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# 2014, № 10 (171)

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.

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