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Acta Mathematica Hungarica

, Volume 156, Issue 1, pp 182–193 | Cite as

Uniform refinable 3D grids of regular convex polyhedrons and balls

  • A. Holhoş
  • D. Roşca
Article

Abstract

We construct a simple volume-preserving map from the cube [0,b]3 to the tetrahedron \({\{(x,y,z)\in \mathbb R^3}\), \({x \geq0}\), \({y \geq 0}\), \({z \geq 0}\), \({x+y+z\leq a\},}\) with \({a=b\sqrt[3]6}\). This map will allow us to construct equal-volume subdivisions of arbitrary tetrahedrons and arbitrary convex polyhedrons into polyhedral cells. Moreover, mapping the regular octahedron onto the ball using a volume-preserving map previously constructed by the authors, one can obtain uniform and refinable grids on the 3D ball by a simple procedure, starting from appropriate grids on the cube.

Mathematics Subject Classification

52B70 65N55 

Key words and phrases

equal volume projection uniform spherical grid hierarchical grid 

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References

  1. 1.
    A. Chow, Orthogonal and symmetric Haar wavelets on the three-dimensional ball, Master’s thesis, University of Toronto (Canada, 2010).Google Scholar
  2. 2.
    Griepentrog J. A., Höppner W., Kaiser H. C., Rehberg J.: A bi-Lipschitz continuous, volume preserving map from the unit ball onto a cube. Note Mat., 1, 177–193 (2008)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Holhoş A., Roşca D.: Area preserving maps and volume preserving maps between a class of polyhedrons and a sphere. Adv. Comput. Math., 43, 677–697 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    I. Pak, Lectures on discrete and polyhedral geometry, http://www.math.ucla.edu/~pak/geompol8.pdf.
  5. 5.
    Roşca D.: Wavelet analysis on some surfaces of revolution via equal area projection. Appl. Comput. Harmon. Anal., 30, 272–282 (2011)CrossRefGoogle Scholar
  6. 6.
    D. Roşca, A. Morawiec and M. De Graef, A new method of constructing a grid in the space of 3D rotations and its applications to texture analysis, Modelling Simul. Mater. Sci. Eng., 22 (2014) 075013, 17 pp.CrossRefGoogle Scholar
  7. 7.
    Pop V., Roşca D.: Generalized piecewise constant orthogonal wavelet bases on 2D-domains. Appl. Anal., 90, 715–723 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Y. Savoye, Cage-based Performance Capture, Springer (Switzerland, 2014).Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Department of MathematicsTechnical University of Cluj-NapocaCluj-NapocaRomania

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