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

Aubrie Kooiker; D. Heddle; H.F. Counts; Mihaela Dobrescu; Walter Pierce

arxiv: 2605.16422 · v1 · pith:MYO5JNARnew · submitted 2026-05-14 · 🌌 astro-ph.IM

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

H.F. Counts , Mihaela Dobrescu , D. Heddle , Aubrie Kooiker , Walter Pierce This is my paper

Pith reviewed 2026-05-20 20:53 UTC · model grok-4.3

classification 🌌 astro-ph.IM

keywords spherical mosaicsCartesian grid intersectionsarea-closed patchescomputational geometrysurface patchingconservative couplingpolar singularity handlinggrid-induced mosaics

0 comments

The pith

A new method slices Cartesian grid intersections with a sphere into final patches that close in area to roundoff with zero splicing failures.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a computational geometry approach for building a surface mosaic on a spherical shell intersected by a Cartesian volume grid. It identifies intersecting cells, forms cell-sphere prepatches, and splices them first along colatitude lines then along azimuth lines to produce indexed final patches. The technique explicitly manages the polar singularity and various degeneracy cases such as doubly-crossing edges and re-entrant arcs. If successful, the resulting patches enable exact area conservation for data exchange between rectangular grids and spherical surfaces in modeling applications.

Core claim

The Mosaic method builds 3618 intersecting Cartesian cells into 3602 ordinary prepatches, 6476 theta patches, and 9714 final phi patches on a representative nonuniform test grid, with all ordinary cells completed successfully, zero theta or phi splicing failures, and final phi-spliced patches closing in normalized area to roundoff relative to their theta-spliced parents.

What carries the argument

Sequential colatitude splicing followed by azimuth splicing of cell-sphere prepatches, with separate handling for polar-derived theta patches and exact great-circle intersections to avoid linear interpolation failures.

If this is right

Final patches indexed by the five-tuple (nx, ny, nz, ntheta, nphi) support conservative coupling without area loss between rectangular data and spherical boundary conditions.
The implementation handles polar singularities by separating polar-derived theta patches from ordinary phi splicing.
Degeneracy cases beyond simple corner straddling are resolved during prepatch construction.
The approach extends to nonuniform Cartesian grids while maintaining area closure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same splicing sequence could be adapted to intersections with other quadric surfaces if the meridian and great-circle intersection routines are generalized.
Patch boundaries exported as JSON could serve as input for flux calculations or remapping in coupled simulation codes without additional area correction steps.
Visualization diagnostics in the implementation allow direct inspection of area closure at each splicing stage for verification on new grids.

Load-bearing premise

All intersection geometries, including doubly-crossing edges, lens-shaped prepatches, secondary closed loops, and re-entrant face arcs, can be correctly identified and spliced by the colatitude then azimuth sequence without missing or double-counting area.

What would settle it

A measured normalized area difference larger than roundoff between any final phi patch and its theta parent, or a splicing failure, when the method is run on a new Cartesian grid containing more complex intersection degeneracies.

Figures

Figures reproduced from arXiv: 2605.16422 by Aubrie Kooiker, D. Heddle, H.F. Counts, Mihaela Dobrescu, Walter Pierce.

**Figure 2.** Figure 2: An ordinary prepatch from an intersection with four corners inside the sphere [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: Monte Carlo visualization of the representative test grid using 20 million random [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

**Figure 4.** Figure 4: The MDI Mosaic application with final patches and mouse-over patch diagnos [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Mosaic delivers a practical algorithm for area-closed spherical patches from Cartesian intersections with polar fixes, but the single test grid leaves the degeneracy handling only partially checked.

read the letter

The core contribution is a step-by-step construction that turns intersecting Cartesian cells into theta-spliced then phi-spliced patches on the sphere, with explicit fixes for the polar singularity via exact great-circle intersections and a list of degeneracy cases such as doubly-crossing edges and re-entrant arcs. The implementation reports building 3602 ordinary prepatches from 3618 cells on one nonuniform test grid, with zero splicing failures and final areas closing to roundoff relative to the parent patches. That is useful for anyone who needs conservative coupling between rectangular volume data and spherical boundary conditions in space-weather or atmospheric codes. The Java code with visualization and JSON export is a concrete deliverable that others could inspect or adapt. The polar handling and the explicit degeneracy list are packaged together in a way that does not appear in the prior literature cited in the abstract. The main limitation is the validation. The results give only aggregate counts and overall closure; they do not show which of the listed degeneracies actually appeared in the test grid or provide separate area diagnostics for each. If the chosen grid simply avoided the harder configurations, the roundoff success does not yet confirm that the colatitude-then-azimuth sequence handles every case without missing or double-counting area. This is a software-methods paper aimed at practitioners who already maintain coupled Cartesian-spherical models. A reader facing the same interface problem would get immediate value from the algorithm description and the released code. It is worth sending to peer review because the central claim is a constructive method backed by an implementation and a reproducible closure check, even though additional test grids with documented degeneracy coverage would make the correctness argument stronger.

