Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeDec 27th 2018
   • PermaLink
   Author: Urs
   Format: MarkdownItexfor the moment just to satisfy links <a href="https://ncatlab.org/nlab/revision/Euler+angle/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Euler+angle">current</a>

   for the moment just to satisfy links

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeJun 7th 2020
   • PermaLink
   Author: zskoda
   Format: MarkdownItexWrote few defining formulas for $SU(2)$ case. <a href="https://ncatlab.org/nlab/revision/diff/Euler+angle/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/Euler+angle/5">v5</a>, <a href="https://ncatlab.org/nlab/show/Euler+angle">current</a>

   Wrote few defining formulas for $SU(2)$ case.

   diff, v5, current

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeJun 7th 2020
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI wanted to write few things about relation to [[Pauli matrices]], but our entry takes unusual conventions disagreeing with [wikipedia](https://en.wikipedia.org/wiki/Pauli_matrices) and [wolframworld](https://mathworld.wolfram.com/PauliMatrices.html), as well as textbooks I have at hand, Sudbery (4.39), Ramond (4.18), Blohincev 1983 (59.9, 59.9'). One is that the square root normalization is put into the definition which then messes the commutation relations stated at [[special unitary group]], another is that the choices themselves are made antihermitian (with role of $x$ and $y$ interchanged in a way). The usual choice is that $i\sigma_x,i\sigma_y,i\sigma_z$ are the antihermitian generators of the real Lie group $su(2)$ and $\sigma_x^2 = \sigma_y^2 = \sigma_z^2 =1$. There is much variation in the literature on which representation is taken for $\gamma$-matrices but I think that for what we call Pauli matrices, the choice is standard, I think. But please let me know how to resolve this.

   I wanted to write few things about relation to Pauli matrices, but our entry takes unusual conventions disagreeing with wikipedia and wolframworld, as well as textbooks I have at hand, Sudbery (4.39), Ramond (4.18), Blohincev 1983 (59.9, 59.9’). One is that the square root normalization is put into the definition which then messes the commutation relations stated at special unitary group, another is that the choices themselves are made antihermitian (with role of $x$ and $y$ interchanged in a way). The usual choice is that $i\sigma_x,i\sigma_y,i\sigma_z$ are the antihermitian generators of the real Lie group $su(2)$ and $\sigma_x^2 = \sigma_y^2 = \sigma_z^2 =1$. There is much variation in the literature on which representation is taken for $\gamma$-matrices but I think that for what we call Pauli matrices, the choice is standard, I think. But please let me know how to resolve this.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeJun 7th 2020
   • PermaLink
   Author: zskoda
   Format: MarkdownItexSketch of the geometric construction for $SO(3)$. <a href="https://ncatlab.org/nlab/revision/diff/Euler+angle/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/Euler+angle/6">v6</a>, <a href="https://ncatlab.org/nlab/show/Euler+angle">current</a>

   Sketch of the geometric construction for $SO(3)$.

   diff, v6, current

5. • CommentRowNumber5.
   • CommentAuthorNikolajK
   • CommentTimeJun 8th 2020
   • (edited Jun 8th 2020)
   • PermaLink
   Author: NikolajK
   Format: MarkdownItexSurprised the word _tori_ doesn't fall on that page. Just two days ago, I incidentally wrote a script to generate a bunch of rotation matrices via a sweep over Euler angles, and I used [this parametrization](https://en.wikipedia.org/wiki/Rotation_formalisms_in_three_dimensions#Euler_angles_%E2%86%94_quaternion) since it's probably the one with the least trigonomentric functions. The "SO(3) article" on Wikipedia is for better or worse actually spread over about 8 articles - whenever you want some matrix representation you gotta check them all. I list a few of them [here at the start](https://gist.github.com/Nikolaj-K/907e0ea3cc958c483572950b746e0145). I have a few hands-on clips about the start of this but also [axis-angle stuff](https://youtu.be/gBMdTSXhYsY). As another recommendation, if you're looking for some more formulas, including the Jacobian matrices as they are computed on your phone, I like [this cheatsheet of an article](http://www.iri.upc.edu/people/jsola/JoanSola/objectes/notes/kinematics.pdf). Btw. I came across [this text](https://math.dartmouth.edu/~jvoight/quat-book.pdf) on quaternion algebras (but that's not just the Hamiltonian quaternions).

   Surprised the word tori doesn’t fall on that page.

   Just two days ago, I incidentally wrote a script to generate a bunch of rotation matrices via a sweep over Euler angles, and I used this parametrization since it’s probably the one with the least trigonomentric functions. The “SO(3) article” on Wikipedia is for better or worse actually spread over about 8 articles - whenever you want some matrix representation you gotta check them all. I list a few of them here at the start. I have a few hands-on clips about the start of this but also axis-angle stuff.

