OON SPHERICAL HARMONICS POSSESSING OCTAHEDRALSYMMETRY
YU. NESTERENKO
Abstract.
In this paper, we present the implicit representation of onespecial class of real-valued spherical harmonics with octahedral symmetry.Based on this representation we construct the rotationally invariant mea-sure of deviation from the specified symmetry. The spherical harmonicswe consider have some applications in the area of directional fields designdue to their ability to represent mutually orthogonal axes in 3D space notrelatively to their order and orientation. Introduction
We consider real-valued spherical harmonics of degree 4 on the unit sphere.These functions form 9D space with standard orthonormal basis Y , − , . . . , Y , (see [6]). Figure 1.
Spherical plots of basis functions Y , − , . . . , Y , .The object of study in this work is the 3D manifold of all normalized spheri-cal harmonics possessing octahedral symmetry. Up to multiplication by -1 allharmonics of this kind may be obtained from the reference harmonic ˜ h withcoordinates ˜ a = (0 , , , , (cid:114) , , , , (cid:114)
512 ) T ∈ R , by rotations a = R x ( α ) × R y ( β ) × R z ( γ ) × ˜ a, where α, β, γ are Euler angles α , β and γ (as well as in cases of different degreesthe space we consider has an important property: it is closed under 3D rotations,i.e. applying a rotation to a harmonic of degree 4 produces another harmonicof the same degree). Appendix A.1 describes the construction of the rotationmatrices R x , R y and R z . a r X i v : . [ c s . G R ] J a n YU. NESTERENKO
Figure 2.
The reference harmonic and its rotation.From geometrical point of view, a ( α, β, γ ) is constrained to be on the manifoldof dimension 3 embedded in the R . Appendix A.2 contains description oftopology of this manifold.The need of effective manipulation with the spherical harmonics of the spec-ified class was recently realized in context of the problem of directional fieldssmoothing (see [1, 7, 10, 11, 12]) due to their ability to represent mutuallyorthogonal axes in 3D space not relatively to their order and orientation.2. Results
The next lemma claims that harmonics manifold we study is simply intersectionof quadrics (hypersurfaces of the second order).
Lemma 1.
The manifold of all normalized spherical harmonics possessing oc-tahedral symmetry is given by the system of equations (2.1) (cid:40) a T a = 1 ,a T S k a = 0 , k = 1 , . . . , , where S , . . . , S are the symmetric matrices defined as follows. (2.2) S = √
28 0 0 0 0 0 0 0 00 7 0 0 0 0 0 0 00 0 − −
17 0 0 0 0 00 0 0 0 −
20 0 0 0 00 0 0 0 0 −
17 0 0 00 0 0 0 0 0 − (2.3) S = √ √ √ √ √ √ √ N SPHERICAL HARMONICS POSSESSING OCTAHEDRAL SYMMETRY 3 (2.4) S = √ √ −
140 0 0 0 0 9 0 − √ √ − √ √ − √ − −
14 0 0 0 0 0 0 0 (2.5) S = √ √ √ √ √ √ −
10 0 3 √ √ √
70 0 0 0 0 3 √ √ (2.6) S = √ √ √ √ − √
70 0 0 0 0 −
10 0 − √ √ √ −
10 0 0 0 0 02 √ − √ − √ Idea of proof.
The given implicit form was obtained by the standard techniquebased on rational parametrization of the unit circle and Gr¨obner basis construc-tion. It can be verified by direct calculations. (cid:3)
Matrices (2.2) - (2.6) were chosen among their different possible linear combi-nations based on the following additional consideration.
Lemma 2.
Real-valued function (2.7) d ( a ) = (cid:88) k =1 ( a T S k a ) , where a ∈ R , defines rotationally invariant measure of harmonic’s deviationfrom octahedral symmetry.Idea of proof. The result was obtained by averaging of some trial non-invariantdeviation measure over SO group. As in the previous lemma, rotational in-variance of the specified function can be verified by direct calculations. (cid:3) YU. NESTERENKO
Figure 3.
Spherical harmonic iterative symmetrization3.
Numerical example
In this section we show how deviation measure (2.7) works. We use the com-bination of the scale controlling term and the symmetry deviation term withpositive weights w and w (3.1) p ( a ; w , w ) = w ( a T a − + w (cid:88) k =1 ( a T S k a ) , w , w > , as the penalty function for simple gradient descent method. Figure 3 shows theconvergence process of the sample initial harmonic to the symmetrical one.The plots below describe the distance to the nearest symmetrical harmonic andsquare root of the penalty value. One can see distance-like behavior of squareroot of p ( a ; w , w ). 0 2 4 6 8 10 12 14 16 18 2010 − − − Iterations D i s t a n ce m e a s u r e s sqrt penaltydistancePenalty (3.1) together with squared gradient term gives a kind of Ginzburg-Landau energy for smoothing of spherical harmonics fields in 3D (see [9, 13]).4. Conclusion
The implicit representation of the manifold of normalized spherical harmonicswith octahedral symmetry is found. Based on this representation the rotation-ally invariant measure of deviation from considered symmetry is constructed.Numerical example describing the behavior of the constructed deviation mea-sure is given. The obtained results have some applications in the area of direc-tional fields design.
