Mathematical Geosciences

, 42:49 | Cite as

Measurement of Areas on a Sphere Using Fibonacci and Latitude–Longitude Lattices

  • Álvaro GonzálezEmail author


The area of a spherical region can be easily measured by considering which sampling points of a lattice are located inside or outside the region. This point-counting technique is frequently used for measuring the Earth coverage of satellite constellations, employing a latitude–longitude lattice. This paper analyzes the numerical errors of such measurements, and shows that they could be greatly reduced if the Fibonacci lattice were used instead. The latter is a mathematical idealization of natural patterns with optimal packing, where the area represented by each point is almost identical. Using the Fibonacci lattice would reduce the root mean squared error by at least 40%. If, as is commonly the case, around a million lattice points are used, the maximum error would be an order of magnitude smaller.

Spherical grid Golden ratio Equal-angle grid Non-standard grid Fibonacci grid Phyllotaxis 


  1. Adler I, Barabe D, Jean RV (1997) A history of the study of phyllotaxis. Ann Bot 80(3):231–244 CrossRefGoogle Scholar
  2. Ahmad R, Deng Y, Vikram DS, Clymer B, Srinivasan P, Zweier JL, Kuppusamy P (2007) Quasi Monte Carlo-based isotropic distribution of gradient directions for improved reconstruction quality of 3D EPR imaging. J Magn Reson 184(2):236–245 CrossRefGoogle Scholar
  3. Baddeley A, Jensen EBV (2004) Stereology for statisticians. Chapman and Hall–CRC Press, Boca Raton Google Scholar
  4. Barclay M, Galton A (2008) Comparison of region approximation techniques based on Delaunay triangulations and Voronoi diagrams. Comput Environ Urban Syst 32(4):261–267 CrossRefGoogle Scholar
  5. Bardsley WE (1983) Random error in point counting. Math Geol 15(3):469–475 CrossRefGoogle Scholar
  6. Bauer R (2000) Distribution of points on a sphere with application to star catalogs. J Guid Control Dyn 23(1):130–137 CrossRefGoogle Scholar
  7. Bevington PR, Robinson DK (1992) Data reduction and error analysis for the physical sciences, 2nd edn. McGraw Hill, Boston Google Scholar
  8. Bevis M, Cambareri G (1987) Computing the area of a spherical polygon of arbitrary shape. Math Geol 19(4):335–346 CrossRefGoogle Scholar
  9. Brauchart JS (2004) Invariance principles for energy functionals on spheres. Monatsh Math 141(2):101–117 CrossRefGoogle Scholar
  10. Bravais L, Bravais A (1837) Essai sur la disposition des feuilles curvisériées. Ann Sci Nat 7:42–110, plates 2–3 Google Scholar
  11. Chukkapalli G, Karpik SR, Ethier CR (1999) A scheme for generating unstructured grids on spheres with application to parallel computation. J Comput Phys 149(1):114–127 CrossRefGoogle Scholar
  12. Conway JH, Sloane NJA (1998) Sphere packings, lattices and groups, 3rd edn. Springer, New York Google Scholar
  13. Cui J, Freeden W (1997) Equidistribution on the sphere. SIAM J Sci Comput 18(2):595–609 CrossRefGoogle Scholar
  14. Damelin SB, Grabner PJ (2003) Energy functionals, numerical integration and asymptotic equidistribution on the sphere. J Complex 19(3):231–246 [Corrigendum 20(6):883–884] CrossRefGoogle Scholar
  15. Dixon R (1987) Mathographics. Basil Blackwell, Oxford Google Scholar
  16. Dixon R (1989) Spiral phyllotaxis. Comput Math Appl 17(4–6):535–538 CrossRefGoogle Scholar
  17. Dixon R (1992) Green spirals. In: Hargittai I, Pickover CA (eds) Spiral symmetry. World Scientific, Singapore, pp 353–368 Google Scholar
  18. Douady S, Couder Y (1992) Phyllotaxis as a physical self-organized growth process. Phys Rev Lett 68(13):2098–2101 CrossRefGoogle Scholar
  19. Earle MA (2006) Sphere to spheroid comparisons. J Navig 59(3):491–496 CrossRefGoogle Scholar
  20. Erikson RO (1973) Tubular packing of spheres in biological fine structure. Science 181(4101):705–716 CrossRefGoogle Scholar
  21. Evans DG, Jones SM (1987) Detecting Voronoi (area-of-influence) polygons. Math Geol 19(6):523–537 CrossRefGoogle Scholar
  22. Feng S, Ochieng WY, Mautz R (2006) An area computation based method for RAIM holes assessment. J Glob Position Syst 5(1–2):11–16 Google Scholar
  23. Fowler DR, Prusinkiewicz P, Battjes J (1992) A collision-based model of spiral phyllotaxis. ACM SIGGRAPH Comput Graph 26(2):361–368 CrossRefGoogle Scholar
