The minimum number of rotations about two axes for constructing an arbitrarily fixed rotation

For any pair of three-dimensional real unit vectors m^ and n^ with |m^Tn^|<1 and any rotation U, let Nm^,n^(U) denote the least value of a positive integer k such that U can be decomposed into a product of k rotations about either m^ or n^. This work gives the number Nm^,n^(U) as a function of U. Here, a rotation means an element D of the special orthogonal group SO(3) or an element of the special unitary group SU(2) that corresponds to D. Decompositions of U attaining the minimum number Nm^,n^(U) are also given explicitly.


Summary
For any pair of three-dimensional real unit vectorsm andn with |m Tn | < 1 and any rotation U, let Nm ,n (U) denote the least value of a positive integer k such that U can be decomposed into a product of k rotations about eitherm orn. This work gives the number Nm ,n (U) as a function of U. Here, a rotation means an element D of the special orthogonal group SO (3) or an element of the special unitary group SU(2) that corresponds to D. Decompositions of U attaining the minimum number Nm ,n (U) are also given explicitly.

Introduction
In this work, an issue on optimal constructions of rotations in the Euclidean space R 3 , under some restriction, is addressed and solved. By a rotation or rotation matrix, we usually mean an element of the special orthogonal group SO (3). However, we follow the custom, in quantum physics, to call not only an element of SO (3) but also that of the special unitary group SU(2) a rotation. This is justified by the well-known homomorphism from SU(2) onto SO(3) ( §3.4). Given a pair of three-dimensional real unit vectorsm andn with |m Tn | < 1, wherem T denotes the transpose ofm, let Nm ,n (A) denote the least value of a positive integer k such that any rotation in A can be decomposed into (constructed as) a product of k rotations about eitherm orn, where A = SU(2), SO (3). It is known that Nm ,n (SO(3)) = Nm ,n (SU(2)) = π/ arccos |m Tn | + 1 for any pair of three-dimensional real unit vectorsm andn with |m Tn | < 1 [1,2].
Then, a natural question arises: What is the least value, Nm ,n (U), of a positive integer k such that an arbitrarily fixed rotation U can be decomposed into a product of k rotations about eitherm orn? In this work, the minimum number Nm ,n (U) is given as an explicit function of U, where U is expressed in terms of parameters known as Euler angles [3,4].

Definitions
The notation to be used includes the following: N denotes the set of strictly positive integers; S 2 = {v ∈ x denotes the smallest integer not less than x ∈ R. As usual, arccos x ∈ [0, π ] and arcsin x ∈ [−π/2, π/2] for x ∈ [−1, 1]. The Hermitian conjugate of a matrix U is denoted by U † .

Parametrizations of the elements in SU(2)
The following lemma presents a well-known parametrization of SU(2) elements.

Lemma 3.2.
For any element U ∈ SU(2), there exist some α, γ ∈ R and β ∈ [0, π ] such that The parameters α, β and γ in this lemma are often called Euler angles. 2 The lemma can be rephrased as follows: any matrix in SU(2) can be written as with some complex numbers a and b such that |a| 2 + |b| 2 = 1 [3]. Hence, any matrix in SU(2) can be written as with some real numbers x,y,z and w such that w 2 + x 2 + y 2 + z 2 = 1. Take a real number θ such that cos(θ/2) = w and sin(θ/2) = 1 − w 2 = x 2 + y 2 + z 2 ; write x, y and z as x = −v x sin(θ/2), y = −v y sin(θ/2) (2) can be written as which is nothing but Rv(θ ) in (

Generic orthogonal axes and coordinate axes
Lemma 3.2 can be generalized as follows.
We also have the following lemma, which is easy but worth recognizing.

The minimum numbers of constituent rotations and optimal constructions of an arbitrary rotation
Here, we present the result establishing Nm ,n (U) with needed definitions.

Limits on constructions
In order to bound Nm ,n (D), etc., from below, we use the geodesic metric on the unit sphere S 2 , which is denoted by d. Specifically, This is the length of the geodesic connectingû andv on S 2 . We have the following lemma.
(Recall we have putRv(θ ) = F(Rv(θ )).) Lemma 5.1. Letn,m be arbitrary vectors in S 2 with δ = d(m,n) = arccosm Tn ∈ (0, π ]. Then, for any k ∈ N and φ 1 , . . . , φ 2k ∈ R, the following inequalities hold: This can be shown easily by induction on k using the triangle inequality for d. In what follows, (5.2) and (5.4) will be used in the following forms: These bounds hold when D and D ∈ SO(3) equal the product of 2k − 1 rotations and that of 2k rotations, respectively, in lemma 5.1 (since k is an integer). It will turn out that these bounds are tight.
6. Proof of the results 6.1. Structure of the proof Here, the structure of the whole proof of the results in this work is described. Theorem 4.3 is obtained as a consequence of lemma 6.2 to be presented. The constructive half of lemma 6.2 is due to propositions 4.4 and 4.7. The other half of lemma 6.2, related to limits on constructions, is due to lemma 5.1. Theorem 3.1 is derived from theorem 4.3 in appendix F.

