S-SHAPED TRANSITION CURVES AS AN ELEMENT OF REVERSE CURVES IN ROAD DESIGN

A road designing involves horizontal and vertical alignment. The horizontal geometry is formed by straight and curvilinear sections that are traditionally formed using circular and transition curves (mainly the clothoid). Different geometric systems that are designed using circular and transition curves are between others circular curves with symmetrical or unsymmetrical clothoids, combined curves, oval curves and reverse curves. Designing these systems is quite complex. Therefore, so-called S-shaped transition curves are an alternative to traditional approaches. These curves are known from literature and are modern geometric tools for the shaping of reverse curves. The paper analyses the basic geometric properties of these curves as well as compare to the geometry of the appropriate geometric systems, which are formed with clothoid or using S-shaped transition curves. In addition, a procedure for designing reverse curves using S-shaped transition curves was proposed. Another research topic was the comparison of the analysed reverse curves (created using polynomial transition curves) with traditional curves (created using the clothoid). The results of the


Introduction
The determining of horizontal alignment plays a key role in road design. The horizontal alignment is usually set using straight lines, circular curves ad transition curves. The most popular transition curve is the spiral curve (also known as the clothoid), whose curvature changes linearly with the increase of the length, making it very easy to shape the curvature (Brockenbrough, 2009). Apart from the spiral curve, other curves are also included. Such a possibility is permitted by the existing regulations (Ministra Transportu i Gospodarki Morskiej, 1999) and road design guidelines (Generalna Dyrekcja Dróg Publicznych, 1995a, 1995b, 1995c. Between various curves that potentially expand the range of design tools used for the geometric design of roads, so-called polynomial transition curves deserve special attention. These curves provide a flexible tool to facilitate the geometric alignment of routes to terrain constraints. Various curves of this type have been presented in the works (Grabowski, 1984;Kobryń, 2016aKobryń, , 2017a. In works (Kobryń, 2016a(Kobryń, , 2017a, it is proposed to divide the polynomial transition curves into several categories: • classical transition curves, • quasi-transition curves, • general transition curves, • S-shaped transition curves, • oval transition curves, • universal transition curves. The categories as mentioned above of transition curves have excellent application potential and are useful in the design of curvilinear sections of road routes, both in the situational plan and in the longitudinal profile. The subjects of the current research are various aspects of application nature. As a result, for example, Kobryń (2017b) describes the appropriate proposals and methodology for dealing with the use of polynomial transition curves in the classical sense. In turn, Kobryń (2016b) presents a methodology and various aspects of using these transition curves for the design of vertical curves.
As is known (American Association of State…, 2018;Brockenbrough, 2009;Easa, 2003;Garber & Hoel, 2014;Kühn, 2013;Lamm, Psarianos, & Mailaender, 1999;Rogers & Enright, 2016;Wolhuter, 2015), different geometric systems are used in the horizontal alignment, e.g. compound, reverse and combined curves. Traditionally shaping this type of geometry requires generally tedious design works, which involve the need for a smooth connecting of several geometric elements (circular arcs and transition curves). The advantage of polynomial transition curves is the description of the entire curvilinear transition between two straight lines using only one equation. Therefore, in the opinion of authors, the use of an appropriate category of polynomial transition curves are a noteworthy alternative to the traditional approach.
Analysis of shaping the geometry of reverse curves will be considered in this article. Such problems were analysed in literature relatively rarely. Generally, reverse curves formed by two clothoids are described in handbooks. Other curves that have been analysed as elements of reverse curves are circular arcs (Easa, 1994) or quantic polynomials (Shebl, 2015). Authors of these papers express the opinion that reverse curves give the possibility of shaping the using so-called S-shaped transition curves that have been presented in Kobryń (2016aKobryń ( , 2017a works. These curves are the subject of further discussion in this article.