  24. González Á (2009) Self-sharpening seismicity maps for forecasting earthquake locations. Abstracts of the sixth international workshop on statistical seismology. Tahoe City, California, 16–19 April 2009.
  25. Gregory MJ, Kimerling AJ, White D, Sahr K (2008) A comparison of intercell metrics on discrete global grid systems. Comput Environ Urban Syst 32(3):188–203 CrossRefGoogle Scholar
  26. Greiner B (1999) Euler rotations in plate-tectonic reconstructions. Comput Geosci 25(3):209–216 CrossRefGoogle Scholar
  27. Gundersen HJG, Jensen EBV, Kiêu K, Nielsen J (1999) The efficiency of systematic sampling in stereology—reconsidered. J Microsc 193(3):199–211 CrossRefGoogle Scholar
  28. Hannay JH, Nye JF (2004) Fibonacci numerical integration on a sphere. J Phys A 37(48):11591–11601 CrossRefGoogle Scholar
  29. Howarth RJ (1998) Improved estimators of uncertainty in proportions, point-counting, and pass-fail test results. Am J Sci 298(7):594–607 Google Scholar
  30. Hüttig C, Stemmer K (2008) The spiral grid: A new approach to discretize the sphere and its application to mantle convection. Geochem Geophys Geosyst 9(2):Q02018 CrossRefGoogle Scholar
  31. Huxley MN (1987) The area within a curve. Proc Indian Acad Sci, Math Sci 97(1–3):111–116 CrossRefGoogle Scholar
  32. Huxley MN (2003) Exponential sums and lattice points III. Proc Lond Math Soc 87(3):591–609 CrossRefGoogle Scholar
  33. Jarái A, Kozák M, Rózsa P (1997) Comparison of the methods of rock-microscopic grain-size determination and quantitative analysis. Math Geol 29(8):977–991 CrossRefGoogle Scholar
  34. Jean RV (1994) Phyllotaxis: a systemic study of plant pattern morphogenesis. Cambridge University Press, Cambridge Google Scholar
  35. Kafka AL (2007) Does seismicity delineate zones where future large earthquakes are likely to occur in intraplate environments? In: Stein S, Mazzotti S (eds) Continental intraplate earthquakes: science, hazard, and policy issues. Geological Society of America special paper 425, Boulder, Colorado, pp 35–48 Google Scholar
  36. Kantsiper B, Weiss S (1998) An analytic approach to calculating Earth coverage. Adv Astronaut Sci 97:313–332 Google Scholar
  37. Kendall DG (1948) On the number of lattice points inside a random oval. Quart J Math Oxford 19(1):1–26 CrossRefGoogle Scholar
  38. Kimerling AJ (1984) Area computation from geodetic coordinates on the spheroid. Surv Mapp 44(4):343–351 Google Scholar
  39. Klíma K, Pick M, Pros Z (1981) On the problem of equal area block on a sphere. Stud Geophys Geod (Praha) 25(1):24–35 CrossRefGoogle Scholar
  40. Knuth DE (1997) Art of computer programming, 3rd edn. Fundamental algorithms, vol 1. Addison-Wesley, Reading Google Scholar
  41. Kossobokov V, Shebalin P (2003) Earthquake prediction. In: Keilis-Borok VI, Soloviev AA (eds) Nonlinear dynamics of the lithosphere and earthquake prediction. Springer, Berlin, pp 141–207. [References in pp 311–332] Google Scholar
  42. Kozin MB, Volkov VV, Svergun DI (1997) ASSA, a program for three-dimensional rendering in solution scattering from biopolymers. J Appl Cryst 30(5):811–815 CrossRefGoogle Scholar
  43. Kuhlemeier C (2007) Phyllotaxis. Trends Plant Sci 12(4):143–150 CrossRefGoogle Scholar
  44. Lean JL, Picone JM, Emmert JT, Moore G (2006) Thermospheric densities derived from spacecraft orbits: application to the Starshine satellites. J Geophys Res 111(4):A04301 CrossRefGoogle Scholar
  45. Li C, Zhang X, Cao Z (2005) Triangular and Fibonacci number patterns driven by stress on core/shell microstructures. Science 309(5736):909–911 CrossRefGoogle Scholar
  46. Maciá E (2006) The role of aperiodic order in science and technology. Rep Prog Phys 69(2):397–441 CrossRefGoogle Scholar
  47. Maley PD, Moore RG, King DJ (2002) Starshine: A student-tracked atmospheric research satellite. Acta Astronaut 51(10):715–721 CrossRefGoogle Scholar
  48. Michalakes JG, Purser RJ, Swinbank R (1999) Data structure and parallel decomposition considerations on a Fibonacci grid. In: Preprints of the 13th conference on numerical weather prediction, Denver, 13–17 September 1999. American Meteorological Society, pp 129–130 Google Scholar
  49. Na HS, Lee CN, Cheong O (2002) Voronoi diagrams on the sphere. Comput Geom Theory Appl 23(2):183–194 Google Scholar
  50. Niederreiter H (1992) Random number generation and quasi-Monte Carlo methods. Society for Industrial and Applied Mathematics, Philadelphia Google Scholar