Proof of propositions 4.4 and 4.7
The following lemma is fundamental to the results in this work.
Lemma 6.1. For any β, θ ∈ R and for anyû,l,m ∈ S 2 such thatl Tm = 0, the following two conditions are equivalent.
I. There exist some α, γ ∈ R such that Proof.
From the viewpoint of construction, we summarize the (most directly) suggested way to obtain an optimal construction of a given element U ∈ SU(2), where we assume δ = arccosm Tn ∈ (0, π/2] without loss of generality. If b(m, U) ≥ b(n, U), choose a construction that attains the minimum in (6.28). The construction is among that of proposition 4.4, that of proposition 4.7 and that of proposition 4.7 applied to U † in place of U [note U † = Rû 1 (φ 1 ) · · · Rû j (φ j ) implies U = Rû j (−φ j ) · · · Rû 1 (−φ 1 )]. If b(m, U) < b(n, U), interchangingm andn, apply the construction just described. 7 See appendix G for a detailed description of the above construction method.

Conclusion
This work has established the least value Nm ,n (U) of a positive integer k such that U can be decomposed into the product of k rotations about eitherm orn for an arbitrarily fixed element U in SU(2), or in SO(3), wherem,n ∈ S 2 are arbitrary real unit vectors with |m Tn | < 1. Decompositions of U attaining the minimum number Nm ,n (U) have also been given explicitly. [10][11][12] In this paper, an algorithm for solving the following unusual optimization problem was presented: minimize length(τ 1 , . . . , τ ν ,m 1 , . . . ,m ν ) subject to Rm 1 (τ 1 )Rm 2 (τ 2 ) · · · Rm ν (τ ν ) = U, ν ∈ N; τ j ∈ R,m j ∈ A for j = 1, . . . , ν where length(τ 1 , . . . , τ ν ,m 1 , . . . ,m ν ) := ν, U is an arbitrary fixed rotation and A ⊂ S 2 with |A| = 2 (the minimum of 'length', the primary part of an optimal solution, has been denoted by Nm ,n (U)). To this author's knowledge, only the work by D'Alessandro [5] and this paper have discussed this optimization problem. Naturally, the present author could not find any (explicit or implicit) indication that Brezov et al. [10][11][12] suggest considering the quantity Nm ,n (U) or analogues. A difference in background between this paper and Brezov et al. [10][11][12] may be understood as follows. While the situation assumed in this paper is that only two axes are available in constructing an arbitrary rotation, assuming a different situation results in problem formulations different from ours. For example, in Leite [7,Lemma 4.2] (attributed to Davenport), a situation where three axes are available but the number of factors in a decomposition is limited to three or less (in words, an equation Rm 1 (τ 1 )Rm 2 (τ 2 )Rm 3 (τ 3 ) = U, i.e. the above equation with ν = 3) is considered. In the series of Brezov et al. [10][11][12], they investigated such decompositions of the Davenport type, seemingly with emphasis on physical aspects. Note that Nm ,n (SU(2)) = max U Nm ,n (U) = π/ arccos |m Tn | + 1,m = ±n, is greater than three except in the classical case, wherem andn are orthogonal to each other.

Comments on Brezov et al.
Despite such differences in essence and background, note in the proof of this paper's formula (6.20) for the minimum even number of factors in lemma 6.2, on which the main theorem (theorem 4.3) relies, the case where the minimum even number is 2 or 4 needs an exceptional treatment (appendix C). This exceptionality would motivate one to read treatments on decompositions into two factors, and such can be found in Brezov et al. [10][11][12].
Funding statement. This work was supported by SCOPE (Ministry of Internal Affairs and Communications) and by Japan Society for the Promotion of Science KAKENHI grant nos. 22540150 and 21244007.

Appendix B. Details on angles in propositions 4.4 and 4.7
Examining the proof of lemma 6.1, we can be specific about α and γ to have the following lemma and corollary. In particular, the corollary gives a sufficient condition, (i), and two necessary conditions, (ii) and (iii), for Rn(θ ) = Rm(α)Rl(β)Rm(γ ), wherel,m andn are set as in propositions 4.4 and 4.7. Remarks 4.6 and 4.9 will be clear from (i). Later, (ii) and (iii) will be used in appendices C and D, respectively, though the use of them is not mandatory. (ii) for any α ∈ R and β ∈ (0, π ], if (B 1) holds for some θ, γ ∈ R, then β ≤ 2δ and there exist some j ∈ Z and t ∈ R such that 8 α = ±H t (β, δ) ± π 2 + π j; (iii) for any γ ∈ R and β ∈ (0, π ], if (B 1) holds for some θ, α ∈ R, then β ≤ 2δ and there exist some j ∈ Z and t ∈ R such that γ = ±H t (β, δ) ± π 2 + π j.
Proof. Setv = (v x , v y , v z ) T with v x = (l ×m) Tn , v y =l Tn and v z =m Tn .