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• CommentRowNumber1.
• CommentAuthorzskoda
• CommentTimeJun 1st 2020

Wrote a minimum of substance to this entry. My interest is prompted by current thinking on $n$-dimensional analogues of Euler angles. There are several articles (mainly in quantum chemistry and mathematical physics literature) around 1969-1974, which introduce some analogues via calculational procedures. In my taste an elementary geometrical introduction using both intrinsic and extrinsic approach, in a spirit of Euler, will be more suggestive.

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeJun 1st 2020

More material, and corrections to the previous.

• CommentRowNumber3.
• CommentAuthorzskoda
• CommentTimeJun 1st 2020

Defined the angle of rotation geometrically in any number of dimensions.

• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeJun 1st 2020

I did not separate the paragraph on 2d case from the rest of the properties section because some $n$-dimensional definitions directly continue and use the 2d definition (e.g. for the angle of rotation).

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJun 1st 2020

Replaced pointer to Wikipedia’s entry on Euler angles with pointer to Euler angles.

• CommentRowNumber6.
• CommentAuthorzskoda
• CommentTimeJun 7th 2020
• (edited Jun 7th 2020)

Thank you, Urs. I will have to extend the stub Euler angles soon.

The statement which I put under rotation (and you likewise wrote in the Euler angles entry) that they supply global coordinates is just a manifold part of the story (besides there is nonuniqueness at a part of the boundary of the domain). More striking is the group part of the story, namely they give an ordered multiplicative decomposition of rotations into elementary ones (around coordinate axes); if we extend the domain of one particular angle to double domain, then we have multivalued coordinates which therefore lift to double cover $SU(2)$. Also I will have to expand on the relation to Pauli matrices in that case, Urs mentioned them under related items in Euler angle. Another striking property, which may be not true for multiplicative decompositions in other multiplicative generators is that both so called extrinsic and intrinsic decompositions may be made with the same angles, but with opposite order. Extrinsic is the decomposition into rotations about fixed coordinate axes, while the intrinsic is about the moving system of a rigid body which moves after each step in the process. This is more in the spirit of active vs. passive operations as far as single step is concerned, but not globally (there is no change in sign which pertains to active/passive points of view, in particular if only one Euler angle is nonzero than intrinsic and extrinsic decompositions are identical).

• CommentRowNumber7.
• CommentAuthorzskoda
• CommentTimeJun 7th 2020
• (edited Jun 7th 2020)

By the way, Urs, you mention “Generalized constructions apply to other Lie groups”. $SU(2)$ and $SO(3)$ are the original case(s) and I have seen that there are several papers for some series like real and complex orthogonal groups and unitary groups (e.g. Bertini et al. J.Math.Phys. 2006, doi). Are you aware of some uniform construction which goes beyond single series of classical groups ?

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeJun 7th 2020

• CommentRowNumber9.
• CommentAuthorzskoda
• CommentTimeJun 9th 2020
• (edited Jun 9th 2020)

I wrote this, but when submitting to the entry the server keeps failing.

Euler’s theorem on rotations: a composition of rotations of a sphere is a rotation of a sphere. It is trivial that it is an isometry (when viewed as a transformation of the space), the nontrivial part of the theorem is that there is a $(n-2)$-hyperplane fixed by the composition, the statement which itself is also called the Euler’s theorem. This is trivial in dimension 2 and nontrivial in dimension 3 proved by Euler and in higher dimension. Each orthogonal transformation in odd dimensional real space has an invariant (1-dimensional) axis, the statement also called the Euler’s theorem (this boils down to the statement that the eigenvalues of the orthogonal matrix are on the unit circle).

## Relation to quaternions in 3d and 4d

Consider the vector space $\mathbf{R}^3\subset\mathbf{H}$ of imaginary quaternions. Let $s\in\mathbf{R}^3\backslash\{0\}$. Then $\sigma_{s^\perp}:q\mapsto -s q s^{-1}$ is a reflection with respect to the plane through origin and orthogonal to $q$. If $s\in\mathbf{H}$ is a quaternion the map $\rho'_s:q\mapsto s q s^{-1}$ leaves the space $\mathbf{R}^3$ of imaginary quaternions invariant and the restriction $\rho_s = \rho'_s|_{\mathbf{R}^3}$ is a rotation. Every rotation arises in that way and the map $s\mapsto\rho_s$ factorizes as the quotient map to the space of real rays $\mathbf{H}\to\mathbf{R}P^3$ and a homeomorphism $\mathbf{R}P^3\cong SO(3)$. The composition of rotations corresponds to the multiplication of quaternions. Components $a,b,c,d$ of quaternions involved fixing $s\in\mathbf{H}$ at the unit sphere $S^3\subset\mathbf{H}$ are called Euler-Rodrigues parameters. There is a redundancy of 1 parameter (plus the matter of 2-element kernel in the map $S^3\to\mathbf{R}P^3$), but this choice has its advantage in numerical modeling as it avoids the special role of the boundaries for the Euler angles.

There is a similar construction for $SO(4)$ (see Berger, 8.9.8). Indeed, there is an epimorphism of Lie groups

$\tau : S^3\times S^3\to SO(4),\,\,\,\,(s,r)\mapsto \{q\mapsto s q\overline{r}\}$

with kernel $\{(1,1),(-1,-1)\}$. The direct product structure of the domain can be used to easily exhibit some nontrivial subgroups in $SO(4)$.

P.S. it worked in the 4th attempt.

• CommentRowNumber10.
• CommentAuthorzskoda
• CommentTimeJun 9th 2020

Added Euler’s theorem on rotations and the role of quaternions in a parametrization of $SO(3)$ (Euler-Rodrigues parameters) and in a parametrization of $SO(4)$.

• CommentRowNumber11.
• CommentAuthorGuest
• CommentTimeJun 19th 2021

Re #9: what exactly does “C. Berger, Geometry” refer to? Google (Books) hasn’t been very useful…

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeJun 19th 2021

The section 8.9.8 that is meant is not in “C. Berger”, but in

See here.