experimental UnitSpherical module that can express geometry on unit sphere

[asinghvi17]

TL;DR

Wrote a new UnitSpherical.jl module (experimental) for working with points on the unit sphere and spherical caps.

What changed?

Created a new file UnitSpherical.jl that implements:

• UnitSphericalPoint struct for representing points on the unit sphere (S²) using Cartesian coordinates in ℝ³
• SphericalCap struct for representing spherical caps
• Coordinate transformations between geographic (lat/long) and unit spherical coordinates
• GeoInterface compatibility for the new types
• Utility functions for spherical geometry:
  • spherical_distance for calculating distances between points
  • slerp for spherical linear interpolation
  • circumcenter_on_unit_sphere for finding the center of a circle passing through three points: still broken
  • angle_between for calculating angles between points
  • randsphere for generating random points on the unit sphere

Quick start

1. Import the module and create unit spherical points:

   using GeometryOps.UnitSpherical
   
   # Create a point from Cartesian coordinates
   p1 = UnitSphericalPoint(1.0, 0.0, 0.0)
   
   # Create a point from geographic coordinates (lat/long)
   p2 = UnitSphericalPoint([45.0, 90.0])
   
   # Calculate spherical distance
   dist = spherical_distance(p1, p2)

2. Test spherical caps and operations:

   cap1 = SphericalCap(p1, 0.5)
   cap2 = SphericalCap(p2, 0.3)
   
   # Test intersection
   _intersects(cap1, cap2)

3. Generate random points on the sphere:

   points = rand(UnitSphericalPoint, 100)

This should provide a nice foundation to start looking at R-trees that are backed by spherical caps, not extents, in preparation for @meggart's work involving this. With the new changes to as/trees SpatialTreeInterface is now representation agnostic, meaning that as long as the predicate you passed in can consume the extent like object you have chosen, any extent like thing (accessible by SpatialTreeInterface.node_extent) can be used by the STI algorithms.

[asinghvi17]

• experimental UnitSpherical module that can express geometry on unit sphere #285 👈 (View in Graphite)
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[asinghvi17]

For this to work well, we'll probably want to port a lot of the stuff from s2, like https://github.com/google/s2geometry/blob/58de4ea1e2f8a294e0c072c602c22232fd1433ad/src/s2/s1chord_angle.h - it's Apache 2.0 licensed so we can directly port it.

They have a good framework to handle this stuff, so if we can get the basics sorted then we're mostly there I believe. That would also be a good foundation to build an s2 indexing framework in Julia, since we would have all the basic components already.

[asinghvi17]

This needs to use robust_cross_product, probably.

[asinghvi17]

So does this