Referee Report

1 major / 2 minor

Summary. The paper presents Mosaic, a computational geometry method for constructing area-closed spherical surface mosaics from Cartesian volume grid intersections with a spherical shell. It identifies intersecting cells, builds cell-sphere prepatches, splices first by colatitude then azimuth (with polar singularity handling via exact great-circle/meridian intersections), and produces final patches indexed by the five-tuple (nx, ny, nz, ntheta, nphi). The method explicitly addresses degeneracy cases including doubly-crossing edges, lens-shaped prepatches, secondary closed loops, and re-entrant face arcs. On a single nonuniform Cartesian test grid it reports 3618 intersecting cells, 3602 ordinary prepatches, 6476 theta patches, 9714 final phi patches, zero splicing failures, and normalized area closure to roundoff.

Significance. If the degeneracy handling is fully validated, the method would support conservative remapping between rectangular grids and spherical surfaces in space-weather, MHD, atmospheric, and geophysical applications. The constructive algorithm, parameter-free area closure, and explicit treatment of polar and degeneracy cases are strengths; however, the current evidence consists only of aggregate counts on one test grid.

major comments (1)

The validation reports only aggregate counts (3618 intersecting cells, 3602 ordinary prepatches, zero theta/phi failures) and overall roundoff area closure for one nonuniform Cartesian grid. No breakdown is given of which degeneracy cases (doubly-crossing edges, lens-shaped prepatches, secondary closed loops, re-entrant face arcs) occurred, how many of each, or per-case area-closure diagnostics. This leaves the central claim that the colatitude-then-azimuth splicing sequence correctly processes every listed degeneracy without area loss or double-counting unverified by the presented results.

minor comments (2)

Notation for the final five-tuple patch index (nx, ny, nz, ntheta, nphi) could be introduced earlier and used consistently in the method description.
The manuscript would benefit from a table or figure summarizing the degeneracy cases and the specific algorithmic steps that resolve each.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful review and for recognizing the method's relevance to conservative remapping in space-weather and geophysical applications. We agree that the validation section would benefit from greater granularity on degeneracy handling and propose a targeted revision to address this directly.

read point-by-point responses

Referee: The validation reports only aggregate counts (3618 intersecting cells, 3602 ordinary prepatches, zero theta/phi failures) and overall roundoff area closure for one nonuniform Cartesian grid. No breakdown is given of which degeneracy cases (doubly-crossing edges, lens-shaped prepatches, secondary closed loops, re-entrant face arcs) occurred, how many of each, or per-case area-closure diagnostics. This leaves the central claim that the colatitude-then-azimuth splicing sequence correctly processes every listed degeneracy without area loss or double-counting unverified by the presented results.

Authors: We agree that aggregate counts and global area closure, while consistent with successful processing (zero failures and roundoff-level closure), do not explicitly document the frequency or individual closure performance of each degeneracy type. In the revised manuscript we will add a table in the numerical results section that enumerates the occurrences of doubly-crossing edges, lens-shaped prepatches, secondary closed loops, and re-entrant face arcs within the 3602 prepatches, together with the normalized area-closure residual for each category. This addition will supply the per-case verification requested while preserving the existing algorithm description and overall statistics. revision: yes

Circularity Check

0 steps flagged

Constructive algorithm with explicit verification shows no circularity

full rationale

The paper describes a computational geometry algorithm that constructs prepatches via colatitude and azimuth splicing, explicitly handling listed degeneracies, then verifies area closure to roundoff and zero failures on one generated test grid of 3618 cells. This is a direct constructive procedure whose output properties are checked by explicit computation rather than predicted from fitted inputs or reduced to self-definitions. No self-citations, uniqueness theorems, or ansatzes appear as load-bearing steps in the abstract or method outline. The area-closure result follows from the splicing implementation itself and is externally falsifiable by re-running the code on the same grid, satisfying the criteria for a self-contained non-circular derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The method rests on standard computational geometry primitives for plane-sphere and great-circle intersections plus the assumption that the input Cartesian grid is axis-aligned and the sphere is a perfect mathematical shell.

