Hexahedral Projections: A Comprehensive Review and Ranking
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThe article is extremely long, 44 pages! It needs to be shortened. For example divide into two. In one, give a presentation of hexahedral projections, and in the other, their good and bad sides with ranking.
As the manuscript is complex, it looks more like a technical report or master's thesis than a scientific article.
It would be good to compare hexahedral projections with classical projections in the plane.
Common designations for latitude and longitude in geodesy and cartography are phi and lambda, while other designations are used in the article. In the article, phi is the symbol for longitude and this can be confusing.
Line 42 and in many places further, instead of „cartographic projection“, „map projection“ is better.
Line 327. Authors use authalic latitude and conformal latitude as auxiliary latitude. Since there are infinitely many ways of mapping an ellipsoid to a sphere, it is not clear why some others were not considered. For example mapping according to normals, i.e. mapping in which nothing is calculated, but geographical coordinates on the sphere are identified with those on the ellipsoid.
Line 346. The authors refer to Karney (2023) who demonstrated significantly greater numerical stability by using the third flattening n. However, this was also established by F. R. Helmert, Die mathematischen und physikalischen Theorien der höheren Geodäsie .., 1880.
Lines 351-397. Can be significantly shortened.
Lines 485-493, cm, mm and meters are mentioned. It is not clear where these values for lengths came from.
Line 501-573 Can be shortened significantly.
Line 634 and on in several places. It is necessary to distinguish between „surface“ and „area“. These two words have different meanings. It is not about surface distortion, but about area distortion.
Line 713 and on. Instead of „parametric projections“, it would be better to use „projections with parameters“.
Line 763, instead of „perfect circles“, circles is enough, because there is no imperfect circle.
Lines 823-927, can be omitted entirely because it is all known
Lines 929-992 should be omitted and replaced with one or two sentences explaining the choice of parameter.
Lines 994-1070, refers only to the TAN projection. It needs to be shortened significantly to just a few sentences.
Lines 1087-1373. Much too detailed.
Line 1419. Instead of Fechs, Sechs should be used.
Author Response
Thank you for taking the time to review our manuscript and for your constructive and positive feedback. Below, you will find detailed responses to the comments, with all revisions highlighted in the re-submitted manuscript using track changes.
Comments 1: The article is extremely long, 44 pages! It needs to be shortened. For example divide into two. In one, give a presentation of hexahedral projections, and in the other, their good and bad sides with ranking.
Response 1: Thank you for your comment. "The manuscript has been shortened. However, due to significant comments from other reviewers that must be addressed, some parts could not be completely removed.
Comments 2: It would be good to compare hexahedral projections with classical projections in the plane.
Response 2: Thank you for your suggestion. While a comparison with classical planar projections would indeed be valuable, the scope of this manuscript is specifically focused on the evaluation and ranking of hexahedral projections. Additionally, due to length constraints and multiple reviewers’ requests for truncation, we don't think it would be acceptable to expand the discussion further.
Comments 3: Common designations for latitude and longitude in geodesy and cartography are phi and lambda, while other designations are used in the article. In the article, phi is the symbol for longitude and this can be confusing.
Response 3: We were already aware of this, but for decades we have been developing program code using the appropriate notations for longitude and latitude, so the formulas followed that notation to prevent errors. However, we have now updated all formulas and references throughout the text. We kindly ask you to refer to the track-changes for a detailed overview.
Comments 4: Line 42 and in many places further, instead of „cartographic projection“, „map projection“ is better.
Response 4: Thank you for your comment. All occurrences of “cartographic projection” have been replaced with “map projection” throughout the manuscript.
Comments 5: Line 327. Authors use authalic latitude and conformal latitude as auxiliary latitude. Since there are infinitely many ways of mapping an ellipsoid to a sphere, it is not clear why some others were not considered. For example mapping according to normals, i.e. mapping in which nothing is calculated, but geographical coordinates on the sphere are identified with those on the ellipsoid.
Response 5: Thank you for the comment. This is another topic that could be explored in more detail, but doing so would further extend an already lengthy manuscript. In short, since the manuscript analyzes both angular and areal distortions, the most logical approach was to use the conformal and authalic auxiliary latitudes when mapping the ellipsoid onto the sphere to fully preserve the corresponding properties. While using geodetic latitude is the simplest method, as it requires no transformations, it slightly compromises the projection's equivalence or conformality.
