research-article

Spherical fibonacci mapping

Authors:

Benjamin Keinert,

Matthias Innmann,

Michael Sänger,

Marc StammingerAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 34, Issue 6

Article No.: 193, Pages 1 - 7

https://doi.org/10.1145/2816795.2818131

Published: 02 November 2015 Publication History

Abstract

Spherical Fibonacci point sets yield nearly uniform point distributions on the unit sphere S² ⊂ R³. The forward generation of these point sets has been widely researched and is easy to implement, such that they have been used in various applications.

Unfortunately, the lack of an efficient mapping from points on the unit sphere to their closest spherical Fibonacci point set neighbors rendered them impractical for a wide range of applications, especially in computer graphics. Therefore, we introduce an inverse mapping from points on the unit sphere which yields the nearest neighbor in an arbitrarily sized spherical Fibonacci point set in constant time, without requiring any precomputations or table lookups.

We show how to implement this inverse mapping on GPUs while addressing arising floating point precision problems. Further, we demonstrate the use of this mapping and its variants, and show how to apply it to fast unit vector quantization. Finally, we illustrate the means by which to modify this inverse mapping for texture mapping with smooth filter kernels and showcase its use in the field of procedural modeling.

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References

[1]

Brockmeyer, E., Poupyrev, I., and Hudson, S. 2013. Papillon: Designing curved display surfaces with printed optics. In Proceedings of the 26th annual ACM symposium on User interface software and technology, ACM, 457--462.

Digital Library

[2]

Cigolle, Z. H., Donow, S., Evangelakos, D., Mara, M., McGuire, M., and Meyer, Q. 2014. A survey of efficient representations for independent unit vectors. Journal of Computer Graphics Techniques (JCGT) 3, 2 (April), 1--30.

[3]

Dammertz, S., and Keller, A. 2008. Image synthesis by rank-1 lattices. In Monte Carlo and Quasi-Monte Carlo Methods 2006. Springer, 217--236.

[4]

Engelhardt, T., and Dachsbacher, C. 2008. Octahedron environment maps. In Proceedings of the Vision, Modeling, and Visualization Conference 2008, VMV 2008, Konstanz, Germany, October 8-10, 2008, 383--388.

[5]

Fowler, D. R., Prusinkiewicz, P., and Battjes, J. 1992. A collision-based model of spiral phyllotaxis. In Proceedings of the 19th Annual Conference on Computer Graphics and Interactive Techniques, ACM, New York, NY, USA, SIGGRAPH '92.

Digital Library

[6]

González, Á. 2010. Measurement of Areas on a Sphere Using Fibonacci and Latitude-Longitude Lattices. Mathematical Geosciences 42, 1, 49--64.

[7]

Greene, N. 1986. Environment mapping and other applications of world projections. Computer Graphics and Applications, IEEE 6, 11 (Nov), 21--29.

Digital Library

[8]

Hannay, J. H., and Nye, J. F. 2004. Fibonacci numerical integration on a sphere. Journal of Physics A: Mathematical and General 37, 48.

[9]

Hart, J. C. 1996. Sphere tracing: A geometric method for the antialiased ray tracing of implicit surfaces. The Visual Computer 12, 10, 527--545.

[10]

Keinert, B., Schäfer, H., Korndörfer, J., Ganse, U., and Stamminger, M. 2014. Enhanced sphere tracing. In Smart Tools and Apps for Graphics - Eurographics Italian Chapter Conference, Eurographics Association, Cagliari, Italy, 1--8.

[11]

Larkins, R. L., Cree, M. J., and Dorrington, A. A. 2012. Analysis of binning of normals for spherical harmonic cross-correlation. In IS&T/SPIE Electronic Imaging, International Society for Optics and Photonics.

[12]

Marques, R., Bouville, C., Ribardière, M., Santos, L. P., and Bouatouch, K. 2013. Spherical Fibonacci Point Sets for Illumination Integrals. In Computer Graphics Forum, vol. 32, Wiley, 134--143.

[13]

Meyer, Q., Süssmuth, J., Sussner, G., Stamminger, M., and Greiner, G. 2010. On floating-point normal vectors. Computer Graphics Forum 29, 4, 1405--1409.

Digital Library

[14]

Newell, A., and Shipman, P. 2005. Plants and fibonacci. Journal of Statistical Physics 121, 5--6, 937--968.

[15]

Smith, J., Petrova, G., and Schaefer, S. 2012. Encoding Normal Vectors using Optimized Spherical Coordinates. Computers & Graphics 36, 5.

Digital Library

[16]

Swinbank, R., and Purser, R. J. 2006. Fibonacci grids: A novel approach to global modelling. Quarterly Journal of the Royal Meteorological Society 132, 619, 1769--1793.

[17]

Vogel, H. 1979. A better way to construct the sunflower head. Mathematical Biosciences 44, 179--189.

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Tang JQu YMa EYue YSun XGan L(2025)Fibonacci array-based temporal-spatial localization with neural networksApplied Acoustics10.1016/j.apacoust.2024.110368228(110368)Online publication date: Jan-2025
https://doi.org/10.1016/j.apacoust.2024.110368
Paulus AEibert T(2024)Inverse Source-Based Three-Antenna Methods in the Near Field2024 18th European Conference on Antennas and Propagation (EuCAP)10.23919/EuCAP60739.2024.10501540(1-5)Online publication date: 17-Mar-2024
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Index Terms

Spherical fibonacci mapping
1. Computing methodologies
  1. Computer graphics

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Published In

cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 34, Issue 6

November 2015

944 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/2816795

Issue’s Table of Contents

Copyright © 2015 ACM.

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Publication History

Published: 02 November 2015

Published in TOG Volume 34, Issue 6

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Tang JQu YMa EYue YSun XGan L(2025)Fibonacci array-based temporal-spatial localization with neural networksApplied Acoustics10.1016/j.apacoust.2024.110368228(110368)Online publication date: Jan-2025
https://doi.org/10.1016/j.apacoust.2024.110368
Paulus AEibert T(2024)Inverse Source-Based Three-Antenna Methods in the Near Field2024 18th European Conference on Antennas and Propagation (EuCAP)10.23919/EuCAP60739.2024.10501540(1-5)Online publication date: 17-Mar-2024
https://doi.org/10.23919/EuCAP60739.2024.10501540
Zhang LHan ZZhong YYu QWu XWang XWang RCai JKankanhalli MPrabhakaran BBoll SSubramanian RZheng LSingh VCesar PXie LXu D(2024)VoCAPTER: Voting-based Pose Tracking for Category-level Articulated Object via Inter-frame PriorsProceedings of the 32nd ACM International Conference on Multimedia10.1145/3664647.3681131(8942-8951)Online publication date: 28-Oct-2024
https://dl.acm.org/doi/10.1145/3664647.3681131
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