Uniform refinable 3D grids of regular convex polyhedrons and balls

Holhoş, Adrian; Rosca, Daniela

doi:10.1007/s10474-018-0855-2

Cited by 5 publications

(6 citation statements)

References 5 publications

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“…For example, a volume-preserving mapping between the cube and tetrahedron [45] combined with a volume-preserving mapping between the tetrahedron and spherical octant [46] allows creating an equal volume spherical grid from one in the cube. While this specific grid is useful for multiresolution analysis and wavelets, the varying cell shapes make it a non-ideal choice for general-purpose data integration.…”

Section: Related Workmentioning

confidence: 99%

General Method for Extending Discrete Global Grid Systems to Three Dimensions

Ulmer

Hall

Samavati

2020

IJGI

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Geospatial sensors are generating increasing amounts of three-dimensional (3D) data. While Discrete Global Grid Systems (DGGS) are a useful tool for integrating geospatial data, they provide no native support for 3D data. Several different 3D global grids have been proposed; however, these approaches are not consistent with state-of-the-art DGGSs. In this paper, we propose a general method that can extend any DGGS to the third dimension to operate as a 3D DGGS. This extension is done carefully to ensure any valid DGGS can be supported, including all refinement factors and non-congruent refinement. We define encoding, decoding, and indexing operations in a way that splits responsibility between the surface DGGS and the 3D component, which allows for easy transference of data between the 2D and 3D versions of a DGGS. As a part of this, we use radial mapping functions that serve a similar purpose as polyhedral projection in a conventional DGGS. We validate our method by creating three different 3D DGGSs tailored for three specific use cases. These use cases demonstrate our ability to quickly generate 3D global grids while achieving desired properties such as support for large ranges of altitudes, volume preservation between cells, and custom cell aspect ratio.

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Section: Related Workmentioning

confidence: 99%

General Method for Extending Discrete Global Grid Systems to Three Dimensions

Ulmer

Hall

Samavati

2020

IJGI

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“…In this case we will obtain a new map, different from the one constructed in [4]. We restrict again to the case x, y, z > 0 because of the symmetry property of the map.…”

Section: The Case P = ∞mentioning

confidence: 99%

“…In our case, we can construct (UR) grids on a ball B p by transporting from a ball B p an already constructed (UR) grid. The simplest example of such a ball with (UR) grids is the cube B ∞ , but we have also constructed such (UR) grids on the regular octahedron B 1 (see [3,4]) and on the 3D Euclidean ball B 2 (see [3,7] ).…”

Section: Possible Applicationsmentioning

confidence: 99%

Volume Preserving Maps Between p-Balls

Holhoş

Rosca

2019

Symmetry

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We construct a volume preserving map U p from the p-ball B p (r) = x ∈ R 3 , x p ≤ r to the regular octahedron B 1 (r ), for arbitrary p > 0. Then we calculate the inverse U −1 p and we also deduce explicit expressions for U ∞ and U −1 ∞ . This allows us to construct volume preserving maps between arbitrary balls B p (r) and B p (r), and also to map uniform and refinable grids between them. Finally we list some possible applications of our maps.

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“…The prism B n is regular with height 2εr and edge ℓ n given in (5). The face B 0 of B n situated in the zone I 0 = I + 0 ∪ I − 0 has the equation…”

Section: Construction Of the Maps (T E N ) −1 And T E Nmentioning

confidence: 99%

“…For the prism B n , we triangularize each face of B n and then construct again 2n tetrahedrons with vertex O and base a triangle. In the second step, we split every of the 4n tetrahedrons into four smaller tetrahedrons having the same volume, this simple procedure being described in [5]. The refinement can be repeated, therefore we end up with a uniform and refinable grid of K n (r ′ , ε), whose image on the ball S 2 (r) will be a uniform and refinable grid.…”

Section: Transporting Uniform and Refinable Grids From The Polyhedrons Kmentioning

confidence: 99%

Area preserving maps and volume preserving maps between a class of polyhedrons and a sphere

Holhoş

Rosca

2016

Adv Comput Math

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For a class of polyhedrons denoted K n (r, ε), we construct a bijective continuous area preserving map from K n (r, ε) to the sphere S 2 (r), together with its inverse. Then we investigate for which polyhedrons K n (r ′ , ε) the area preserving map can be used for constructing a bijective continuous volume preserving map from K n (r ′ , ε) to the ball S 2 (r). These maps can be further used in constructing uniform and refinable grids on the sphere and on the ball, starting from uniform and refinable grids of the polyhedrons K n (r, ε) and K n (r ′ , ε), respectively. In particular, we show that HEALPix grids can be obtained by mappings polyhedrons K n (r, ε) onto the sphere.

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Uniform refinable 3D grids of regular convex polyhedrons and balls

Cited by 5 publications

References 5 publications

General Method for Extending Discrete Global Grid Systems to Three Dimensions

General Method for Extending Discrete Global Grid Systems to Three Dimensions

Volume Preserving Maps Between p-Balls

Area preserving maps and volume preserving maps between a class of polyhedrons and a sphere

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