Comments 6: Line 346. The authors refer to Karney (2023) who demonstrated significantly greater numerical stability by using the third flattening n. However, this was also established by F. R. Helmert, Die mathematischen und physikalischen Theorien der höheren Geodäsie .., 1880..
Response 6: Thank you for the valuable reference. It has been added as [43], and the text has been updated as follows:
“The significance of applying the third flattening in geodesy was established by F. R. Helmert in 1880 [43], who demonstrated its role in achieving faster convergence of the corresponding series. In his work, however, n is not explicitly identified under a specific term but is treated as one of the auxiliary quantities. Even before Helmert, this parameter was used in calculations by Bessel in 1841 [44] and Encke in 1852 [45].”
Comments 7: Lines 351-397. Can be significantly shortened.
Response 7: Thank you for the comment. Table 1, along with a portion of the text, has been removed.
Comments 8: Lines 485-493, cm, mm and meters are mentioned. It is not clear where these values for lengths came from.
Response 8: These are arc distances calculated for the authalic sphere. However, this is no longer relevant, as the corresponding images and a portion of the text have been removed to shorten the manuscript.
Comments 9: Line 501-573 Can be shortened significantly.
Response 9: Agree. This section has been significantly shortened.
Comments 10: Line 634 and on in several places. It is necessary to distinguish between „surface“ and „area“. These two words have different meanings. It is not about surface distortion, but about area distortion.
Response 10: Thank you for pointing this out. It was an oversight during the editing process. The terms have now been aligned.
Comments 11: Line 713 and on. Instead of „parametric projections“, it would be better to use „projections with parameters“.
Response 11: Thank you for your suggestion. However, we believe that the term “parametric projection” is more widely recognized in the literature and aligns better with the established terminology in this field. Therefore, we have decided to retain the original term.
Comments 12: Line 763, instead of „perfect circles“, circles is enough, because there is no imperfect circle.
Response 12: Thank you for pointing this out. The issue has been corrected.
Comments 13: Lines 823-927, can be omitted entirely because it is all known
Response 13: This section has been revised. However, numerous comments from other reviewers not only required a response but also influenced our understanding, making it important to retain this part in the manuscript..
Comments 14: Lines 929-992 should be omitted and replaced with one or two sentences explaining the choice of parameter.
Response 14: Thank you for the comment. This section has been revised and significantly shortened.
Comments 15: Lines 994-1070, refers only to the TAN projection. It needs to be shortened significantly to just a few sentences.
Response 15: The TAN projection is included for illustration purposes only. We believe it is important for readers to familiarize themselves with the shape of the curves that depict the dependence of metrics on the parameter p. The text has been significantly revised, and some figures have been removed.
Comments 16: Lines 1087-1373. Much too detailed.
Response 16: Although other reviewers suggested that more detail was needed, this section has been significantly revised and slightly shortened. All unrelated comments have been removed, leaving only information pertinent to the metrics and rankings.
Comments 17: Line 1419. Instead of Fechs, Sechs should be used.
Response 17: Thank you for the correction. I have updated the reference as per your recommendation. Although this is the official name on the atlas itself, as shown in the attached image.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsThis paper is a review of several map projections on a cube. Although no new results are achieved, a significant contribution is to summarize previous papers, and comparing merits of such mappings to each other using a consistent method. The study can be, therefore, considered as novel.
The minor details of the paper is fine. The only major problem is that the authors failed to read and reference any studies about the *general theory* of map projections. Thus, their use of terminology is a bit clumsy, and they made some false general statements (without reference, of course). A reference to some basic theoretical papers is a must in every scientific paper.
I shall refer to line numbers in my detailed comments:
19–24. I feel that the tone of this paragraph is slightly too pathetic for a serious scientific paper.
27. The result of the difference in topology is that all map projections are discontinuous somewhere (e.g. you need to cut the cube along edges to unfold it to the plane). It is completely unrelated to distortions. The cube is homeomorphic to the sphere, but no distortion-free mapping is possible, because their Gaussian curvature differs.
28. No map projection can preserve shapes. For example, the Mercator projection is conformal, but the shape of Greenland is quite distorted. Map projections may only preserve the shape of infinitely small objects.
33–38. The authors list examples of polyhedra used for map projections, but fail to list references to strengthen their statements.
