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Core Concepts & Geometries

PyEQSP is built on the mathematical framework for partitioning the unit sphere Sᵈ into regions of equal area.

The EQ Algorithm

The EQ algorithm partitions the unit sphere $S^2$ (the 2-sphere in $\mathbb{R}^3$) into $N$ equal-area regions. It works by dividing the sphere into "collars" (latitudinal zones) and then further subdividing each collar into equal-area regions.

Key Properties

  • Equal Area: Every region in a partition has exactly the same measure (area).
  • Small Diameter: The regions are designed to be as "round" as possible, with a diameter that remains small as $N$ increases.
  • Recursive Logic: The partitioning process is defined recursively across dimensions.

Supported Manifolds

While most applications focus on the physical 2-sphere ($S^2$), PyEQSP supports a wide range of manifolds:

  • $S^1$ (Circle): Simple angular partitioning.
  • $S^2$ (Sphere): The world-standard sphere, used in geophysics and climate modelling.
  • $S^3$ (3-sphere): Essential for applications involving quaternions and rotations (SO(3)).
  • $S^d$ (High-dimensional spheres): Generalized logic for any $d \ge 1$.

Coordinate Systems

To support different research backgrounds, PyEQSP provides utilities for converting between common coordinate conventions:

Spherical Coordinates

  • Latitude/Longitude: Standard for geospatial applications.
  • Colatitude/Azimuth: Common in physics and mathematical modelling.

Euclidean / Cartesian Coordinates

Points on the sphere are represented as unit vectors in $\mathbb{R}^{d+1}$:

  • For $S^2$, points are $(x, y, z)$ where $x^2 + y^2 + z^2 = 1$.

The core logic resides in the eqsp package.

Even Collar Symmetry

A unique feature of the Python implementation is the even_collars option. When enabled, this ensures that the Equatorial plane ($z=0$) aligns exactly with a collar boundary. This is critical for:

  • Splitting the sphere into identical North/South hemispheres.
  • Precise mapping of $S^3$ to SO(3) for quaternion sampling.