Symmetry group of the equiangular cubed sphere

Abstract

The equiangular cubed sphere is a spherical grid, widely used in computational physics. This paper deals with mathematical properties of this grid. We identify the symmetry group, i.e. the group of the orthogonal transformations that leave the cubed sphere invariant. The main result is that it coincides with the symmetry group of a cube. The proposed proof emphasizes metric properties of the cubed sphere. We study the geodesic distance on the grid, which reveals that the shortest geodesic arcs match with the vertices of a cuboctahedron. The results of this paper lay the foundation for future numerical schemes, based on rotational invariance of the cubed sphere.

Least squares spherical harmonics approximation on the Cubed Sphere

“…The proof principle is close to the one of [5, Corollary 10]. To begin with, [3] proves that the group G of the cube, described by G = 1 e σ 1 2 e σ 2 3 e σ 3 , σ ∈ S 3 , ∈ {−1, 1} 3 , e 1 = (1, 0, 0), e 2 = (0, 1, 0), e 3 = (0, 0, 1), (30) is also the symmetry group of CS N . Therefore, the quadrature error defines a linear form…”

“…Then Case (ii) further divides by 2 this number. And finally Cases (iii-iv) multiply this number by 3 8 . Then, two facts suggest the approximation…”

Interpolation on the cubed sphere with spherical harmonics

“…Another issue is the symmetry properties of the interpolation space. This includes studying the invariance under the action of the group of the sphere, has to be undertaken, in the line of [2].…”

“…The orthogonal matrices U n , V n and the block diagonal matrix S n are dened by the SVD of the block in position (2,2)…”

“…One may wonder how (49) compares to the value N = 3N in (45). The analysis in [2] has etablished that the shortest geodesic distance sep(CS N ) is realized for any short arc 2 This is the principle behind the rank function in Matlab.…”