5. Symmetric EQ Partitions (even_collars)

PyEQSP provides a specialized mode for generating partitions that are perfectly symmetric about the sphere’s equator. This is controlled via the even_collars parameter in the core API.

5.1. Motivation

Many research applications require the equator (\(\theta = \pi/2\)) to fall exactly on a region boundary:

  • Robotics & Orientation Estimation: Hyperhemispherical grid filters (Pfaff et al., 2020) construct one half of a partition and mirror it.

  • Biomedical RNA Folding: Sampling the rotation groups \(SO(3)\) via quaternions on \(S^3\). Due to double-covering, researchers often only need to sample the upper hyperhemisphere, requiring an exact equatorial split.

5.2. Using the API

To force a symmetric partition, set even_collars=True in eq_regions or eq_point_set.

from eqsp.partitions import eq_regions

# N must be even for even_collars=True
regions = eq_regions(dim=2, N=100, even_collars=True)

Tip

See examples/user-guide/src/example_symmetric_partitions.py for a complete example including symmetry verification.

Important

If \(N\) is odd, setting even_collars=True will raise a ValueError. This is because for odd \(N\), no single hyperplane can perfectly divide the partition into two equal hemispheres.

5.3. Effectiveness and Research Basis

The symmetric modification preserves the Equal Area property exactly and maintains the \(O(N^{-1/d})\) Diameter Bound established in Leopardi (2009). While the fitting angle may deviate slightly from the “asymmetric ideal,” the difference becomes negligible as \(N\) increases.

For technical details on the underlying optimizations (KDTrees, Symmetry exploitation, and Process Pools), see the Algorithmic Optimizations guide.