64. It is impossible to map the ellipsoid directly to a regular polyhedron because the ellipsoid of revolution does not have the regular symmetries the polyhedra have. The only "work-around" solution is to use auxiliary latitudes on a sphere, but that is technically a double mapping (spheroid→sphere→polyhedron). Strictly speaking, hexahedral projections cannot be defined for the ellipsoid, only for the sphere.
98. That is not a rotation of the graticule, but changing the Prime Meridian.
114. Lee listed not only power series, but also closed formulae in terms of elliptic functions. Given that standard implementations exist (e.g. Burlisch's method), those are fairly usable.
162. I doubt that the projection of Rančić would be different from that of Lee, as Lee has demonstrated that (apart from scaling and rotation) there is only one conformal map of a sphere on the cube. Is not it just a reformulation with better numerical stability?
212. The mapping with the lowest mean angular distortion is the conformal one, i.e., the projection of Lee. I bet the authors cannot list a map projection having less mean angular deviation than zero…
417–436. This has been studied previously, but the authors fail to give references. Wray (DOI 10.3138/E382-8522-4783-28K5) deals with the cube on pp. 41–43, but the whole work is relevant from the perspective of the study. It gives proper terminology on graticule rotation. Furthermore, eqs. (6) and (7) can also be expressed in terms of geographic coordinates. See DOI 10.1080/23729333.2016.1184554
445. The 'LatLon' is properly called as the Plate Carrée map projection.
476. It is a long tradition in cartography to denote latitude by curly phi and longitude by lambda. Please consider to adapt, as your notation made it hard to me to follow your derivations.
483. You should never calculate longitudes by arcsine/arccosine. This is the reason of your numerical instability. Deriving formulae (9) and (11) from either the cotangent four-part formula or the Napier rules, would make it possible to express the *tangent* of the longitude. The arctan2 function is numerically stable.
619. Please consider to stick to well-known terms intead of 'area-preserving'. Any of the three other terms listed on line 658 would be acceptable.
629. Do not use plural, as there is only one single conformal map on the cube.
747–748. See comment for 27.
768–771. These formulae appear without reference to previous literature. Furthermore, the terminology is unusal. omega is the maximum angular *deviation*, sigma is *areal scale*, alpha is *angular* distortion. You must not use the term angular distortion for omega, as it contradicts the terminology of previous papers. Please stick to angular deviation.
777. omega and alpha are not *correlated*, they have *functional relationship*, which is a stronger dependence than just a 'close correlation': omega=2arcsin((alpha-1)/(alpha+1)), i.e., they measure exactly the same property, just on a different scale. Fig. 20. strengthens my statements.
799. Again, please do not confuse correlation with functional dependence. They are completely different concepts.
802–806. If you decide not drawing distribution of GOF (as it is redundant with areal scale sigma), why draw distribution of omega, which is also functionally dependent on alpha (so is also redundant).
861. You are plainly wrong here. The reverse is always true: If *all* indicatrices are circles then the map projection is conformal. It is well-known, read any basic reference material on theory of map projections…
873. Tissot's indicatrices are always ellipses. Please check the original monograph of Tissot. Thouse strange shapes in Fig. 18 are *not* Tissot's indicatrices. Anyway, the formulae defining the semi-axes of Tissot's ellipses contain the partial derivatives of the mapping functions. How could you even evaluate these formulae, where the mapping is not C1 differentiable? Did you cheat with some numerical method? If yes, your results can be misleading!
937. Areal scale sigma is not additive, but multiplicative. This means that the proper measure of dispersion is not the standard deviation, but the *geometric* standard deviation. See DOI 10.1080/15230406.2020.1768439 and https://www.jstor.org/stable/2530139
973. The proper terminology for this 'debate' is already settled by Meshcheryakov, G. A. (1968). Teoreticheskie osnovy matematicheskoj kartografii. Nedra. This is a basic reference that is ought to be cited here. Meshcheryakov calls minimization of the (geometric) standard deviation as 'variational method' and the minimization of ratio (or difference) between extrema as the 'minimax method'. These terms are accepted and well-understood by the community.
I appreciate the authors for indicating that the ranking method in Sec. 5 is partially subjective. I rarely read such honest self-assessment in similar papers. I have no objections to this section.
The conclusions are perfect.
Author Response
Thank you very much for your extensive and extremely useful review. Your comments have contributed most to improving the quality of the manuscript itself, but also to a deeper understanding of the problem and to a more formal and professional tone in the manuscript. Below, you will find detailed responses to the comments, with all revisions highlighted in the re-submitted manuscript using track changes.
