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An area preserving projection from the regular octahedron to the sphere. (English) Zbl 1260.86015

Summary: In this paper, we propose an area preserving bijective map from the regular octahedron to the unit sphere \({\mathbb{S}^2}\), both centered at the origin. The construction scheme consists of two steps. First, each face \(F_i\) of the octahedron is mapped to a curved planar triangle \({\mathcal{T}_i}\) of the same area. Afterwards, each \({\mathcal{T}_i}\) is mapped onto the sphere using the inverse Lambert azimuthal equal area projection with respect to a certain point of \({\mathbb{S}^2}\). The proposed map is then used to construct uniform and refinable grids on a sphere, starting from any triangular uniform and refinable grid on the triangular faces of the octahedron.

MSC:

86A30 Geodesy, mapping problems
85-08 Computational methods for problems pertaining to astronomy and astrophysics
86-08 Computational methods for problems pertaining to geophysics

Software:

Healpix
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Full Text: DOI

References:

[1] Alexander R.: On the sume of distances between N points on the sphere. Acta Mathematica. 23, 443–448 (1972) · Zbl 0265.52009
[2] Grafared E.W., Krumm F.W.: Map Projections, Cartographic Information Systems. Springer, Berlin (2006) · Zbl 1134.86001
[3] Górski K.M., Wandelt B.D., Hivon E., Banday A.J., Hansen F.K., Reinecke M., Bartelmann M.: HEALPix: A framework for high-resolution discretization and fast analysis of data distributed on the sphere. Astrophys. J. 622(2), 759 (2005) · doi:10.1086/427976
[4] Leopardi P.: A partition of the unit sphere into regions of equal area and small diameter. Electron. Trans. Numer. Anal. 25, 309–327 (2006) · Zbl 1160.51304
[5] Roşca D.: New uniform grids on the sphere. Astron. Astrophys. 520, A63 (2010) · doi:10.1051/0004-6361/201015278
[6] Roşca D.: Uniform and refinable grids on elliptic domains and on some surfaces of revolution. Appl. Math. Comput. 217(19), 7812–7817 (2011) · Zbl 1218.65141 · doi:10.1016/j.amc.2011.02.095
[7] Roşca D., Plonka G.: Uniform spherical grids via equal area projection from the cube to the sphere. J. Comput. Appl. Math. 236, 1033–1041 (2011) · Zbl 1231.65044 · doi:10.1016/j.cam.2011.07.009
[8] Song, L., Kimerling, A.J., Sahr K.: Developing an equal area global grid by small circle subdivision. In: Goodchild, M., Kimerling, A.J. (eds.) Discrete Global Grids. National Center for Geographic Information & Analysis, Santa Barbara (2002)
[9] Snyder J.P.: Flattening the Earth. University of Chicago Press, Chicago (1990)
[10] Teanby N.A.: An icosahedron-based method for even binning of globally distributed remote sensing data. Comput. Geosci. 32(9), 1442–1450 (2006) · doi:10.1016/j.cageo.2006.01.007
[11] Tegmark M.: An icosahedron-based method for pixelizing the celestial sphere. Astrophys. J. 470, L81–L84 (1996) · doi:10.1086/310310
[12] Zhou,Y.M.: Arrangements of points on the sphere. PhD thesis, Mathematics, Tampa, FL (1995)
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