Mosaic: Area-Closed Spherical Surface Mosaics Induced by Cartesian Grids

TL;DR

Mosaic is a computational geometry method that constructs spherical surface mosaics from Cartesian grid intersections, facilitating data coupling in space-weather and geophysical models.

Contribution

The paper introduces a novel algorithm for constructing spherical surface mosaics from Cartesian grids, explicitly handling polar singularities and degeneracies.

Findings

Successfully constructed 9714 final patches with minimal splicing failures.

Built 3602 ordinary prepatches with all Cartesian cells intersecting the shell.

Final patches closely approximate round areas, indicating high accuracy.

Abstract

We describe Mosaic, a computational geometry method for constructing the surface mosaic induced when a Cartesian volume grid intersects a spherical shell. The motivating application is conservative coupling between data produced on rectangular grids and diagnostics or boundary conditions defined on spherical surfaces, as occurs in space-weather, magnetohydrodynamic, atmospheric, and geophysical models. The method identifies Cartesian cells that intersect the shell, constructs cell-sphere prepatches, splices those regions by the spherical colatitude grid, and then splices by azimuth to produce final patches indexed by the five-tuple (nx, ny, nz, ntheta, nphi). The implementation explicitly treats the polar singularity by separating polar-derived theta patches from ordinary phi splicing. A near-pole numerical failure mode, caused by linear interpolation in azimuth, is removed by computing exact intersections between great-circle boundary segments and meridian planes. The prepatch construction also handles several degeneracy cases that occur beyond the ordinary corner-straddling geometry, including doubly-crossing edges, lens-shaped prepatches, secondary closed loops, and re-entrant face arcs. For a representative nonuniform Cartesian test grid, the current implementation builds 3618 intersecting Cartesian cells, 3602 ordinary prepatches, 6476 theta patches, and 9714 final phi patches, with all 3602 ordinary cells built successfully and zero theta or phi splicing failures. The final phi-spliced patches close in normalized area to roundoff relative to their theta-spliced parents. The method is implemented in the Java/Maven application mdi-mosaic, which provides visualization, diagnostics, mouse-over patch inspection, and JSON export of final patch boundaries and normalized areas.