   As another recommendation, if you’re looking for some more formulas, including the Jacobian matrices as they are computed on your phone, I like this cheatsheet of an article. Btw. I came across this text on quaternion algebras (but that’s not just the Hamiltonian quaternions).

6. • CommentRowNumber6.
   • CommentAuthorzskoda
   • CommentTimeJun 9th 2020
   • PermaLink
   Author: zskoda
   Format: MarkdownItexIt seems the furthest general analogue in the setup of compact Lie groups so far is the scheme recently discovered in * S. L. Cacciatori, F. Dalla Piazza, A. Scotti, _Compact Lie groups: Euler constructions and generalized Dyson conjecture_, Trans. Amer. Math. Soc. 369 (2017), 4709-4724 [doi](https://doi.org/10.1090/tran/6795)

   It seems the furthest general analogue in the setup of compact Lie groups so far is the scheme recently discovered in

   • S. L. Cacciatori, F. Dalla Piazza, A. Scotti, Compact Lie groups: Euler constructions and generalized Dyson conjecture, Trans. Amer. Math. Soc. 369 (2017), 4709-4724 doi
7. • CommentRowNumber7.
   • CommentAuthorzskoda
   • CommentTimeJun 9th 2020
   • PermaLink
   Author: zskoda
   Format: MarkdownItex5, NikolajK, thank you. Quaternionic presentation of rotations to which you allude deserve and will eventually have a page separate from [[Euler angles]]. Although Euler invented both the Euler rotation formula and the Euler-Rodrigues parameters, which are quaternions in a disguise. The latter have a hyperbolic analogue * D. Brezov, C. Mladenova, I. Mladenov, _Vector parameters in classical hyperbolic geometry_, Proc. 15th Int. Conf. on Geometry, Integrability and Quantization, Ivaïlo M. Mladenov, A. Ludu, A. Yoshioka, eds. (Sofia: Avangard Prima, 2014) 79-105 [euclid](https://projecteuclid.org/euclid.pgiq/1436815703) Euler rotation formula is often mistakenly attributed to also to Rodrigues, see * Hui Cheng, K. C. Gupta, _An historical note on finite rotations_, J. Appl. Mech. Mar 1989, 56(1): 139-145 (7 pages) [doi](https://doi.org/10.1115/1.3176034)

   5, NikolajK, thank you.

   Quaternionic presentation of rotations to which you allude deserve and will eventually have a page separate from Euler angles. Although Euler invented both the Euler rotation formula and the Euler-Rodrigues parameters, which are quaternions in a disguise. The latter have a hyperbolic analogue

   • D. Brezov, C. Mladenova, I. Mladenov, Vector parameters in classical hyperbolic geometry, Proc. 15th Int. Conf. on Geometry, Integrability and Quantization, Ivaïlo M. Mladenov, A. Ludu, A. Yoshioka, eds. (Sofia: Avangard Prima, 2014) 79-105 euclid

   Euler rotation formula is often mistakenly attributed to also to Rodrigues, see

   • Hui Cheng, K. C. Gupta, An historical note on finite rotations, J. Appl. Mech. Mar 1989, 56(1): 139-145 (7 pages) doi
8. • CommentRowNumber8.
   • CommentAuthorzskoda
   • CommentTimeJun 9th 2020
   • PermaLink
   Author: zskoda
   Format: MarkdownItex5 NIkolajK, I wrote few lines on the role of quaternions under [[rotation]], following Berger's book, chapter 8.

   5 NIkolajK, I wrote few lines on the role of quaternions under rotation, following Berger’s book, chapter 8.

9. • CommentRowNumber9.
   • CommentAuthornLab edit announcer
   • CommentTimeJun 17th 2021
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexThere was a missing minus sign in the parameterization of $u$. It was not equal to the decomposition below. Due to this error, the determinant of $u$ was not $1$. I.e., the upper right and bottom left entries were not complex conjugates of each other. Anonymous <a href="https://ncatlab.org/nlab/revision/diff/Euler+angle/10">diff</a>, <a href="https://ncatlab.org/nlab/revision/Euler+angle/10">v10</a>, <a href="https://ncatlab.org/nlab/show/Euler+angle">current</a>

   There was a missing minus sign in the parameterization of $u$. It was not equal to the decomposition below. Due to this error, the determinant of $u$ was not $1$. I.e., the upper right and bottom left entries were not complex conjugates of each other.

   Anonymous

   diff, v10, current