{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:21:10Z","timestamp":1772252470559,"version":"3.50.1"},"reference-count":16,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,5,24]],"date-time":"2022-05-24T00:00:00Z","timestamp":1653350400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"European Regional Development Fund","award":["313011AUP1"],"award-info":[{"award-number":["313011AUP1"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Free Route Airspace (FRA), a new concept implemented across European airspace, is designed to eliminate the adverse effects of air traffic, reduce fuel consumption, simplify and expand flight planning. In our research, we model FRA system by using graph theory and we use networks and convex analysis to study it. We work with 2-dimensional space defined by latitude a longitude. We know that the basic mathematical properties of a scientific discipline can be expressed using mathematical axioms. Therefore, in our work, we sought to answer the question, what are the basic mathematical properties of airspace? We found a mathematical object that can describe airspace in general. Using this object, we uttered an axiom describing airspace. Our axiom is given by means of practical, significant entry and entry-exit points in the airspace.<\/jats:p>","DOI":"10.3390\/axioms11060244","type":"journal-article","created":{"date-parts":[[2022,5,24]],"date-time":"2022-05-24T22:04:06Z","timestamp":1653429846000},"page":"244","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Spatially Bounded Airspace Axiom"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6986-5753","authenticated-orcid":false,"given":"Peter","family":"Szab\u00f3","sequence":"first","affiliation":[{"name":"Department of Aviation Technical Studies, Faculty of Aeronautics, Technical University of Ko\u0161ice, 040 01 Ko\u0161ice, Slovakia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8967-0582","authenticated-orcid":false,"given":"Miroslava","family":"Ferencov\u00e1","sequence":"additional","affiliation":[{"name":"Department of Aviation Technical Studies, Faculty of Aeronautics, Technical University of Ko\u0161ice, 040 01 Ko\u0161ice, Slovakia"},{"name":"Eurocontrol, Rue de la Fus\u00e9e 96, 1130 Brussels, Belgium"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0349-1864","authenticated-orcid":false,"given":"Monika","family":"Bli\u0161\u0165anov\u00e1","sequence":"additional","affiliation":[{"name":"Department of Air Transport Management, Faculty of Aeronautics, Technical University of Ko\u0161ice, 040 01 Ko\u0161ice, Slovakia"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Szab\u00f3, P., Ferencov\u00e1, M., and \u017delezn\u00edk, V. (2022). Cloud Computing in Free Route Airspace Research. Algorithms, 15.","DOI":"10.3390\/a15040123"},{"key":"ref_2","first-page":"5","article-title":"An Airspace Model Aplicable for Automatic Flight Route Planning inside Free Route Airspace","volume":"35","author":"Drupka","year":"2018","journal-title":"Zesz. Nauk. Politech. Rzesz."},{"key":"ref_3","first-page":"139","article-title":"Air Space Routing and Flights Planning: A Problem Statement and Discussion Of approaches to Solution","volume":"2","author":"Alieksieiev","year":"2018","journal-title":"Math. Model."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1488","DOI":"10.1108\/AEAT-11-2020-0272","article-title":"Regular graph-based free route flight planning approach","volume":"93","author":"Samolej","year":"2021","journal-title":"Aircr. Eng. Aerosp. Technol."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Lawrence, J.L. (2019). Celestial Calculations\u2014A Gentle Introduction to Computational Astronomy, The MIT Press.","DOI":"10.7551\/mitpress\/11401.001.0001"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Smart, W.M. (1977). Textbook on Spherical Astronomy, Cambridge University Press. [6th ed.].","DOI":"10.1017\/CBO9781139167574"},{"key":"ref_7","unstructured":"(2022, April 14). Digital Imaging and Remote Sensing Image Generation. Coordinate Systems. Available online: http:\/\/dirsig.cis.rit.edu\/docs\/new\/coordinates.html."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Van Sickle, J. (2017). Basic GIS Coordinates, CRC Press. Taylor & Francis Group.","DOI":"10.1201\/9781315154671"},{"key":"ref_9","unstructured":"FAA (2022, May 15). Federal Aviation Administration. Airspace. Available online: https:\/\/tinyurl.cc\/airspace-faa."},{"key":"ref_10","unstructured":"IVAO (2022, April 14). International Virtual Aviation Organisation. Available online: https:\/\/tinyurl.com\/fl-IVAO."},{"key":"ref_11","unstructured":"Ferencov\u00e1, M., and Szab\u00f3, P. (2022, April 14). FRA Research Cloud. Available online: https:\/\/tinyurl.com\/fra-research-cloud\/."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Ferencov\u00e1, M., F\u0151z\u0151, L., and Szab\u00f3, P. (2021, January 25\u201326). Free Route Airspace\u2014Graphs, Networks, Algorithms. Proceedings of the 2021 New Trends in Aviation Development (NTAD), Ko\u0161ice, Slovakia.","DOI":"10.1109\/NTAD54074.2021.9746504"},{"key":"ref_13","unstructured":"Newman, M.E.J. (2010). Networks: An introduction, Oxford University Press Inc."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Sallan, J.M., and Lordan, O. (2019). Air Route Networks through Complex Networks Theory, Elsevier Science Publishing Co. Inc.","DOI":"10.1016\/B978-0-12-812665-3.00010-4"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Ferencov\u00e1, M., \u0160ulej, R., and Szab\u00f3, P. (2018, January 30\u201331). The follow-up on the Free Route Airspace Implementation in Slovakia with the proposed optimization model. Proceedings of the 2018 XIII International Scientific Conference\u2014New Trends in Aviation Development (NTAD), Ko\u0161ice, Slovakia.","DOI":"10.1109\/NTAD.2018.8551689"},{"key":"ref_16","first-page":"469","article-title":"The quickhull algorithm for convex hulls","volume":"1996 22","author":"Bradford","year":"1996","journal-title":"ACM Trans. Math. Softw."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/6\/244\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:17:31Z","timestamp":1760138251000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/6\/244"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,24]]},"references-count":16,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["axioms11060244"],"URL":"https:\/\/doi.org\/10.3390\/axioms11060244","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints202203.0318.v2","asserted-by":"object"},{"id-type":"doi","id":"10.20944\/preprints202203.0318.v1","asserted-by":"object"}]},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,5,24]]}}}