  51. Niederreiter H, Sloan IH (1994) Integration of nonperiodic functions of two variables by Fibonacci lattice rules. J Comput Appl Math 51(1):57–70 CrossRefGoogle Scholar
  52. Nye JF (2003) A simple method of spherical near-field scanning to measure the far fields of antennas or passive scatterers. IEEE Trans Antenna Propag 51(8):2091–2098 CrossRefGoogle Scholar
  53. Ochieng WY, Sheridan KF, Sauer K, Han X (2002) An assessment of the RAIM performance of a combined Galileo/GPS navigation system using the marginally detectable errors (MDE) algorithm. GPS Solut 5(3):42–51 CrossRefGoogle Scholar
  54. Prusinkiewicz P, Lindenmayer A (1990) The algorithmic beauty of plants. Springer, New York Google Scholar
  55. Purser RJ (2008) Generalized Fibonacci grids: A new class of structured, smoothly adaptive multi-dimensional computational lattices. Office Note 455, National Centers for Environmental Prediction, Camp Springs, MD, USA Google Scholar
  56. Purser RJ, Swinbank R (2006) Generalized Euler-Maclaurin formulae and end corrections for accurate quadrature on Fibonacci grids. Office Note 448, National Centers for Environmental Prediction, Camp Springs, MD, USA Google Scholar
  57. Rakhmanov EA, Saff EB, Zhou YM (1994) Minimal discrete energy on the sphere. Math Res Lett 1(6):647–662 Google Scholar
  58. Ridley JN (1982) Packing efficiency in sunflower heads. Math Biosci 58(1):129–139 CrossRefGoogle Scholar
  59. Ridley JN (1986) Ideal phyllotaxis on general surfaces of revolution. Math Biosci 79(1):1–24 CrossRefGoogle Scholar
  60. Saff EB, Kuijlaars ABJ (1997) Distributing many points on a sphere. Math Intell 19(1):5–11 CrossRefGoogle Scholar
  61. Sigler LE (2002) Fibonacci’s Liber Abaci: a translation into modern English of Leonardo Pisano’s book of calculation. Springer, New York Google Scholar
  62. Singh P (1985) The so-called Fibonacci numbers in ancient and medieval India. Hist Math 12(3):229–244 CrossRefGoogle Scholar
  63. Sjöberg LE (2006) Determination of areas on the plane, sphere and ellipsoid. Surv Rev 38(301):583–593 Google Scholar
  64. Sloan IH, Joe S (1994) Lattice methods for multiple integration. Oxford University Press, London Google Scholar
  65. Svergun DI (1994) Solution scattering from biopolymers: advanced contrast-variation data analysis. Acta Cryst A 50(3):391–402 CrossRefGoogle Scholar
  66. Swinbank R, Purser RJ (1999) Fibonacci grids. In: Preprints of the 13th conference on numerical weather prediction, Denver, 13–17 September 1999. American Meteorological Society, pp 125–128 Google Scholar
  67. Swinbank R, Purser RJ (2006a) Standard test results for a shallow water equation model on the Fibonacci grid. Forecasting Research Technical Report 480, Met Office, Exeter, UK Google Scholar
  68. Swinbank R, Purser RJ (2006b) Fibonacci grids: a novel approach to global modelling. Q J R Meteorol Soc 132(619):1769–1793 CrossRefGoogle Scholar
  69. Van den Dool HM (2007) Empirical methods in short-term climate prediction. Oxford University Press, London Google Scholar
  70. Vogel H (1979) A better way to construct the sunflower head. Math Biosci 44(3–4):179–189 CrossRefGoogle Scholar
  71. Vriend G (1990) WHAT IF: a molecular modeling and drug design program. J Mol Graph 8(1):52–56 CrossRefGoogle Scholar
  72. Weiller AR (1966) Probleme de l’implantation d’une grille sur une sphere, deuxième partie. Bull Géod 80(1):99–111 CrossRefGoogle Scholar
  73. Weisstein EW (2002) CRC concise encyclopedia of mathematics CD-ROM, 2nd edn. CRC Press, Boca Raton Google Scholar
  74. Wessel P, Smith WHF (1998) New, improved version of Generic Mapping Tools released. Eos Trans Am Geophys Union 79(47):579 CrossRefGoogle Scholar
  75. Williamson DL (2007) The evolution of dynamical cores for global atmospheric models. J Meteorol Soc Jpn B 85:241–269 CrossRefGoogle Scholar
  76. Winfield DC, Harris KM (2001) Phyllotaxis-based dimple patterns. Patent number WO 01/26749 Al. World Intellectual Property Organization Google Scholar
  77. Zaremba SK (1966) Good lattice points, discrepancy, and numerical integration. Ann Mat Pura Appl 73(1):293–317 CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geosciences 2009

Authors and Affiliations

  1. 1.Departamento de Ciencias de la TierraUniversidad de ZaragozaZaragozaSpain

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