A Spatially Bounded Airspace Axiom
Round 1
Reviewer 1 Report
This paper presents a mathematical definition of airspace in the context of graph theory and is well-written.
Although the application possibilities are manifold and the paper may be used as basis for other research papers, it has to be improved. It is not unusual to use graph theory to model airspace – not only free airspace – and this has been done earlier in many contexts. Here, it is only stated that the authors use graph theory in their FRA studies but they introduce neither advantages of applying graph theory in this context nor examples of applications. I would appreciate a paper clearly defining such a graph-theoretic model that could be referred to by others. But here I miss examples of applications (FRA is far too general), a clear description of the goal of this paper (Why is this axiom defined? What is it good for?) and a relation to the numerous earlier works where graph theory has been applied in the context of airspace, see e.g. “An Airspace Model Applicable for Automatic Flight Route Planning Inside Free Route Airspace” by Drupka et. al., “Regular graph-based free route flight planning approach” by Samolej et. al. or “Air Space Routing and Flights Planning: A Problem Statement and Discussion of Approaches to Solution” by Alieksieiev.
I strongly recommend to rework the paper and start with a thorough literature review showing the value of graph theory in airspace applications. The manifold applications can be a motivation for the clear definition of an axiom as presented. The benefits of graph theory should be clearly explained. An outlook on how the theory could be enhanced by taking the third (and eventually fourth – that is time horizon) dimension into account could complete the paper.
I also have some more direct comments:
The definition of FRA significant point in section 3.4 seem not to be restricted to free route airspace. The same definition would be valid for classic air route networks.
In chapter 4 (line 215/216) the authors state that a flight can only enter and exit an airspace once. This would require convexity of the airspace. Although later in the paper an airspace polygon is defined as convex hull of the airspace, the axiom itself (defining the spatially bounded airspace) does not require convexity. In the conclusions it is again referred to the axiom not to the convex airspace polygon. It is not stated why the convexity is necessary. It seems to result only from the restriction that a flight is allowed to enter the airspace only once. But what about entry points on the concave part of the airspace? How are those considered by the presented model? Are there any mappings to the convex hull?
In section 4.2 the no loop property has at least one mandatory Intermediate Point (line 242). Why is an intermediate point necessary? A flight might take a direct route from entry to exit, especially in a free route environment.
Besides all these critical comments, the application of graph theory is undoubtedly valuable in airspace applications. I think that the enhancement of the presented work towards an overview of clear definition of graph theory, its advantages and application possibilities in relation to the airspace domain would lead to a very valuable contribution to the Axioms journal.
Author Response
You will find the review answers in the attached file
Reviewer 2 Report
This paper has some interested, but is, in my opinion, out of the scope of the journal since no new mathematical result is presented. I suggest to resubmit to an application-oriented journal like transportation research part B.
Some points of concern are given below:
p7, l228. A curve is by definition a 1d topological object. It is the ambient space that is is two-dimensional. The definition given here is not the usual one as a polygon may have intersecting segments.
p7, l231. The right term for "without holes" is "simply connected".
p7, l249. This is not a proposition, but an assumption. A proposition has to be proved or a reference to the proof given.
p8, l257. Same as above.
p8, l260. Lemma is useless.
p8, l272. Finding the convex hull of a finite set of points is a well-known problem in algorithmic geometry. A reference to one of the algorithms is enough here.
p9, l300. Polygons on manifolds cannot be defined using straight lines. The natural extension is a piecewise geodesic loop. Be careful that on a compact manifold like the sphere, the notion of enclosed region is not always well defined (think of a torus and a sphere).
Author Response
This paper has some interested, but is, in my opinion, out of the scope of the journal since no new mathematical result is presented. I suggest to resubmit to an application-oriented journal like transportation research part B.
We were thinking about transportation research part B journal, but we chose this journal due to the nature of the article. Of course, we are also preparing other publications that have a classic mathematical character.
Some points of concern are given below:
p7, l228. A curve is by definition a 1d topological object. It is the ambient space that is is two-dimensional. The definition given here is not the usual one as a polygon may have intersecting segments. Instead of word „curve”, which term should we use? Polyline ? The topology is not our strength, but we know the peano curve.
p7, l231. The right term for "without holes" is "simply connected".
- Corrected. The same term is simply connected.
p7, l249. This is not a proposition, but an assumption. A proposition has to be proved or a reference to the proof given.
- Corrected. Instead of the proposition, we used assumption.
p8, l257. Same as above. OK. Corrected.
p8, l260. Lemma is useless. (Lemma was omitted)
p8, l272. Finding the convex hull of a finite set of points is a well-known problem in algorithmic geometry. A reference to one of the algorithms is enough here.
- The reference has been added.
p9, l300. Polygons on manifolds cannot be defined using straight lines. The natural extension is a piecewise geodesic loop. Be careful that on a compact manifold like the sphere, the notion of enclosed region is not always well defined (think of a torus and a sphere). We are considering adding axioms to 3D, but it is not that simple. Thank you for warning.
The authors would like to thank the anonymous referees for their helpful comments and constructive advices to improve the article.
Round 2
Reviewer 1 Report
Thank you for improving the manuscript. I have no further comments.
Reviewer 2 Report
The paper has improved over the first version, and is of publishable quality. However, I still think that it is not within the scope of the journal. I thus emit an "accept" advice, but inform the editor about my concern and let him make the final decision.
