Trial Direction Sampling

Trial geometry generation begins by choosing directions on the unit sphere. This spatial sampling problem is independent of the element types and should be assessed separately from molecular rotation and local optimization.

Benchmarking

Run the coverage benchmark with:

pyar-benchmark-orientations -N 8 12 20 28

The table reports:

  • min_sep_deg: smallest angular separation between sampled directions. Higher values reduce duplicate approaches.

  • cover_radius_deg: approximate worst unsampled angular gap, measured on a dense deterministic reference grid. Lower values indicate better full-sphere coverage.

  • mean_cover_deg: average angular distance to the closest sampled direction. Lower values are better.

  • centroid_norm: directional imbalance. Values near zero are preferred.

Methods

fibonacci is deterministic equal-area spherical Fibonacci sampling. lhs and random are randomized baselines. lhs_maximin and fibonacci_maximin select a farthest-point subset from an oversized candidate pool.

The aggregate workflow default uses spherical Fibonacci placement directions. For multi-atom monomers it pairs those directions with deterministic low-discrepancy quaternion rotations, converted to the existing Euler-angle rotation API.

Standalone commands that request multiple generated populations use a deterministic sequence_offset for each population. This retains reproducibility while avoiding repeated copies of the same point design.

The implementation is exposed by pyar.trial_generation and the pyar-trial-generation command.

Literature Basis

Spherical Fibonacci point sets are established deterministic point sets for nearly uniform sampling on the unit sphere and can be generated for arbitrary numbers of points:

  • R. Marques, C. Bouville, M. Ribardiere, L. P. Santos, and K. Bouatouch, Spherical Fibonacci Point Sets for Illumination Integrals, Computer Graphics Forum 32 (2013) 134-143, DOI: 10.1111/cgf.12190.

  • B. Keinert, M. Innmann, M. Saenger, and M. Stamminger, Spherical Fibonacci Mapping, ACM Transactions on Graphics 34 (2015), article 193, DOI: 10.1145/2816795.2818131.

The present benchmark focuses on separation and approximate covering radius because PyAR needs directionally distinct approaches with no large unsampled region of the sphere.