Comments 1: The only major problem is that the authors failed to read and reference any studies about the *general theory* of map projections. Thus, their use of terminology is a bit clumsy, and they made some false general statements (without reference, of course).
Response 1: Thank you for your comment. We fully agree with your observation. The total number of references has been increased by 17, with careful attention to aligning the terminology with the formal terms used in those references, as well as incorporating the terms suggested in your further comments.
Comments 2: 19–24. I feel that the tone of this paragraph is slightly too pathetic for a serious scientific paper.
Response 2: The paragraph has been revised and is now presented as follows:
“Cartographic maps have played a crucial role in human civilization, facilitating navigation, planning, and understanding of the world. From early representations of landscapes, such as carvings on mammoth tusks over 25,000 years ago, and Babylonian clay tablets, to the mathematical theories of Earth's shape developed in ancient Greece, cartography has undergone significant evolution to reach its modern forms.”
Comments 3: 27. The result of the difference in topology is that all map projections are discontinuous somewhere (e.g. you need to cut the cube along edges to unfold it to the plane). It is completely unrelated to distortions. The cube is homeomorphic to the sphere, but no distortion-free mapping is possible, because their Gaussian curvature differs.
Response 3: The sentence in line 27 has been rephrased and now reads as follows:
“Since the sphere and the plane have different Gaussian curvature, no mapping can preserve both angles and areas perfectly.”
Comments 4: 28. No map projection can preserve shapes. For example, the Mercator projection is conformal, but the shape of Greenland is quite distorted. Map projections may only preserve the shape of infinitely small objects.
Response 4: We completely agree, and it was an inadvertent mistake. The sentences in lines 27 and 28 have been rephrased and now reads as follows:
“Since the sphere and the plane have different Gaussian curvature, no mapping can preserve both angles and areas perfectly. Projections that locally preserve angles are known as conformal or orthomorphic projections, whereas those that preserve areas are referred to as equal-area or equivalent projections.”
Comments 5: 33–38. The authors list examples of polyhedra used for map projections, but fail to list references to strengthen their statements.
Response 5: The text has been revised as follows:
“Commonly used Platonic solids include the tetrahedron [9,51,52], octahedron [9,53], and icosahedron [9,54,55], with 4, 8, and 20 triangular faces, respectively, as well as the cube [1-9,11-18], which has 6 square faces. The truncated icosahedron—a semi-regular Platonic solid with 32 faces, consisting of 20 hexagons and 12 pentagons—is also frequently employed [9].”
Comments 6: 64.It is impossible to map the ellipsoid directly to a regular polyhedron because the ellipsoid of revolution does not have the regular symmetries the polyhedra have. The only "work-around" solution is to use auxiliary latitudes on a sphere, but that is technically a double mapping (spheroid→sphere→polyhedron). Strictly speaking, hexahedral projections cannot be defined for the ellipsoid, only for the sphere.
Response 6: Agree. The sentence has been revised based on your comment and now reads as follows:
“Specifically, by employing auxiliary latitudes to map a spheroid onto a sphere, it becomes possible to indirectly project an ellipsoid onto a cube.”
Comments 7: 98. That is not a rotation of the graticule, but changing the Prime Meridian.
Response 7: The sentence was initially reformulated, but following another reviewer's request to significantly shorten the section, the entire paragraph was condensed and now reads as follows:
“In 1803, German cartographer Christian Gottlieb Reichard produced a six-sheet world atlas [1] employing a gnomonic projection, also known as Tangential Spherical Cube (TSC projection), aligned with the Ferro meridian.”
Comments 8: 114. Lee listed not only power series, but also closed formulae in terms of elliptic functions. Given that standard implementations exist (e.g. Burlisch's method), those are fairly usable.
Response 8: Agree.
“Around the same time, though not published until 1976, L. P. Lee explored conformal projections based on Jacobian elliptic functions [4], including the projection of a sphere onto a cube. His study derived the relevant formulas and computed coordinates both in closed form and through series expansions.”
Comments 9: 162. I doubt that the projection of Rančić would be different from that of Lee, as Lee has demonstrated that (apart from scaling and rotation) there is only one conformal map of a sphere on the cube. Is not it just a reformulation with better numerical stability?
Response 9: Agree.
“In 1995, M. Rančić et al. reformulated the conformal hexahedral projection intro-duced by Lee two decades earlier, enhancing its numerical stability (RAN projection) and applying it in the global shallow-water model [17].”
