A New Derivation of the Formula for the Length of a Loxodrome Arc on a Sphere Using Cylindrical Projections
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThe author abstracts the Earth as a standard sphere, which makes the calculation formula relatively straightforward. It is suggested that the author consider adding the calculation of the loxodrome distance when abstracting the Earth as an ellipsoid. Additionally, it is generally more meaningful in aviation to calculate the shortest distance (great circle distance) on Earth. The author is requested to further elaborate on the practical significance of calculating the loxodrome distance.
Author Response
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Reviewer 2 Report
Comments and Suggestions for AuthorsNo comments!
Author Response
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Reviewer 3 Report
Comments and Suggestions for AuthorsThe author did a great job. There is only an error in Eq. (42), which does not need a "d".
Author Response
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Reviewer 4 Report
Comments and Suggestions for Authors
Nice paper, detailed solution of a well-defined problem.
My comments:
line 21-25, and 39-41: the formatting is different.
line 184: Equation 42: instead of ds’: s’
line 212:, the comma after the formula is required by grammar, but it is not part of the formula, however first it seems to be misleading.
line 246: prijections?
line 58: A short (one paragraph long) explanation is required, why loxodromes are important for us. It was important in the previous centuries, as it made possible the navigation, but why is it important now?
Author Response
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Reviewer 5 Report
Comments and Suggestions for AuthorsThe first stated goal of the paper is self-evident, namely that one must account for distortion when measuring map distance. The first two sentences of the abstract state this as a basic and well-accepted fact, thus it is not clear why this same statement becomes a first goal of the research. The second goal appears to be more interesting, on first glance. However, I question why anyone would have an interest in measuring allegedly accurate distances using a loxodrome, which by definition is intended to measure routes that follow a constant compass bearing. The path of constant compass bearing between two points is rarely equivalent to the shortest distance between them. Even a short inspection of a globe (the closest spherical approximation that one might encounter to a true representation of the geoid, on which distances might be measured accurately) will show this to be the case. For example, travel from Toronto Canada to Augusta, Western Australia (antipodes) and one will travel roughly north for some distance, then roughly south. My point is that constant compass bearing is not equivalent to shortest distance.
The author does not make clear why someone would want to measure distance along a loxodrome, which is a line of constant compass bearing. Navigational distances are rarely measured on cylindrical projections. As a matter of fact, it is dubious that anyone would want to measure distance by means of a loxodrome, when formulae for arc distance directly from latitude and longitude are readily available in textbooks and on the Internet. Spherical calculations will be accurate within the limits of one’s computing device, of course, but certainly arc distances would work on digital computers, cell phones, or even a slide rule or hand calculator, if those are the only available option. A stronger paper might be achieved if the author were to explain one or several situations in which loxodrome distance calculations might be needed. It might also help for the author to explain what is a loxodrome, and an orthodrome, at the outset of the paper.
The equations appear to be correct, and I say this verifying them against several projections texts (Snyder’s books, which the author cites, as well as Richardus and Adler’s text, and some of Tobler’s work).
Line 26 – The statement that no map projection preserves correct scale throughout is not true. The planimetric projection of topographic maps produced by national mapping agencies in most nations preserves scale throughout. Possibly the author is referring to smaller scale thematic maps, and if so that needs to be stated to clarify the false statement.
Line 33 – The incorrect statement about all projections introducing distortion is repeated. I must emphasize these statements need to be corrected. Likewise, the reference to distortion of surfaces needs clarification, since most thematic world maps do not record third dimension shapes or areas, even if they display them. That is to say, even if mountains are incorporated into thematic map display, distance measurement is still most often planar.
Line 37 – It is correct that Euler proved the impossibility of projecting a sphere onto a plane without some type of distortion. Once again however, this is true only for map projections that are simple, meaning only one projection formula is applied. The planimetric projection is a composite of an infinite number of spherical-to-planar projections, accomplished through integral calculus, whose purpose is to project Earth features onto a plane without distortion. Let me encourage the author to qualify statements made using this gross generalization. Once qualified, most of the statements in this article about distortion being applicable in certain map projections become valid. Without this qualification, the article appears to have been written by someone who is unfamiliar with advanced map projections.
Line 93 – the author states skepticism that a loxodrome on Mercator’s projection does not follow the standard equation for Euclidean distance. This contradicts the purpose of a Mercator projection, that loxodromes are represented by straight lines. By definition, the loxodrome will not represent the shortest distance between two points. However, it will represent the shortest distance between two points along a route that follows constant compass bearing; and it will do so with accuracy within the limits of the nominal map scale. One has only to measure the orthodrome on a gnomonic projection to establish the shortest distance between those same two points, if that must be done on a planar projection.
Mathematical results are clearly presented, however the premise of the paper is flawed, and thus the findings do not reflect the stated goal of the paper. I cannot recommend accepting the manuscript.
Author Response
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Author Response File: Author Response.pdf
Round 2
Reviewer 5 Report
Comments and Suggestions for AuthorsReview of IJGI-3414624 R1
New Derivation of the Formula for the Length of a Loxodrome Arc on a Sphere Using Cylindrical Projections
Author comment “… is unfortunately not self-evident. An example of this is the Law on State Surveying and Real Estate Cadastre in Croatia, which stipulates that area is determined in the map projection (Transverse Mercator) without considering distortions.”
Reviewer comment: if this in fact is correct, then the author should state this with a full citation in the revised version.
Author comment: I agree with the Reviewer that constant compass bearing is not equivalent to shortest distance. The article does not mention the shortest distance, nor does it claim that loxodrome navigation gives the shortest path. The article does not deal with the orthodrome, that is, the great circle, but with the loxodrome.
Reviewer comment: That is exactly my point, that the author has avoided responding to. Let me repeat my question: why would anyone have an interest is measuring allegedly accurate distances using a loxodrome? At a minimum, the revision needs to address this question. And pointing this reviewer to run a literature search in another journal misses the point that what the author has not done in the original or in the revised version is to cite these papers and expand the discussion to clarify why anyone would want to measure distance along a loxodrome. In navigation, the conventional practice is to approximate the loxodrome with a series of orthodromes (great circle arcs) and then tally the distances along the series.
Author comment: Distances in navigation are almost always determined from maps made in the Mercator projection, which is one of the cylindrical projections.
Reviewer comment: This is simply not correct in current practice. Distances in navigation are determined from satellite or other forms of digital data, not from projected map data and certainly not from Mercator projected data. Perhaps the author is referring to historical practice, but in current practice, even my first semester students would not make such a mistake.
Author comment: The Reviewer is wrong. The statement that no map projection preserves the correct scale throughout is true.
Reviewer comment: I won’t get into a debate about this point, having worked on cartographic research with national mapping agencies in several countries. Planimetric projections are designed to preserve correct scale throughout the extent of the data. And the author’s repeated response on this point does not change the truth.
Author’s comment: The proof was first given by Euler in 1777. His proof does not apply to "simple projections", but to any projections.
Reviewer comment: I direct the author to read any reliable text on the derivation of a planimetric projection, which is a composite of an infinite number of infinitely small projections. Continuous mathematics allows many tasks involving infinite numbers of infinitely small items. Taking one example from map projections, consider Tissot’s Indicatrix that measures distortion in projections by transforming an infinitely small circle from a globe onto a projected surface. Other examples exist in many branches of analytic geometry. I suggest that the author do as he or she has proposed, namely “… immediately retract [the] article.”
Author Response
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