Three-Dimensional Design Theory And Method For Highway Geometric Alignment Under Invariant Constraints Of Spatial Curve

Highway alignment design is an important part of highway construction, its design quality not only relates to the highway construction quality and costs, but also directly influences safety operation and driving comfort. Due to computer performance, mathematical models and information technology limitations, current highway centerlines are usually designed and displayed by using the traditional two-dimensional(2-D) separated way by superimposing the horizontal and vertical projections. This artificial separated and combined design practice ignores the coordination and the interaction between horizontal and vertical geometric elements, which makes the final quality mainly depend on the designers’ experience and ability. Consequently, some potential design faults and deficiencies are brought into operation, thus leading to traffic accidents easily.In order to define the deficiencies of traditional highway horizontal and vertical separated design practice and improve design quality and safety, new indexes, curvature and torsion, are adopted to describe the geometric properties of spatial curves and to design three-dimensional(3-D) alignment on the basis of highway 3-D nature and the differential geometric principles. The curvature and torsion are two basic intrinsic invariants, which can entirely determine the geometric shape of the spatial curve. And the relationship between current 2-D design indexes and new 3-D alignment indexes, curvature and torsion, is developed to analysis the spatial geometric properties of the combined alignments. On the basis of the above research, a new design methodology and framework for 3-D highway geometric design are proposed, in which, the 3-D alignment is created by functions of curvature and torsion with arc-length as a variable, and using geometrical continuity requirement, stability of vehicle motion, and driving comfort as engineering constraints. The constraints of 3-D alignment indexes and the development of 3-D alignment model are discussed in detail in this paper. The chief research contents and achievements can be summarized as follows:(1)Continuity is a measure of the smoothness of the curve, and a basic design requirement of highway alignment design. Thus, geometrical continuity(denoted as nG) was proposed here to describe and dissect the continuity of the combined alignments objectively, and the geometrical continuity at the connection points is usually determined by the continuity of the tangent vector, curvature, torsion and the Frenet frame. Two indexes, curvature and torsion differences, were established to quantify the curvature and torsion mutations between the consecutive combined sections. Using the regression analysis of the SPSS software, the relationship model of the curvature and torsion differences with accident rates on consecutive segments of expressways and two-lane highways, and the corresponding evaluation criteria were developed. The results show that superposition of horizontal and vertical alignments can’t guarantee geometrical continuity in 3-D space, and the geometrical continuity will degradation in different degrees when 2-D design elements change. The mutation amounts of curvature and torsion have a strong relationship with traffic safety, in which curvature continuity plays a leading role, while the torsion has little influence. Compared with operating speed consistency, new 3-D consistency measures and criteria can identify these locations with improper horizontal and vertical combination, and higher accident rates.(2)The vehicle was treated as a rigid body and a new spatial motion model was established from the microscopic view with the curvature and torsion as the control variables base on the vehicle coordination(Frenet Frame). To ensure vehicle stability and driving comfort, it is necessary to control and restrict the ranges of curvature and torsion for 3-D alignment design. Then, with the engineering constrains: geometric continuity, vehicle stability, and driving comfort, the constraining model was established. Finally, design values of 3-D alignment indexes, curvature and torsion, under different design speeds were presented on the basis of the current design guidelines, experiences and the constraining model. It is found that the setting of curvature plays a leading control role in the spatial motion, while the torsion only affects the vertical jerk. The 3-D alignment of highway must keep the second-order geometrical continuity(2G) for safety and comfort.(3)The paper discussed the types of different alignment combinations of horizontal and vertical alignment elements and the corresponding curvature and torsion, and found that the combined alignments consisted of different spatial curves, including spatial tangent, helix, and approximate helix, which formed the main cause of geometrical continuity degradation. Through the contrastive analysis of 3-D splines, combined curves and Pythagorean-Hodograph(PH) curves with the geometrical characteristics, the calculation requirements and the information transformation, PH quintic was finally selected to design and construct 3-D highway alignment. The PH curve was represented in fifth-order Bezier polynomial form, thus the problem of the 3-D PH curve can be simply convertible to the solution problem of the polynomial coefficients. And the path length, curvature and torsion of the curve can be described with the polynomial and calculated precisely without any approximation. The detailed process of the PH curve calculation can be found in this paper.(4)For 3-D highway alignment design, first, select a series of main control points between the design starting point and finishing point and determine the direction of the tangent vectors at these main control points. Then, interpolate the minor control points along the tangent vectors. The initial shape, curvature and torsion of the designed alignment can be determined according to the PH curve algorithm. To satisfy the design constraints, including curvaturecontinuous, the maximum curvature, curvature change rate and torsion bounds, the distances between the main and minor control points need to be adjusted constantly to arrive at satisfactory values of curvature and torsion. These adjustable spacing of the control points provided a convenient way for highway designers to control geometric shape and 3-D indexes. This will ensure the final designed 3-D PH curve is a curvature-continuous, safe and travellable path for the vehicle, besides, it has a unified mathematical expression.The aforementioned research findings will provide new theoretical reference for the improvement of traditional separated horizontal and vertical alignments design practice, and highway safety level, and establish sound theoretical basis for the design system of true 3-D highway alignment in future.