Comments 10: 212. The mapping with the lowest mean angular distortion is the conformal one, i.e., the projection of Lee. I bet the authors cannot list a map projection having less mean angular deviation than zero
Response 10: Agree. This error occurred during the editing process while attempting to improve readability, resulting in the term “compromise projection” being mistakenly replaced with “compromise design.” Now, the text reads as follows:
“The Outerra Spherical Cube (OTC projection) is a compromise hexahedral projection that minimizes mean angular distortion but suffers from significant areal distortion at the cube's vertices.”
Comments 11: 417–436. This has been studied previously, but the authors fail to give references. Wray (DOI 10.3138/E382-8522-4783-28K5) deals with the cube on pp. 41–43, but the whole work is relevant from the perspective of the study. It gives proper terminology on graticule rotation. Furthermore, eqs. (6) and (7) can also be expressed in terms of geographic coordinates. See DOI 10.1080/23729333.2016.1184554].
Response 11: Thank you very much for suggesting the two highly valuable references ([46] and [47]) included in the manuscript. Changes have been made in several paragraphs, so we kindly ask you to refer to the track-changes for a detailed overview.
Comments 12: 445. The 'LatLon' is properly called as the Plate Carrée map projection.
Response 12: Thank you for your comment. The change has been made as suggested.
Comments 13: 476. It is a long tradition in cartography to denote latitude by curly phi and longitude by lambda. Please consider to adapt, as your notation made it hard to me to follow your derivations.
Response 13: We were already aware of this, but for decades we have been developing program code using the appropriate notations for longitude and latitude, so the formulas followed that notation to prevent errors. However, we have now updated all formulas and references throughout the text. Instead of the slanted phi, the standard phi is now used, as it is more common in older literature. We kindly ask you to refer to the track-changes for a detailed overview.
Comments 14: 483. You should never calculate longitudes by arcsine/arccosine. This is the reason of your numerical instability. Deriving formulae (9) and (11) from either the cotangent four-part formula or the Napier rules, would make it possible to express the *tangent* of the longitude. The arctan2 function is numerically stable.
Response 14: The formulae have been updated, but this change is unrelated to the error that occurred, as the image displayed an error in latitude (where lambda equals 0). In any case, using atan2 is preferable. For the revised formulae, please refer to the manuscript. The entire section has been significantly shortened in response to a request from one of the reviewers.
Comments 15: Please consider to stick to well-known terms instead of 'area-preserving'. Any of the three other terms listed on line 658 would be acceptable.
Response 15: Thank you for your comment. We have now adopted the term “equal-area”.
Comments 16: 629. Do not use plural, as there is only one single conformal map on the cube.
Response 16: Thank you for the comment. It has been revised throughout the manuscript.
Comments 18: 747–748. See comment for 27.
Response 18: Agree.
“A spheroidal surface, such as that of the Earth, possesses intrinsic curvature, whereas a map is a flat surface with zero Gaussian curvature. According to Gauss's Theorema Egregium [58], curvature is an invariant property that cannot be altered through any isometric mapping. Consequently, since a curved surface cannot be flattened without modifying distances, angles, or areas, every map projection necessarily introduces some form of distortion.”
Comments 19: 768–771. These formulae appear without reference to previous literature. Furthermore, the terminology is unusal. omega is the maximum angular *deviation*, sigma is *areal scale*, alpha is *angular* distortion. You must not use the term angular distortion for omega, as it contradicts the terminology of previous papers. Please stick to angular deviation.
Response 19: Thank you for your comment. Please refer to the revised manuscript to review the changes.
Comments 20: 777. omega and alpha are not *correlated*, they have *functional relationship*, which is a stronger dependence than just a 'close correlation': omega=2arcsin((alpha-1)/(alpha+1)), i.e., they measure exactly the same property, just on a different scale. Fig. 20. strengthens my statements.
Response 20: Thank you very much. This was one of the key comments that led to significant revisions in the manuscript. The changes are widespread, so they cannot be listed here. Please refer to the manuscript to review the updates.
Comments 21: 799. Again, please do not confuse correlation with functional dependence. They are completely different concepts.
Response 21: We apologize for the mistake and thank you for your comment. The error has been corrected.
Comments 22: 802–806. If you decide not drawing distribution of GOF (as it is redundant with areal scale sigma), why draw distribution of omega, which is also functionally dependent on alpha (so is also redundant).
