In the fragment "Rotations of Space" (1819) in which Gauss outlined the general properties of a quaternions algebra, Gauss stated the following:
Given three consecutive scales with the relavent pivot points being $P,P',P''$ and angles of rotation being $\lambda,\lambda',\lambda''$, then the spherical triangle $PP'P''$ has angles $\frac{\lambda}{2},\frac{\lambda'}{2},\frac{\lambda''}{2}$.
["Scales" are Gauss's term for quaternions. Gauss does not say that explicitly but it is clear from the context that $P''$ and $\lambda''$ are the equivalent axis and angle of rotation to the composition of the first two rotations.]
After some effort, I succeeded in proving Gauss's statement up to a constant. Let us denote the first two quaternions as: $$q_1= \mathbb{cos}(\frac{\lambda}{2})+\mathbb{sin}(\frac{\lambda}{2})\vec{A}, q_2= \mathbb{cos}(\frac{\lambda'}{2})+\mathbb{sin}(\frac{\lambda'}{2})\vec{B}$$
and denote the resulting product of these two rotation quaternions is:
$$q_3 = q_1q_2 = \mathbb{cos}(\frac{\lambda''}{2})+\mathbb{sin}(\frac{\lambda''}{2})\vec{C}$$
After expansion of the product we get that the equivalent rotation $q_3$ is:
$$q_3 = (\mathbb{cos}(\frac{\lambda}{2})\mathbb{cos}(\frac{\lambda'}{2})-\mathbb{sin}(\frac{\lambda}{2})\mathbb{sin}(\frac{\lambda'}{2})\vec{A}\cdot \vec{B}) + (\mathbb{sin}(\frac{\lambda'}{2})\mathbb{cos}(\frac{\lambda}{2})\vec{B}+\mathbb{sin}(\frac{\lambda}{2})\mathbb{cos}(\frac{\lambda'}{2})\vec{A}+\mathbb{sin}(\frac{\lambda'}{2})\mathbb{sin}(\frac{\lambda}{2})\vec{A}\times \vec{B})$$
Now let us attack this problem using the sine law of spherical trigonometry. Denote by $\alpha,\beta,\gamma$ the sides $PP',P'P'',PP''$ of the spherical triangle. Then it is clear that:
$$sin\alpha=|\vec{A}\times\vec{B}|,sin\beta=|\vec{B}\times\vec{C}|, sin\gamma=|\vec{A}\times\vec{C}|$$
Let us find, for example, the value of $\mathbb{sin}\beta$. Then:
$$\mathbb{sin}\beta = |\vec{B}\times\vec{C}| = |\vec{B}\times(\frac{1}{\mathbb{sin}(\frac{\lambda''}{2})}(\mathbb{sin}(\frac{\lambda'}{2})\mathbb{cos}(\frac{\lambda}{2})\vec{B}+\mathbb{sin}(\frac{\lambda}{2})\mathbb{cos}(\frac{\lambda'}{2})\vec{A}+\mathbb{sin}(\frac{\lambda'}{2})\mathbb{sin}(\frac{\lambda}{2})\vec{A}\times \vec{B}))|$$
I will not detail all the algebra, but just mention that I used the following vector identites in the expansion of the product:
- $\vec{B}\times\vec{B}=0$ and $\vec{A}\cdot\vec{B} = cos\alpha$.
- The identity for the triple vector product: $\vec{A}\times(\vec{B}\times\vec{C}) = (\vec{A}\cdot\vec{C})\vec{B}-(\vec{A}\cdot\vec{B})\vec{C}$.
- After using these identities to simplify the resulting expression, one should take the norm of the vector in $||$.
Finally this leads to the expression $\mathbb{sin}(\frac{\lambda''}{2})\mathbb{sin}(\beta) = \mathbb{sin}(\frac{\lambda}{2})\mathbb{sin}(\alpha) \implies \frac{\mathbb{sin}(\alpha)}{\mathbb{sin}(\frac{\lambda''}{2})} = \frac{\mathbb{sin}(\beta)}{\mathbb{sin}(\frac{\lambda}{2})}$, which (together with the relation for $\gamma$ which can be derived in a similar way) is the spherical sine theorem up to a constant.
Questions
- The (partial) derivation here is quite cumbersome algebraically, so I would like to know how to streamline it and prove the full result.
- This derivation is very long while Gauss stated the result in 2-3 lines, so this made me wonder if I am missing something fundamental in my general understanding of versors (rotation quaternions). Perhaps there is an interpretation of versors that makes Gauss's remark much more transparent.