Analysed solutions of S-shaped transition curves
Polynomial solutions of S-shaped transition curves have been defined by Kobryń (2016aKobryń ( , 2017a as curves whose curvature varies continuously from zero at the starting point P to zero at the endpoint K. Within these curves, there is one inflexion point S (where the value of the curvature is also zero). The inflexion point divides the S-shaped transition curve into two parts, each with one extreme of curvature ( Figure 1). Various solutions of polynomial transition curves, presented in Kobryń (2016aKobryń ( , 2017a works, are based on the appropriate polynomial function in general form: Appropriate boundary conditions (presented in Kobryń (2016aKobryń ( , 2017a works) for the values of this function and its derivativesy ′ y , ′′ y and ′′′ y at points, P and K are taken into account in the function (1). Consideration of these conditions is required to define the desired distribution of curvature within the curve. In agreement to Kobryń (2016aKobryń ( , 2017a, appropriate Equations for two families of S-shaped transition curves result from Eq. (1) for n = 5 and n = 7.
In agreement to Kobryń (2016aKobryń ( , 2017a, the Eq. (1) of the S-shaped transition curves that were obtained using Eq. (1) for n = 5, has the following form where M t t t and in agreement to Figure 1: x K − abscissa of the endpoint K in the local coordinates system; α − angle of chord inclination connecting points P and K ; uP, uK − angles of tangent inclination respectively at points P and K .
The variable x in Eq.
(2) is replaced by the variable t x x K with the values t ∈ 0 1 ; . As a result, the abscissa xK of the point K plays the role of a size, which scales the curve (the design parameter, which determines the span of the curve). Other designing parameters for these curves are values tanα, tanuP, and tanuK.  (Kobryń, 2016a) In the case of the solution (2), obtaining of the curvature distribution, which is appropriate for S-shaped transition curves, requires to meet of following conditions (in agreement to Kobryń (2017a)): allow obtaining a correct distribution of curvature within the curve (Kobryń, 2011).
The second equation of S-shaped transition curves obtained on the basis Eq. (1) for n = 7 has the following form: where G t t t t The meaning of symbols, which were used in Eq. (5) is analogous to Eq. (2).
As in the case of curves (2), it is necessary to meet the appropriate conditions to obtain the curvature distribution suitable for S-shaped transition curves. The first condition is analogous to (3), while the second condition is the equivalent of condition (4) that has the form (in agreement to Kobryń (2017a)):   (Kobryń, 2011).
In practical applications related to the layout of reverse curves, it is justified to write Equations of curves (2) and (5) in a more straightforward form that is possible after adopting tanα = 0 ( Figure 2). The Eq. (2) takes the form: In turn, the Eq. (5) takes the form In light of design conditions, it is essential below the permissible values of curvature. Due to the strictly defined distribution of curvature, controlling this condition requires determining the position and value of the curvature extremes that for any function y = f(x) is described by Eq. (9) where ′ y , ′′ y and ′′′ y are derivatives of the corresponding function (2) or (5).

