Measurement of areas on a sphere using Fibonacci and latitude-longitude lattices

TL;DR

This paper compares Fibonacci and latitude-longitude lattices for measuring spherical regions, demonstrating that Fibonacci lattices significantly reduce numerical errors and improve measurement accuracy.

Contribution

It introduces the use of Fibonacci lattices for spherical area measurement, showing they outperform latitude-longitude lattices in reducing numerical errors.

Findings

Fibonacci lattices reduce root mean squared error by at least 40%.

Maximum error is an order of magnitude smaller with Fibonacci lattices at one million points.

Fibonacci lattices have nearly uniform area representation per point.

Abstract

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.