is actually the unit quaternion (the former) rather than the vector quaternion (the latter). (At least the Wikipedia article mentions that “a quaternion representation of rotation is written as a versor (normalized quaternion)”.) Commented Mar 27 at 16:20
• @NicholasTodoroff The Wikipedia articles Using quaternions as rotations (the third paragraph) and Quaternions and three-dimensional geometry (the first paragraph) appear to suggest that a vector quaternion (not a unit quaternion) does denote a three-dimensional point. So it seems that such analogy is still reasonable? Commented Mar 27 at 16:58

I wonder if there exists some case where the so-called vector quaternions can be used to address non-trivial problems in solid geometry.

If you consider representation of rotations as addressing a non-trivial problem in solid geometry, then I think the following counts as well.

The subset of unit length quaternions that have real part zero can be used as reflections in a way that is almost like how the unit-length quaternions are used for rotations.

In detail, if $$n$$ is a unit-length quaternion with real part $$0$$, it defines a reflection $$x\mapsto nxn$$, and $$n$$ is the normal vector to the plane of reflection. ($$x$$ is, again, from the subspace of $$\mathbb H$$ representing 3-space.)

Books that talk about applications of Clifford algebras also explain how to do projective geometry and conformal geometry and other things with higher dimensional analogues of $$\mathbb H$$. There are really a ton of applications out there.