2. The Applicability of PyEQSP: Use Cases

Based on: “The applicability of equal area partitions of the unit sphere”, Leopardi (2024), Journal of Approximation Software, 1(2).

2.1. History: From Toolbox to Library

The journey of Equal Area Sphere Partitioning (EQ) code began with the 2005 Recursive Zonal Equal Area Sphere Partitioning Toolbox for MATLAB. Over nearly two decades, the toolbox supported many research projects. PyEQSP is the modern Python-based successor, designed for high performance and integration with the scientific Python ecosystem (NumPy, SciPy, Matplotlib).

2.2. Cross-Disciplinary Applications

The EQ algorithm provides a robust way to partition the unit sphere \(\mathbb{S}^d\) into regions of equal area and small diameter. This property is critical in many scientific domains.

2.2.1. Biology and Bioinformatics

  • Sampling Rotations (Chu et al., 2009): EQSP partitions were used to sample rotations of protein domains (\(S^3\)). This allowed for an efficient and unbiased exploration of the conformational space to simulate electrostatic interactions and protein flexibility.

  • Species Diversity Visualization (Arrigo et al., 2012): The R2G2 package incorporates EQSP partitions to provide global grids for 3D histograms in Google Earth, enabling quantitative visualization of species richness across equal-area regions.

2.2.2. Medicine and MRI

  • SPARKLING MRI Trajectories (Lazarus et al., 2020): EQSP partitioning was used to sample k-space trajectories on a sphere, ensuring trajectories are distributed according to a target density while maintaining globally uniform sampling.

  • Automated Brain Parcellation (Das and Maharatna, 2020): EQ partitions were used to create a non-anatomical, equal-area “igloo” grid of the brain, facilitating the analysis of structural and functional connectivity.

2.2.3. Climate and Geoscience

  • Arctic Temperature Gridding (Werner et al., 2018): EQSP partitions were used as a global equal-area grid to construct millennium-length summer temperature reconstructions, avoiding high-latitude distortions.

  • Empirical Mode Decomposition (Fauchereau et al., 2008): EQSP provided the mandatory equal-area grid for applying 2D EMD to geophysical fields, separating spatial scales in climate variations.

2.2.4. Planetary Science and Geophysics

  • Lunar Tectonic Patterns (Matsuyama et al., 2021): Digitized fault segments were sampled into 400 equal-area regions to distinguish between isotropic contraction and other stress-generating mechanisms on the moon.

  • Geomagnetic Virtual Observatories (Hammer et al., 2021): EQSP partitions were used to define a global grid of GVOs for studying sub-decadal variations in the Earth’s core field.

2.2.5. Engineering and Materials Science

  • Composite Fiber Orientations (Sabiston et al., 2021): EQSP partitioning (specifically with 1200 partitions) was used to define representative fiber orientations for micromechanics modelling of injection-molded composites.

2.2.6. Numerical Weather Prediction

  • Parallel Load Balancing (Mozdzynski et al., 2015): EQ-regions were used to decompose reduced Gaussian grids into equal-area regions for efficient parallel processing in ECMWF’s IFS model.

2.2.7. Mathematical Physics and Estimation

  • Orientation Estimation (Pfaff et al., 2020): Developed hyperhemispherical grid filters based on the EQSP algorithm for robust orientation estimation in robotics.

  • Fermi Gas Correlation Energy (Benedikter et al., 2021): The EQSP construction was used to partition the Fermi surface in momentum space into patches, a key step in calculating correlation energy.

2.3. Evaluating Performance

In comparative studies, PyEQSP’s recursive zonal partitioning has shown distinct advantages over other methods like \(k\)-means clustering, spiral points, and icosahedral partitions, particularly for area consistency and bounded-diameter regions for large \(N\).