3. Core Concepts & Geometries

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

3.1. 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.

3.1.1. 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.

3.2. 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 \(d \ge 1\).

3.3. Coordinate Systems

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

3.3.1. Spherical Coordinates

  • Latitude/Longitude: Standard for geospatial applications.

  • Colatitude/Azimuth: Common in physics and mathematical modelling.

3.3.2. 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.

3.4. 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.