{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,30]],"date-time":"2025-12-30T05:26:35Z","timestamp":1767072395166,"version":"3.48.0"},"reference-count":30,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,12,25]],"date-time":"2025-12-25T00:00:00Z","timestamp":1766620800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Research Funds of Sichuan Province Engineering Technology Research Center of BIM+ Application and Intelligent Visualization","award":["BIM-2024(2)-Z-01"],"award-info":[{"award-number":["BIM-2024(2)-Z-01"]}]},{"name":"Key Laboratory of Digitalization and Dissemination of Sichuan-Yunnan Ethnic Cultures","award":["CDSZCBZD202501"],"award-info":[{"award-number":["CDSZCBZD202501"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Embankments are extensively applied in civil engineering and exist naturally, characterized by a constant slope ensuring stable material stacking. Despite their ubiquity, research on embankment surface geometric modeling remains limited. Three novel construction methods for embankment surfaces are proposed: the envelope of a one-parameter family of cones, the envelope of constant-slope tangent planes, and the geometric limit. Subsequently, the properties of embankment surfaces and their geometric interpretation are elaborated. Specific examples are provided to illustrate that these methods offer convenient and feasible alternatives for engineering geometric design related to these surfaces.<\/jats:p>","DOI":"10.3390\/axioms15010013","type":"journal-article","created":{"date-parts":[[2025,12,25]],"date-time":"2025-12-25T23:52:27Z","timestamp":1766706747000},"page":"13","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Novel Geometric Construction Methods and Application for Embankment Surface"],"prefix":"10.3390","volume":"15","author":[{"given":"Fangxiao","family":"Zhou","sequence":"first","affiliation":[{"name":"School of Mathematics and Computer Science, Panzhihua University, Panzhihua 617000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhiming","family":"Zou","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science, Panzhihua University, Panzhihua 617000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gang","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science, Panzhihua University, Panzhihua 617000, China"},{"name":"Sichuan Province Engineering Technology Research Center of BIM+ Application and Intelligent Visualization, Panzhihua 617000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jingshun","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science, Panzhihua University, Panzhihua 617000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,12,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3818","DOI":"10.1007\/s11629-025-9652-x","article-title":"Mechanical properties and failure mechanisms of coarse carbonaceous mudstone particles as highway embankment filler under coupled temperature-moisture conditions","volume":"22","author":"Zeng","year":"2025","journal-title":"J. Mt. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1134\/S1069351325700272","article-title":"New Possibilities of Combining Calculations and In-Situ Seismic Data in Railway Embankment State Survey","volume":"61","author":"Antonovskaya","year":"2025","journal-title":"Izv. Phys. Solid Earth"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"67","DOI":"10.3997\/1873-0604.2010058","article-title":"Zero-offset profiling using frequency cross-hole radar in a layered embankment test site: Antenna design, simulation and experimental results","volume":"9","author":"Rejiba","year":"2011","journal-title":"Surf. Geophys."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Glaeser, G. (2020). Geometry and Its Applications in Arts, Nature and Technology, Springer Nature.","DOI":"10.1007\/978-3-030-61398-3"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"04025309","DOI":"10.1061\/IJGNAI.GMENG-11370","article-title":"Probabilistic Stability Frameworks of Stone Column\u2013Supported Embankments Considering Failure Mechanisms","volume":"26","author":"Zheng","year":"2026","journal-title":"Int. J. Geomech."},{"key":"ref_6","first-page":"463","article-title":"NUMERICAL ANALYSIS OF SLOPE STABILITY AND SETTLEMENT OF LOCAL ROAD EMBANKMENT CONSTRUCTED WITH CLAY AND MUNICIPAL SOLID WASTE","volume":"3","author":"Perwaiz","year":"2025","journal-title":"Annu. Methodol. Arch. Res. Rev."},{"key":"ref_7","first-page":"726","article-title":"Parametric Representation of Embankments","volume":"75","year":"1995","journal-title":"ZAMM-J. Appl. Math. Mech.\/Z. Angew. Math. Mech."},{"key":"ref_8","unstructured":"Hartmann, E. (2003). Geometry and Algorithms for Computer Aided Design, Darmstadt University of Technology."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Pottmann, H., Wallner, J., Pottmann, H., and Wallner, J. (2001). Computational Line Geometry, Springer.","DOI":"10.1007\/978-3-642-04018-4"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"617","DOI":"10.26713\/cma.v10i3.916","article-title":"Embankment Surfaces in Euclidean 3-space and their visualizations","volume":"10","author":"Kazan","year":"2019","journal-title":"Commun. Math. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"20979","DOI":"10.3934\/math.2025937","article-title":"Embankment surfaces with the Darboux frame in Euclidean 3-space","volume":"10","year":"2025","journal-title":"AIMS Math."},{"key":"ref_12","unstructured":"Pottmann, H., and Peternell, M. (2000, January 3\u20136). Envelopes-computational theory and applications. Proceedings of the Spring Conference on Computer Graphics, Budmerice, Slovakia."},{"key":"ref_13","first-page":"345","article-title":"Envelopes\u2013notion and definiteness","volume":"48","author":"Kock","year":"2007","journal-title":"Beitr\u00e4ge Algebra Geom."},{"key":"ref_14","first-page":"101","article-title":"Exploration of envelopes of parameterized families of surfaces in a technology-rich environment","volume":"17","year":"2023","journal-title":"Electron. J. Math. Technol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1016\/j.cam.2005.08.002","article-title":"Analytic and algebraic properties of canal surfaces","volume":"195","author":"Xu","year":"2006","journal-title":"J. Comput. Appl. Math."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Landsmann, G., Schicho, J., Winkler, F., and Hillgarter, E. (2000, January 7\u20139). Symbolic parametrization of pipe and canal surfaces. Proceedings of the 2000 International Symposium on Symbolic and Algebraic Computation, St. Andrews, Scotland, UK.","DOI":"10.1145\/345542.345631"},{"key":"ref_17","first-page":"206","article-title":"A STEAM approach to canal surfaces","volume":"17","year":"2023","journal-title":"Electron. J. Technol. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1080\/16864360.2005.10738324","article-title":"Swept volumes","volume":"2","author":"Peternell","year":"2005","journal-title":"Comput.-Aided Des. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1177\/027836499000900507","article-title":"Geometric Representation of Swept Volumes with Application to Polyhedral Objects","volume":"9","author":"Weld","year":"1990","journal-title":"Int. J. Robot. Res."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1016\/S0010-4485(96)00101-7","article-title":"The sweep-envelope differential equation algorithm and its application to NC machining verification","volume":"29","author":"Blackmore","year":"1997","journal-title":"Comput.-Aided Des."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1016\/S0010-4485(96)00097-8","article-title":"Geometric representation of the swept volume using Jacobian rank-deficiency conditions","volume":"29","author":"Yeh","year":"1997","journal-title":"Comput.-Aided Des."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1063\/1.3459064","article-title":"From golden spirals to constant slope surfaces","volume":"51","author":"Munteanu","year":"2010","journal-title":"J. Math. Phys."},{"key":"ref_23","first-page":"79","article-title":"Certain constant angle surfaces constructed on curves","volume":"4","author":"Nistor","year":"2011","journal-title":"Int. Electron. J. Geom."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"6341","DOI":"10.3934\/math.2020408","article-title":"Construction of the spacelike constant angle surface family in Minkowski 3-space","volume":"5","author":"Li","year":"2020","journal-title":"AIMS Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"989","DOI":"10.1007\/s11464-023-0009-x","article-title":"Constant Angle Ruled Surfaces Associated with Slant Ruled Surfaces","volume":"19","author":"Kaya","year":"2024","journal-title":"Front. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"8364","DOI":"10.3934\/math.2025385","article-title":"Novel theorems on constant angle ruled surfaces with Sasai\u2019s interpretation","volume":"10","author":"Eren","year":"2025","journal-title":"AIMS Math."},{"key":"ref_27","unstructured":"Marschner, S., and Shirley, P. (2016). Fundamentals of Computer Graphics, CRC Press."},{"key":"ref_28","unstructured":"Do Carmo, M.P. (2016). Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition, Courier Dover Publications."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Zhou, F. (2025). Adaptive Terrain Modeling for Side-Slope Surfaces. Symmetry, 17.","DOI":"10.3390\/sym17020191"},{"key":"ref_30","unstructured":"Zhang, J., and Zhang, T. (2015). Road Survey and Design, Tongji University Press. [3rd ed.]."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/1\/13\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,30]],"date-time":"2025-12-30T05:25:02Z","timestamp":1767072302000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/1\/13"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,25]]},"references-count":30,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1]]}},"alternative-id":["axioms15010013"],"URL":"https:\/\/doi.org\/10.3390\/axioms15010013","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,12,25]]}}}