axioms (2)

domain assumption Cartesian cells are axis-aligned rectangular boxes.
Invoked when identifying intersecting cells and constructing cell-sphere prepatches.
domain assumption The spherical shell is a perfect mathematical sphere with no thickness or surface irregularities.
Required for exact great-circle and meridian-plane intersections.

pith-pipeline@v0.9.0 · 5844 in / 1334 out tokens · 42848 ms · 2026-05-20T20:53:39.067978+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

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).
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The final phi-spliced patches close in normalized area to roundoff relative to their theta-spliced parents.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, UK, 2002.doi:10.1017/ CBO9780511791253

work page 2002
[2]

P. H. Lauritzen, R. D. Nair, P. A. Ullrich, A conservative semi- lagrangian multi-tracer transport scheme (cslam) on the cubed-sphere grid, Monthly Weather Review 138 (8) (2010) 3052–3065.doi:10.1175/ 2010MWR3433.1

work page 2010
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W. W. White, G. L. Siscoe, G. M. Erickson, Z. Kaymaz, N. C. Maynard, K. D. Siebert, B. U. Ö. Sonnerup, D. R. Weimer, The magnetospheric sash and the cross-tail s, Geophysical Research Letters 25 (10) (1998) 1605–1608.doi:10.1029/98GL50865

work page doi:10.1029/98gl50865 1998
[4]

G. Tóth, B. van der Holst, I. V. Sokolov, et al., Adaptive solutions of the space weather modeling framework on structured and unstructured grids, Journal of Computational Physics 231 (3) (2012) 870–903.doi: 10.1016/j.jcp.2011.02.006

work page doi:10.1016/j.jcp.2011.02.006 2012
[5]

T. I. Gombosi, Y. Chen, A. Glocer, Z. Huang, X. Jia, M. W. Liemohn, W. B. Manchester, T. Pulkkinen, N. Sachdeva, Q. Al Shidi, I. V. Sokolov, J. Szente, V. Tenishev, G. Tóth, B. van der Holst, D. T. Welling, L. Zhao, S. Zou, What sustained multi-disciplinary research can achieve: Thespaceweathermodelingframework, arXivpreprint.https: //arxiv.org/abs/2105.1...

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work page doi:10.1006/jcph 2002
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[8]

J. M. Stone, T. A. Gardiner, P. Teuben, J. F. Hawley, J. B. Simon, Athena: A new code for astrophysical mhd, The Astrophysical Journal Supplement Series 178 (1) (2008) 137–177.doi:10.1086/588755

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Ringler, M

T. Ringler, M. Petersen, R. L. Higdon, D. Jacobsen, M. Maltrud, P. Jones, A multi-resolution approach to global ocean modeling, Ocean Modelling 69 (2013) 211–232.doi:10.1016/j.ocemod.2013.04.010

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[10]

J. A. Benek, J. L. Steger, F. C. Dougherty, A flexible grid embedding technique with application to the euler equations, in: AIAA 6th Compu- tational Fluid Dynamics Conference, American Institute of Aeronautics and Astronautics, Hampton, VA, 1986

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[11]

M. J. Berger, J. Oliger, Adaptive mesh refinement for hyperbolic partial differential equations, Journal of Computational Physics 53 (3) (1984) 484–512.doi:10.1016/0021-9991(84)90073-1

work page doi:10.1016/0021-9991(84)90073-1 1984
[12]

Pierce, Chimera grid: The geometry of intersecting cartesian and spherical grids, Unpublished master’s thesis, Christopher Newport Uni- versity, Newport News, Virginia (2013)

W. Pierce, Chimera grid: The geometry of intersecting cartesian and spherical grids, Unpublished master’s thesis, Christopher Newport Uni- versity, Newport News, Virginia (2013)

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[13]

Heddle, A modular multi-document framework for scientific visu- alization and simulation in java (2026).arXiv:2602.21026,doi: 10.48550/arXiv.2602.21026

D. Heddle, A modular multi-document framework for scientific visu- alization and simulation in java (2026).arXiv:2602.21026,doi: 10.48550/arXiv.2602.21026. URLhttps://arxiv.org/abs/2602.21026 19

work page doi:10.48550/arxiv.2602.21026 2026

[1] [1]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, UK, 2002.doi:10.1017/ CBO9780511791253

work page 2002

[2] [2]

