This manuscript provides a systematic treatment of the geometric algorithms underpinning spherical regridding operations. The paper addresses a genuine and longstanding gap in the geoscience modeling literature: these algorithms are widely used but poorly documented, implemented with varying degrees of rigor, and rarely compared systematically. The text will be a valuable reference for model developers.

Although I have developed an appreciation for this work, I cannot recommend publication until the following major revisions are made.

Major Comments:

1. The paper’s central prescriptive recommendations, such as the use of Kahan’s formula for edge length calculations, the preference for Eriksson’s formula over L’Huilier and quadrature-based methods for face area computation, and the conclusion that k-d trees and ball trees exhibit broadly similar performance, are supported by benchmark results presented only in the supplementary material. Readers should not need to consult the supplement to assess whether the paper’s core numerical claims are well supported.

I encourage the authors to incorporate the key quantitative findings from Supplement into the main text, including, at a minimum, representative error magnitudes and relative performance comparisons for the recommended approaches. Supplement can still serve an important role by presenting complete tables and additional comparisons.

2. The AccuCross and AccuXGCA algorithms represent key novel contributions of the paper. However, the manuscript does not currently include error bounds or empirical demonstrations of their accuracy, noting instead that such analysis is deferred to future work. To strengthen the paper, it would be helpful for the authors to include, at a minimum, either asymptotic error bounds or a concise empirical assessment of accuracy for AccuCross and AccuXGCA in the main text or appendix.



3. Several contributions are described as being implemented in existing tools such as TempestRemap but documented here for the first time, including the extremal latitude formula and the sweep-line bounding rectangle algorithm. While documenting previously undescribed algorithms is valuable, the paper would benefit from a clearer and more explicit delineation, ideally in the introduction or in a dedicated summary, of which elements are algorithmically novel and which are being formally described for the first time.

Minor Comment:

The manuscript is well written, and the English is of a high standard throughout. Nevertheless, a careful final proofreading pass is recommended to catch isolated typographical errors before resubmission. As one example, Section 5.4 contains a duplicated word: "we we find from (3) that". 

This paper describes the procedures and algorithms used to remap gridded data (mainly) on a sphere.  It is appropriately submitted as a “review and perspectives” paper in that it describes and, in some cases, evaluates existing methods for accomplishing each of the elements of a remapping procedure.  Included are some descriptions of previously undocumented algorithms used in existing tools.  I recommend publication subject to consideration of the following feedback.

For all but a relatively small number of experts actively developing regridding tools, the paper presents a challenge due to its technical nature.  I think the paper could be published in its present form, but the authors might consider attempting to make it more accessible and useful to the large number of analysts who are faced with a choice of remapping tools and would like to choose the one most appropriate for their research.  

I can imagine that at the end of each section (or subsection) describing a particular operation, a summary could be provided indicating, for example, the relative merits of different approaches including 1) accuracy, 2) stability, and computational efficiency.  Of course, in some cases, the differences might be minor, so a statement that all the approaches appear to be equally reliable and computationally efficient might be sufficient.  The summaries provided could be supplemented by a table indicating for each of the (4?) remapping tools considered, which method is used for each of the required operations.

Even for expert developers of remapping tools, that kind of summary information could be exceptionally helpful.  They could more easily assimilate the exceptional expertise of the authors and point them to areas where improvements and exploration of alternatives might be most productive.