Response 22: We completely agree. The entire sentence has been deleted. However, Figure 13 (formerly Figure 20) has been retained to illustrate the difference in isoline density.
Comments 23: 861. You are plainly wrong here. The reverse is always true: If *all* indicatrices are circles then the map projection is conformal. It is well-known, read any basic reference material on theory of map projections…
Response 23: Agree.
“If all indicatrices are circles, regardless of their size, the projection is conformal. If all indicatrices have the same area, regardless of their shape, the projection is equal-area.”
Comments 24: 873. Tissot's indicatrices are always ellipses. Please check the original monograph of Tissot. Thouse strange shapes in Fig. 18 are *not* Tissot's indicatrices. Anyway, the formulae defining the semi-axes of Tissot's ellipses contain the partial derivatives of the mapping functions. How could you even evaluate these formulae, where the mapping is not C1 differentiable? Did you cheat with some numerical method? If yes, your results can be misleading!
Response 24: Agree.
“Moreover, Tissot's formulas cannot be applied at discontinuities, such as along the face diagonals of the QSC projection. In these cases, numerical methods must be used to determine the shape into which the infinitesimal circle is mapped.”
Comments 25: 937. Areal scale sigma is not additive, but multiplicative. This means that the proper measure of dispersion is not the standard deviation, but the *geometric* standard deviation. See DOI 10.1080/15230406.2020.1768439 [48] and https://www.jstor.org/stable/2530139 [49]
Response 25: Thank you for the valuable references, which have been added as [48] and [49]. Your comment has had a significant impact on the manuscript. Please review the paper to see the changes.
Comments 26: The proper terminology for this 'debate' is already settled by Meshcheryakov, G. A. (1968). Teoreticheskie osnovy matematicheskoj kartografii. Nedra. [50] This is a basic reference that is ought to be cited here. Meshcheryakov calls minimization of the (geometric) standard deviation as 'variational method' and the minimization of ratio (or difference) between extrema as the 'minimax method'. These terms are accepted and well-understood by the community.
Response 26: Thank you for the valuable reference, which has been added as [50].
“In previous studies [33,15,16], the parameterization of compromise hexahedral projections has consistently aimed to minimize areal distortion, employing variational and minimax methods [50]. The variational method minimizes dispersion by reducing the standard deviation, while the minimax method minimizes the ratio of extreme values for multiplicative metrics or the difference of extreme values for additive metrics.”
Author Response File: Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for AuthorsDear colleagues,
thank you for your valuable contribution on hexahedral projections for geomapping. Indeed, this is an important topic. Especially in terms of global reference frames, hexahedral projections support processing of big data, which by far extents geovisualisation.
Your contribution is understandable and your arguments are easy to follow.
A clear research question is hard to find. Is it a recommendation for a better hexahedral projection with lower distortion values?
From my point of view this paper, as also stated by the authors, gives an overview on the different approaches of hexahedral projections and compares their classification. The most important sections from my point of view are chapter 4 that focuses on distortion metrics and chapter 5 that evaluates and ranks different hexahedral projections.
The introduction is fine. Chapter 2 on the history is much too long for a scientific paper. Chapter 3 is a fundamental insight into the hexahedral projection, which could be a paper on its own. It is quite extensive.
Thank you for your effort and important contribution.
Author Response
Thank you for taking the time to review our manuscript and for your constructive and positive feedback. Below, you will find detailed responses to the comments, with all revisions highlighted in the re-submitted manuscript using track changes.
Comments 1: A clear research question is hard to find. Is it a recommendation for a better hexahedral projection with lower distortion values?
Response 1: Thank you for your comment. To clarify, this manuscript evaluates hexahedral projections based on multiple criteria and introduces a ranking system designed to provide recommendations for specific applications while identifying the most versatile hexahedral projection. Of course, minimizing distortion is an important aspect of this, but it is not the only one.
Comments 2: Chapter 2 on the history is much too long for a scientific paper.
Response 2: Agree. Section 2 has been substantially shortened while preserving all relevant references. Hexahedral projections referenced later in the manuscript were retained to define their abbreviated names and maintain their order of appearance. Comments and descriptions have been considerably condensed.
Comments 3: Chapter 3 is a fundamental insight into the hexahedral projection, which could be a paper on its own. It is quite extensive.