Basic geometric characteristics of the S-shaped transition curves
Taking into account the Eq. (9), follows that the condition of the existence of extremes of curvature (Eq. (10)) is replaced − with some simplification − by a necessary condition in the form ccc y 0. This condition results from the fact that cc cc t t y t y t k t M . The necessary condition ccc y 0 has the form: • for curves (7 In turn, the necessary condition of the existence of the inflexion point cc y 0 has the form (in agreement to Kobryń (2016a)):  Some specific geometric properties of these curves are relevant considering the usefulness of the S-shaped transition curves for forming the reverse curves. Namely, with the increase of inclinations tan u P and tan u K , there are relatively longer sections with a small curvature (bolded in Figure 3) in the vicinity of points P, K and S.
It is, therefore, necessary to determine the maximum values of tangent inclinations (tan u P and tan u K ) at points P and K, where changes of curvature in those sections of the curve do not exceed certain limit values that were resulting from standard guidelines, e.g. Generalna Dyrekcja Dróg Publicznych (1995a, 1995b, 1995c). As a reference curve, the spiral curve is used. Its Equation has the form (Kobryń, 2017a): where t l L = ; l − natural parameter; L − total length of the spiral curve; R − designed minimum radius of the curvature at the point K, which is for t = 1.
For example, will be compared the radius of curvature at two points, whose position is described by values t and 't, respectively. Assuming ' t 0 05 . is obtained: . for t = 0 05 . ; • r r t t t ' 1 5 . for t = 0 10 . ; • …;   resulting from these calculations are summarised in Table 2. Presented results are illustrative. Therefore, they were here limited to the vicinity of points of P and K, omitting results regarding neighbourhood of the inflexion point S.
In the light of presented values, it is reasonable to use in the designing practice of curves (7) and (8) under certain conditions: • in the case of curves (7) -if angles of tangent inclinations are below 40 °; • in the case of curves (8) -if angles of tangent inclinations are below 55 °. It is possible, that − in design practice -it is necessary to have, for example, appropriate nomograms that allow rapid evaluation of the fulfilment of the above requirements by S-shaped transition curves. Piotr Stachera

Designing of reverse curves using S-shaped transition curves
In practical applications, which are associated with laying out of reverse curves using S-shaped transition curves, generally there are two design options: • 1 st option − a connecting of two points P X Y P P , and K X Y K K , with a given position in the superordinate coordinates system (in that a whole route is designed) as well with given tangent directions at P and K ( Figure 4); • 2 nd option − a positioning of points ′ P and ′ K on tangents with given directions in such a way as not to exceed permissibly, i.e. resulting from proper guidelines, values of curvature within the curve ( Figure 5). In addition -regardless of the above design options -it is important that the parameter of the transition (A 2 = RL) meets certain conditions that are defined in the relevant regulations.
In agreement to Figures 4 and 5, the curvilinear transition created by the S-shaped transition curves in both options will be designed in the local coordinate system of these curves (xy). A defining this transition in connection with the entire route is possible by appropriate coordinate transformation to the superordinated coordinate system.
In the first option, the starting point is a calculation of values tan u P and tan u K (in the local coordinate system as in Figure 4) based on present positions of points P and K and given tangent directions in these points. Then, based on the Eq. (10), values tM are calculated. These values However, it is to be expected that the radii calculated based on Eqs (11) or (12) would exceed the limit values. In this case, correction of positions of the start and endpoint of the curve will be required in agreement to a procedure described below.
In the second option, it is essential to determine if a minimum value of the abscissa xK of the point K In the local coordinates system of the S-shaped transition curve. It is important to ensure that the permissible (i.e. resulting from suitable guidelines) values of curvature at points M P and M K are not exceeded. As in the first option, the values of inclination tan u P sand tan u K must be calculated first based on positions of points P and K and tangent directions given in these points. Subsequently, the position of points M

S-Shaped Transition Curves as an Element of Reverse Curves in Road Design
• for curves (8) Additionally, a distance between points P X Y P P , and K X Y K K , in the super-ordinated coordinates system is calculated: If d x PK K ≥ , it means that no point of the S-shaped transition curve will exceed the permissible value of the curvature, i.e. value resulting from proper guidelines. Otherwise, points P and K on tangents are determined so that their connection using the S-shaped transition curve with the permissible values of curvature will be obtained.
Based on Figure 5, it is stated that the abscissa of the point C in the local coordinates system (defined by points P and K) will be expressed as: From the above follows the ordinate y x u C C P = tan as well as the distance between points P and C that is equal to d x y PC C C 2 2 . Based on Figure 5, it follows: As a result, taking into account d d d , coordinates of the point ′ P in the local coordinates system defined by points P and K will be expressed as: Further steps regarding connecting of points P and K using S-shaped transition curve (7) or (8) are related to the use of the Equations given in the end part of Section 2. It is necessary to calculate appropriate radii of curvature and control whether they do not exceed values resulting from proper guidelines.

