Amiridis K., Psarianos B. APPLICATION OF DIFFERENTIAL GEOMETRY TO SOLVE THE HIGHWAY ALIGNMENT LOCATION PROBLEM The objective of this paper is to develop a mathematical methodology that allows the design of the 3-d alignment of a road directly in space thus overcoming the conventional approach of studying the three-dimensional road configuration separately and sequentially in its horizontal and vertical alignment. Initially, the proposed methodology is based on the fact that the road centerline is actually a three-dimensional curve, whereas the road as a whole is in turn a three-dimensional surface. In other words, the geometric road alignment is faced as the mathematical definition of a three-dimensional curve (road centerline) and a three-dimensional surface (road surface). There are many ways to represent and study a three-dimensional curve. However, in the suggested methodology the road centerline (3-d curve) is selected to be approached as a three-dimensional B-spline curve on the grounds that it has certain significant properties regarding highway engineering which will be described in the discussion that follows. Indeed, the road surface (3-d surface) is selected to be approached as a three-dimensional B-spline surface. The field of mathematics which is most relevant in studying curves and surfaces is differential geometry. This is why the mathematical background used in the methodology is directly associated with concepts incorporated in differential geometry which are expressed and linked to terms of highway engineering. The computational part of this approached is attained by expressing the functions in code through the programming language offered by the software Mathematica, whereas the whole process is demonstrated as a case study.Key words: 3-d highway design, B-Spline curves, B-Spline surfaces, pseudogeodesic curvature, pseudonormal curvature, pseudonormal unit vector, Procrustes transformation.
1. Gray, A. (1998), Modern Differential Geometry of Curves and Surfaces with MATHEMATICA, CRC Press, New York.
2. Brauer, P. (1942), Zur raumlichen Theorie der Strabe. Ingenieur Archiv Band XIII. [In German]
3. Lorenz, H. (1943), Raumliche Gestaltung von Raumkurven. Die Strabe Jahrgang 28, 1943. [In German]
4. Freising, F. (1949), Folgerungen aus der Untersuchung des perspektivischen Bildes von Linienelementen der Strabe. Technische Hochschule Stuttgart, Dissertation. [In German]
5. Scheck,H.J. (1973), Optimierungsberechnungen und Sensitiitatsanalyse als Hilfsmittel bei der Entwurfsbearbeitun von Straben. Strabenbau und Strabenverkehrstechnik, H. 153. [In German]
6. Borgmann, H. (1976), Zur Trassierung mit Hilfe von Biege-(Spline-) Linien statisch bestimmt gelagerter Elementarstabe. Zeitschrift fur Vermessungstechnik 101, 3. [In German]
7. Psarianos, B. (1982), Ein Beitrag zur Entwicklung des raumlichen Trassierungsprozesses von Verkehrswegen und insbesondere von Straben. Universitat Hannover. [In German]
8. Kuhn, W. (2002). Neuartige Modellvorstellungen und Verfahren — Ein Beitrag zur Weiterentwicklung der Entwurfsmethodik fur Strasen. Advanced model concepts and methods — A contribution to the development of design methodology for roads. Technische Universitat Dresden, Habilitationsschrift. [In German]
9. Kuhn, W. (2012), 3D Methodology for the design process of safe rural highways, TRB, Washington DC.
10. Zuo, W., L. Jian-Pei, and Teng-Feng, G. (2007), Calculating Method and Appraising Technique of Highway 3-D View Sight Distance, Journal of Chang'an University (Natural Science Edition), 27(6), 44-48. [In Chinese]
11. Hao, P., S. Zhan-feng, and Teng-Feng, G. (2007), Integration and Visualization of Highway Alignment Design, J. Cent. South Univ. (Science and Technology), 35(5), 830-835. [In Chinese]
12. Makanae, K. (2000), Three-dimensional highway alignment design systems using stereoscopy of aerial photographs and computer graphics. Proceedings on the Eighth International Conference on: Computing in Civil and Building Engineering, Stanford CA, August 2000.
13. Makanae, K. (1997), Development of a 3D Highway Modeler using a Driving Game Interface, proceedings of the 7th International Conference on Construction Applications of Virtual Reality: October 22-23, 2007.
14. Makanae, K. (2002) Functionalization of Highway Alignment for Computer-Based Design, proceedings of the Ninth International Conference on Computing in Civil and Building Engineering, April 3-5, 2002, Taipei, Taiwan.
15. Karri, G. & Jha, M.K. (2007), A New Method for 3–Dimensional Roadway Design Using Visualization Techniques, in Urban Transport XIII (Urban Transport 2007), C.A. Brebbia et al. (eds.), WIT Press, Southampton, U.K.
16. Kim, D. G. & Lovell, D.J. (2010), A Procedure for 3-D Sight Distance Evaluation using Thin Plate Splines, proceedings of the 4th International Symposium on Highway Geometric Design, Valencia, Spain, June 2-5, 2010.
17. Jha, M., Karri, G. & Kuhn, W. (November 2010), A New 3-Dimensional Highway Design Methodology for Sight Distance Measurement, TRB, Washington DC.
18. Kuhn, W. & Jha, M. (January 2012), Methodology for checking shortcomings in the three — dimensional alignment. Washington DC, TRB.
19. Karri, K., Maji, A. & Jha, M. (January 2012), Optimizing geometric elements of a 3-dimensional alignment in a single stage highway design process. Washington, TRB.
20. Prautzsch, H., Boehm, W. & Paluszny, M. (March 2002), Bezier and B-Spline Techniques, Springer, Berlin
21. Lipschutz, M. (1982), Differential Geometry, ESPI, Athens. [In Greek]
22. Polyrakis, I. (2008), Elementary Differential Geometry, National Technical University of Athens, Athens [In Greek]
23. Markatis, S. (1989), Differential Geometry, National Technical University of Athens, Athens [In Greek]
24. Karras, G. (1998), Linear Transformations of Coordinates in Photogrammetry, National Technical University of Athens, Athens. [In Greek]
25. Kraus, K. (2010), Photogrammetry, Technical Chamber of Greece (translated into Greek), Athens. [In Greek]
26. Patias, P. (1991), Introduction to Photogrammetry, Ziti, Thessaloniki. [In Greek]
27. Awange, J., Grafarend, E., Palancz, B. & Zaletnyik, P. (2010), Algebraic Geodesy and Geoinformatics, Springer, New York.
28. Kadianakis, N. & Karanasios, S. (2003), Linear Algebra Analytic Geometry and Applications, National Technical University of Athens, Athens. [In Greek]
29. Kadianakis, N. (2013), Differential Geometry of Curves and Surfaces, National Technical University of Athens, Athens [In Greek]
30. Amiridis, K. (2014), Three Dimensional Road Alignment by Applying Differential Geometry and B-spline Interpolation Curves, M.Sc. Thesis, National Technical University of Athens, Athens. [In Greek]
31. AASHTO (2011), A Policy on Geometric Design of Highways and Streets, 6th Edition, Washington, D.C.
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Authors: Rsarianos V., Amiridis K.
|Sergey Aleksandrovich |