Survey Review, 43, 319 pp.67-79 (January 2011) Contact: Said Easa e-mail: seasa@gwemail.ryerson.ca © 2011 Survey Review Ltd 67 DOI 10.1179/003962611X12894696204669 FITTING COMPOSITE HORIZONTAL CURVES USING THE TOTAL LEAST-SQUARES METHOD Said M. Easa 1 and Fujian Wang 2 1 Professor, Department of Civil Engineering, Ryerson University, Toronto, Ontario, Canada 2 Associate Professor, Department of Civil Engineering, Zhejiang University, Hangzhou 310027, China ABSTRACT The parameters of horizontal curves are sometimes missing or need to be updated using observed x and y coordinate data. The observed data for existing alignments may be obtained using global positioning systems (GPS) or extracted from digital imagery. Previous methods have addressed the case where only the circular or the transition curves are estimated. This paper presents an optimisation model for composite horizontal curves that simultaneously fits the circular curve, its transition curves and the two tangents using the total least-squares method. The objective function minimises the squared deviations between the observed and predicted values. The model can be applied to fitting single or multiple composite horizontal curves. Since the model is nonlinear and non-convex, a close initial solution is first developed and then used to obtain the global optimal solution. The model is validated and applied using actual data of a horizontal alignment. The proposed model presents an important extension to existing methods for estimating horizontal alignments and therefore should be of interest to surveying professionals. KEYWORDS: Transition curve. Circular curve. Horizontal alignment. Estimation. Data. Optimisation. INTRODUCTION Horizontal curves (with or without transition curves) are important elements in the design of highways and railways. A simple horizontal curve consists of a circular curve that connects two straight lines (tangents), while a composite horizontal curve consists of a circular curve, a transition curve at each end and two tangents. The geometry of horizontal curves can be found in most surveying engineering texts [1], [2], [9], [13]. The parameters of the simple horizontal curve are the curve radius and deflection angle. The parameters of the composite horizontal curve are curve radius, transition curve parameter and deflection angle. Throughout the paper, a composite horizontal curve is simply referred to as a horizontal curve. The estimation of the parameters of horizontal curves may be necessary in the following practical situations [13]: (a) the parameters may not be available on file, especially for older routes, (b) it may be necessary to check the adherence of an as-built highway to design guidelines and (c) in railroad abandonment programs, where rails have been used to delineate the right-of-way lines, the parameters of the railroad alignment must be established before the rails are removed. Another important application is establishing the parameters of horizontal alignments through road extraction from aerial or satellite images [5], [7], [14]. Several methods are available in the literature for fitting horizontal alignments. The fitting of a simple horizontal curve, where a circular curve is first fitted to the data points to determine its radius and the centre using the Least-Squares (LS) method is addressed in [13]. Given the curve centre and a single data point on each tangent, the point of curvature (PC) and the point of tangency (PT) of the curve are then determined. In the method used by [7], the two tangents are assumed fixed and the best circular curve is established using an iterative method, where the PC and PT points are directly determined. The method given in [6] assumes that the tangents are fixed and