Geometry comparison of traditional and proposed reverse curves
A comparison of the geometry of traditional curves with curves formed by S-shaped transition curves is very valuable. Therefore, traditional curves formed by two fragments of curves in the form of two clothoids sequence, also known as a vertex clothoid, was adopted as a reference ( Figure 6).
Using known formulas (Brockenbrough, 2009;Wolhuter, 2015), for the above angles and the given distance between points B and C (dBC = 730.81m), fundamental values for curves formed by symmetric vertex clothoids were first determined: • variant I: τ1 = 26°, τ2 = 20.5°, the parameter α = 389.169; • variant II: τ1 = 13°, τ2 = 10°, the parameter α = 572.045; • variant III: τ1 = 52°, τ2 = 41°, the parameter α = 235.102. As a result, other geometric sizes were also calculated, describing the symmetrical curves of two vertex clothoids. In this way, it was possible to determine the position of points P and K that were assumed as footing points also for the main chord of the S-shaped transition curve (7) and (8).  Note: *in the case of vertex clothoids, this is the size referring to the length of entire curve connecting points P and t = l L , analogous as for curves (7) and (8). Then, using the Equations given in Section 2, the geometrical sizes of curves formed by curves (7) and (8) were calculated. Besides, lengths of curves connecting points P and K, formed by both vertex clothoids and S-shaped transition curves (7) and (8), were calculated. Obtained values were summarised in Table 3.
Additionally, Figures 7−9 show a typical illustration of curves formed by vertex clothoids and curves in the form of S-shaped transition curves  (7) and (8). From Table 6 and Figures 7−9 follows that, in the case of small values of inclinations of the main tangents, all three types of curves have a very similar geometry. In addition, in the case of small tangent inclinations, S-shaped transition curves allow obtaining larger radii in points with the maximum curvature. Additionally, the increase in tangent inclination results in noticeable elongation of curves formed by S-shaped transition curves relative to the length of the traditional curves, as well as the greater deviation of these curves relative to the curves formed by vertex clothoids.

Conclusions
1. This article presents alternative geometric tools in the form of so-called S-shaped transition curves that are useful for the routing of reverse curves. An essential feature of these curves compared to traditional highway curves (formed by the sequence of spiral curves) is that they allow a description of the entire curvilinear transition between two points using only one equation. The proposed solutions are useful in horizontal alignment wherever appropriate regulations provide the use of curvilinear transitions. 2. The analyses allowed to determine limit values of angles of tangent inclinations, for whose S-shaped transition curves are useful in practical applications in the routing of reverse curves. These angles are no more than 40 ° for the first family of considered curves and no more than 55 ° for the second family of curves. When using higher values of tangent inclinations, the geometry of these curves becomes inadequate from design practice. Within the curves, there are too long sections with relatively small changes in curvature. For this reason, the above restrictions on the values of tangent inclinations are a specific drawback of the proposed curves. 3. The article also presents the method of forming of reverse curves using S-shaped transition curves so that the permissible, i.e. resulting from proper guidelines, values of curvature are not exceeded. The proposed methodology has an algorithmic character, so it is suitable for possible implementation in appropriate computer software for road design. 4. The article also contains examples to compare a geometry of curves in the form of two vertex clothoids with corresponding curves formed using S-shaped transition curves. These examples have shown that, for smaller angles of tangent inclinations, the geometry of traditional and proposed curves is very similar. This feature of the proposed solutions is important in light of design practice. Therefore, S-shaped transition curves are an alternative to traditional ways of forming reverse curves. It should be noted that these curves describe the whole curvilinear transition between two points using only one equation. 5. Whereas, for larger angles of tangent inclinations, greater deviations of S-shaped transition curves relative to traditional curves are observed. It is a desirable feature of S-shaped transition curves. As a result, in the case of possible terrain restrictions, a designer has more different types of geometric tools that give wider possibilities to an adjustment of a route to these restrictions.