N SPHERICAL HARMONICS POSSESSING OCTAHEDRAL SYMMETRY 5 Appendix A.1
The rotational matrices R x , R y and R z for spherical harmonics of degree 4 aredefined as follows. (5.1) R z ( γ ) = cos 4 γ γ γ γ
00 0 cos 2 γ γ γ γ − sin γ γ − sin 2 γ γ − sin 3 γ γ − sin 4 γ γ (5.2) R x ( π √
14 0 − √ − √ √ √
14 00 2 √ √ √ − √
14 0 − √ √ − √ √ − √
14 0 0 0 0 0 00 0 0 0 √
35 0 − √ (5.3) R y ( β ) = R x ( π × R z ( β ) × R x ( π T (5.4) R x ( α ) = R y ( π T × R z ( α ) × R y ( π YU. NESTERENKO Appendix A.2
Topologically, the manifold of all rotations of the reference harmonic is thequotient space SO / S , where S denotes the group of order 24 of all octahedrasymmetries.It may be clearly described in Rodriguez representation of 3 D rotations (see[2]). Fundamental zone for octahedral symmetry in this representation has formof truncated cube with 6 regular octagonal faces and 8 regular triangular facesas shown in the picture below. x yz Figure 4.
Fundamental zone for rotations with octahedral symmetry.The topology we consider is obtained by gluing the opposite octagons and theopposite triangles with 45 ◦ and 60 ◦ turn respectively. Colors in the pictureindicate how the vertices map to each other. N SPHERICAL HARMONICS POSSESSING OCTAHEDRAL SYMMETRY 7
References [1] P.A. Beaufort, J. Lambrechts, C. Geuzaine, and J.F. Remacle. Quaternionic octahedralfields: SU(2) parameterization of 3D frames.
ArXiv , 1910.06240, 2019.[2] R. Becker and S. Panchanadeeswaran. Crystal rotations represented as Rodrigues vectors.
Texture, Stress, and Microstructure , 10:167–194, 1989.[3] M.A. Blanco, M. Florez, and M. Bermejo. Evaluation of the rotation matrices in the basisof real spherical harmonics.
Journal of Molecular Structure: THEOCHEM , 419(1-3):19–27, 1997.[4] C.H. Choi, J. Ivanic, M.S. Gordon, and K. Ruedenberg. Rapid and stable determinationof rotation matrices between spherical harmonics by direct recursion.
The Journal ofChemical Physics , 111(19):8825–8831, 1999.[5] J.R.A. Collado, J.F. Rico, R. Lopez, M. Paniagua, and G. Ramirez. Rotation of realspherical harmonics.
Computer Physics Communications , 52(3):323–331, 1989.[6] C. G¨orller-Walrand and K. Binnemans. Rationalization of crystal-field parametrization.
Handbook on the Physics and Chemistry of Rare Earths , 23:121–283, 1996.[7] J. Huang, Y. Tong, H. Wei, and H. Bao. Boundary aligned smooth 3D cross-frame field.
ACM Transactions on Graphics , 30(6):143:1–143:8, 2011.[8] J. Ivanic and K. Ruedenberg. Rotation matrices for real spherical harmonics. Directdetermination by recursion.
The Journal of Chemical Physics , 100(15):6342–6347, 1996.[9] A. Macq, M. Reberol, F. Henrotte, P.A. Beaufort, A. Chemin, J.F. Remacle, and J.V.Schaftingen. Ginzburg-Landau energy and placement of singularities in generated crossfields.
ArXiv , 2010.16381, 2020.[10] D. Palmer, D. Bommes, and J. Solomon. Algebraic representations for volumetric framefields.
ACM Transactions on Graphics , 39(2):16:1–16:17, 2020.[11] N. Ray and D. Sokolov. On smooth 3D frame field design.
ArXiv , 1507.03351, 2015.[12] J. Solomon, A. Vaxman, and D. Bommes. Boundary element octahedral fields in volumes.
ACM Transactions on Graphics , 36(3):28:1–28:16, 2017.[13] R. Vietel and B. Osting. An approach to quad meshing based on harmonic cross-valued maps and the Ginzburg-Landau theory.
SIAM Journal on Scientific Computing ,41(1):A452–A479, 2019.
Mechanical Analysis Division, Mentor, a Siemens Business
Email address ::