Response 3: Agree. Section 3 has been condensed even further than Section 2.
Author Response File: Author Response.pdf
Reviewer 4 Report
Comments and Suggestions for AuthorsThis paper systematically combs the hexahedral projections developed in the past 50 years, covering a variety of types and characteristics, and classifies them in detail, providing a clear historical context and genre framework for research in this field.
The complete implementation process of the projection from earth surface to cube and back projection is described in detail, including the mapping from ellipsoid to sphere, the rotation of coordinate grid, the determination of cube side and the mapping from sphere part to cube surface. The influence of each step on accuracy and distortion is analyzed in detail.
This paper also deeply analyzes the numerical and graphical characteristics of hexahedral projection, focusing on the distortion index, and visualizes the distortion through a variety of methods, which provides a multi-dimensional perspective for projection quality evaluation. For the parameterized hexahedral projection, an optimization strategy is proposed to determine the parameter values under different optimization criteria to improve the projection performance.
A comprehensive hierarchical ranking system is constructed to rank hexahedral projections by single index and comprehensive index based on multiple criteria such as structural characteristics, numerical characteristics and distortion indicators, which provides a powerful reference for selecting appropriate projections for different application scenarios.
Some issues and suggestions:
1) In the experiment part, the specific configuration of the parameters, and the hardware equipment and software environment used in the experiment, as well as the experimental data generation and the samples selection could be described in detail to enhance the repeatability and reliability of the experimental results.
2) In the ranking method, the basis for determining the weight of each index in the comprehensive ranking process need to be further explained to ensure the rationality and scientific nature of the ranking results, so that readers can better understand the advantages and disadvantages of different projections in the overall evaluation.
Author Response
Thank you for taking the time to review our manuscript and for your constructive and positive feedback. Below, you will find detailed responses to the comments, with all revisions highlighted in the re-submitted manuscript using track changes.
Comments 1: In the experiment part, the specific configuration of the parameters, and the hardware equipment and software environment used in the experiment, as well as the experimental data generation and the samples selection could be described in detail to enhance the repeatability and reliability of the experimental results.
Response 1: Thank you for pointing this out, but it was already defined in the manuscript. For example, in the section where performance and accuracy are presented, it states:
“Measurements are taken from 64 million samples uniformly distributed over the S0 side. The tests were conducted on a system equipped with an Intel Core i5-11400H CPU at 2.70GHz and 16GB of DDR4 dual-channel memory. The program code was developed in ISO C++ 14, using the Visual Studio 2022 (v143) platform toolset, and executed on a Windows 11 Pro Education operating system. Precise time measurements were obtained using the QueryPerformanceCounter() Win32 profiler API function, which offers a resolution of 0.1 microseconds on the tested platform.”
Comments 2: In the ranking method, the basis for determining the weight of each index in the comprehensive ranking process need to be further explained to ensure the rationality and scientific nature of the ranking results, so that readers can better understand the advantages and disadvantages of different projections in the overall evaluation.
Response 2: Agree. This section has been extensively revised and significantly shortened per another reviewer's request. We hope it now strikes a good balance.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
Comments and Suggestions for AuthorsLine 609: „According to Gauss's Theorema Egregium [58], curvature is an invariant property that cannot be altered through any isometric mapping. Consequently, since a curved surface cannot be flattened without modifying distances, angles, or areas, every map projection necessarily introduces some form of distortion.“ The statement is correct, but Euler proved and published the same statement in 1777, the year Gauss was born. See: Euler L (1777) De repraesentatione superficiei sphaericae super plano, Acta Academiae Scientiarum Imperialis Petropolitanae. Translated into German in: Drei Abhandlungen uber Kartenprojection, Ostwald's Klassiker der exakten Wissenschaften, no. 93, pp. 3–37, Leipzig, Wilhelm Engelmann, 1898.
Line 1183: If you look a little bit more closely, you will notice that the letter ſ is very similar to the letter f, but is still different from it. At the time that atlas was published, the ſ character was used in Germany instead of the current s.
The manuscript is too long. There is material for two articles in the manuscript. If I were the editor, I would ask that this one be shortened in half. But since I am not the editor of the journal, I leave it to the editor to make the decision.
Author Response
Thank you for taking the time to review our revised manuscript and for providing additional constructive feedback. Below, we have provided detailed responses to your comments.
