Abstract
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.
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The work has been funded by a Grant for bilateral cooperation (PL 170/14-1) of the German Research Foundation (DFG). This is gratefully acknowledged.
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Dedicated to Werner Haußmann in memoriam
This work has been supported by the German Research Foundation, Grant PL 170/14-1.
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Roşca, D., Plonka, G. An Area Preserving Projection from the Regular Octahedron to the Sphere. Results. Math. 62, 429–444 (2012). https://doi.org/10.1007/s00025-012-0286-2
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DOI: https://doi.org/10.1007/s00025-012-0286-2