P. H. Lauritzen, R. D. Nair, P. A. Ullrich, A conservative semi- lagrangian multi-tracer transport scheme (cslam) on the cubed-sphere grid, Monthly Weather Review 138 (8) (2010) 3052–3065.doi:10.1175/ 2010MWR3433.1

work page 2010

[3] [3]

W. W. White, G. L. Siscoe, G. M. Erickson, Z. Kaymaz, N. C. Maynard, K. D. Siebert, B. U. Ö. Sonnerup, D. R. Weimer, The magnetospheric sash and the cross-tail s, Geophysical Research Letters 25 (10) (1998) 1605–1608.doi:10.1029/98GL50865

work page doi:10.1029/98gl50865 1998

[4] [4]

G. Tóth, B. van der Holst, I. V. Sokolov, et al., Adaptive solutions of the space weather modeling framework on structured and unstructured grids, Journal of Computational Physics 231 (3) (2012) 870–903.doi: 10.1016/j.jcp.2011.02.006

work page doi:10.1016/j.jcp.2011.02.006 2012

[5] [5]

T. I. Gombosi, Y. Chen, A. Glocer, Z. Huang, X. Jia, M. W. Liemohn, W. B. Manchester, T. Pulkkinen, N. Sachdeva, Q. Al Shidi, I. V. Sokolov, J. Szente, V. Tenishev, G. Tóth, B. van der Holst, D. T. Welling, L. Zhao, S. Zou, What sustained multi-disciplinary research can achieve: Thespaceweathermodelingframework, arXivpreprint.https: //arxiv.org/abs/2105.1...

work page arXiv 2021

[6] [6]

Sussman, P

A. Dedner, F. Kemm, D. Kröner, C. D. Munz, T. Schnitzer, M. We- senberg, Hyperbolic divergence cleaning for the mhd equations, Journal of Computational Physics 175 (2) (2002) 645–673.doi:10.1006/jcph. 2001.6961

work page doi:10.1006/jcph 2002

[7] [7]

W. M. Putman, S. J. Lin, Finite-volume transport on various cubed- sphere grids, Journal of Computational Physics 227 (1) (2007) 55–78. doi:10.1016/j.jcp.2007.07.022. 18

work page doi:10.1016/j.jcp.2007.07.022 2007

[8] [8]

J. M. Stone, T. A. Gardiner, P. Teuben, J. F. Hawley, J. B. Simon, Athena: A new code for astrophysical mhd, The Astrophysical Journal Supplement Series 178 (1) (2008) 137–177.doi:10.1086/588755

work page doi:10.1086/588755 2008

[9] [9]

Ringler, M

T. Ringler, M. Petersen, R. L. Higdon, D. Jacobsen, M. Maltrud, P. Jones, A multi-resolution approach to global ocean modeling, Ocean Modelling 69 (2013) 211–232.doi:10.1016/j.ocemod.2013.04.010

work page doi:10.1016/j.ocemod.2013.04.010 2013

[10] [10]

J. A. Benek, J. L. Steger, F. C. Dougherty, A flexible grid embedding technique with application to the euler equations, in: AIAA 6th Compu- tational Fluid Dynamics Conference, American Institute of Aeronautics and Astronautics, Hampton, VA, 1986

work page 1986

[11] [11]

M. J. Berger, J. Oliger, Adaptive mesh refinement for hyperbolic partial differential equations, Journal of Computational Physics 53 (3) (1984) 484–512.doi:10.1016/0021-9991(84)90073-1

work page doi:10.1016/0021-9991(84)90073-1 1984

[12] [12]

Pierce, Chimera grid: The geometry of intersecting cartesian and spherical grids, Unpublished master’s thesis, Christopher Newport Uni- versity, Newport News, Virginia (2013)

W. Pierce, Chimera grid: The geometry of intersecting cartesian and spherical grids, Unpublished master’s thesis, Christopher Newport Uni- versity, Newport News, Virginia (2013)

work page 2013

[13] [13]

Heddle, A modular multi-document framework for scientific visu- alization and simulation in java (2026).arXiv:2602.21026,doi: 10.48550/arXiv.2602.21026

D. Heddle, A modular multi-document framework for scientific visu- alization and simulation in java (2026).arXiv:2602.21026,doi: 10.48550/arXiv.2602.21026. URLhttps://arxiv.org/abs/2602.21026 19

work page doi:10.48550/arxiv.2602.21026 2026