Comments 1: Line 609: „According to Gauss's Theorema Egregium [58], curvature is an invariant property that cannot be altered through any isometric mapping. Consequently, since a curved surface cannot be flattened without modifying distances, angles, or areas, every map projection necessarily introduces some form of distortion.“ The statement is correct, but Euler proved and published the same statement in 1777, the year Gauss was born. See: Euler L (1777) De repraesentatione superficiei sphaericae super plano, Acta Academiae Scientiarum Imperialis Petropolitanae. Translated into German in: Drei Abhandlungen uber Kartenprojection, Ostwald's Klassiker der exakten Wissenschaften, no. 93, pp. 3–37, Leipzig, Wilhelm Engelmann, 1898.
Response 1: Thank you very much for pointing out this important reference. After the revision, the paragraph has the following form:
"A spheroidal surface, such as that of the Earth, possesses intrinsic curvature, whereas a map is a flat surface with zero Gaussian curvature. The impossibility of perfectly mapping a sphere onto a plane was first demonstrated by Euler half a century before Gauss's Theorema Egregium [52], in his work De repraesentatione superficiei sphaericae super plano [53]. Gauss later showed that curvature is an invariant property that cannot be altered through any isometric mapping. Consequently, since a curved surface cannot be flattened without modifying distances, angles, or areas, every map projection necessarily introduces some form of distortion."
This way, Euler's precedence over Gauss is highlighted, while still preserving Gauss's theorem and his broader theoretical framework and contributions to this field.
Added reference:
[53] Euler L. De repraesentatione superficiei sphaericae super plano, Acta Academiae Scientiarum Imperialis Petropolitanae, 1777, 1777(1), 107–132.
Comments 2: Line 1183: If you look a little bit more closely, you will notice that the letter ſ is very similar to the letter f, but is still different from it. At the time that atlas was published, the ſ character was used in Germany instead of the current s.
Response 2: I apologize for my lack of knowledge of the German language. You are absolutely right.
Comments 3: The manuscript is too long. There is material for two articles in the manuscript. If I were the editor, I would ask that this one be shortened in half. But since I am not the editor of the journal, I leave it to the editor to make the decision.
Response 3: We do acknowledge that the manuscript contains an extensive amount of material. However, since the reviewers evaluated it in its complete form, significantly shortening it after incorporating their suggestions was not feasible.
Author Response File: Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsThe authors improved the paper based on the previous round.
I am satisfied in all responses but number 10.
The new version still reads that the OTC projection minimizes angular deviation. However, if a conformal map projection exists (and it does exist in our case) then that one minimizes angular distortion, and no other can.
Regarding Response 24, DOI 10.1559/152304006779500687 in their Fig. 21. show an analityc method to deal with this situation: They suggest plotting two half indicatrices.
Author Response
Thank you for taking the time to review our revised manuscript and for providing additional constructive feedback. Below, we have provided detailed responses to your comments.
Comments 1: I am satisfied in all responses but number 10. The new version still reads that the OTC projection minimizes angular deviation. However, if a conformal map projection exists (and it does exist in our case) then that one minimizes angular distortion, and no other can.
Response 1: Thank you for your comment. We completely agree that the term “minimization of angular distortion” is not appropriate when referring to a compromise projection. Therefore, the sentence has been reformulated and now reads:
“The Outerra Spherical Cube (OTC projection) is a projection with relatively low mean angular distortion—lower than that of all other compromise hexahedral projections—but with significant areal distortion at the cube's vertices.”
Comments 2: Regarding Response 24, DOI 10.1559/152304006779500687 in their Fig. 21. show an analityc method to deal with this situation: They suggest plotting two half indicatrices.
Response 2: Thank you for the suggested reference. We would only generalize the case so that, instead of the composition of two parts as stated in reference [61], we refer to the composition of multiple segments bounded by discontinuities. Thus, the paragraph now reads:
“Moreover, Tissot's formulas cannot be applied at discontinuities, such as along the face diagonals of the QSC projection. In these cases, numerical methods can be used to determine the shape into which the infinitesimal circle is mapped, or the shape can be reconstructed as a composition of segments of the indicatrix bounded by discontinuities [58].”
Added reference:
[58] Van Leeuwen, D.; Strebe, D. A "Slice-and-Dice" Approach to Area Equivalence in Polyhedral Map Projections. Cartography and Geographic Information Science, 2006, 33(4), 269–286. https://doi.org/10.1559/152304006779500687
Author Response File: Author Response.pdf
