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Towards Uniform Online Spherical Tessellations

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Computing with Foresight and Industry (CiE 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11558))

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Abstract

The problem of uniformly placing N points onto a sphere finds applications in many areas. An online version of this problem was recently studied with respect to the gap ratio as a measure of uniformity. The proposed online algorithm of Chen et al. was upper-bounded by 5.99 and then improved to 3.69, which is achieved by considering a circumscribed dodecahedron followed by a recursive decomposition of each face. We analyse a simple tessellation technique based on the regular icosahedron, which decreases the upper-bound for the online version of this problem to around 2.84. Moreover, we show that the lower bound for the gap ratio of placing up to three points is \(\frac{1+\sqrt{5}}{2} \approx 1.618\). The uniform distribution of points on a sphere also corresponds to uniform distribution of unit quaternions which represent rotations in 3D space and has numerous applications in many areas.

I. Potapov has been partially supported by EPSRC grant EP/R018472/1.

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Notes

  1. 1.

    Note by abuse of notation that we write \(\rho ^T(T)\) rather than the more formal \(\rho ^T(\mu (T))\), as explained previously.

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Authors and Affiliations

  1. Department of Computer Science, Liverpool John Moores University, James Parsons Building, Byrom Street, Liverpool, L3-3AF, UK

    Paul C. Bell

  2. Department of Computer Science, University of Liverpool, Ashton Building, Ashton Street, Liverpool, L69-3BX, UK

    Igor Potapov

Authors
  1. Paul C. Bell
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  2. Igor Potapov
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Correspondence to Paul C. Bell .

Editor information

Editors and Affiliations

  1. Christian Albrechts University of Kiel, Kiel, Germany

    Florin Manea

  2. Durham University, Durham, UK

    Barnaby Martin

  3. Durham University, Durham, UK

    Daniël Paulusma

  4. Università degli Studi di Milano, Milan, Italy

    Giuseppe Primiero

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Bell, P.C., Potapov, I. (2019). Towards Uniform Online Spherical Tessellations. In: Manea, F., Martin, B., Paulusma, D., Primiero, G. (eds) Computing with Foresight and Industry. CiE 2019. Lecture Notes in Computer Science(), vol 11558. Springer, Cham. https://doi.org/10.1007/978-3-030-22996-2_11

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  • DOI: https://doi.org/10.1007/978-3-030-22996-2_11

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-22995-5

  • Online ISBN: 978-3-030-22996-2

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