## 1 Introduction  
A suspension bridge is one of the most advantageous bridge types for long bridges spanning over one thousand meters. The main cable is the core load-bearing component of a suspension bridge, mainly regarding tension. In the design and construction of suspension bridges, the main cable curve directly affects the blanking length of the cable, the installation location of the cable clamp, the length of the suspenders, and other parameters. In addition, the cable curve has large effects on the mechanical performance of the bridge after construction [1]. Since the main cable curve is difficult to adjust during construction, it is essential to calculate and control the curve accurately before construction to control the cable curve and to make it consistent with the design. In addition, before the construction of a suspension bridge, the alignment and internal force need to be analyzed using the forward installation and reverse dismantling method based on the alignment of the completed bridge [2]. Therefore, researchers and constructors need to pay attention to the design and calculation of the main cable curve of a suspension bridge (i.e., the main cable shape findings) to better design and construct suspension bridges and to completion their construction [3, 4]. In a completed suspension bridge, the main cable curve is calculated according to three controlled parameters--the rise-span ratio, the theoretical elevation of the main cable mid-span, and the theoretical elevation of the bridge tower top--combined with the loads on the main cable and an appropriate line shape calculation theory. Obviously, according to these controlled parameters, the main cable curve results are different when calculated based on different curve calculation theories. Hence, it is very important to choose an appropriate calculation theory. Regarding theories behind the calculation of the main cable curve, the linear calculation theory was the earliest to be presented and can be classified into two categories: the analytical method and the finite element method (FEM). The analytical method mainly includes the parabola method and the catenary method. With an in-depth study of the mechanical characteristics of suspension bridges and a large number of calculations, it was found that the main cable curve of each span of a completed suspension bridge is not an ideal parabola or catenary under the distributed load along the main cable and the concentrated load at the cable clamp. Additionally, each cable segment of the main cable between the suspenders is catenary under of its own weight, also known as segmented catenary. [PERSON] et al. (2003) [PERSON] et al. (2003) proposed the segmented catenary method, which determines the cable force and main cable curve of each part according to the mechanical balance and deformation coordination conditions, and the nonlinear problem of the main cable curve was considered. Then, the main cable curve can be solved in the form of an analytical expression, which is an accurate calculation method for the main cable curve. [PERSON] et al. (2009) [PERSON] et al. (2009) then developed the segmented catenary method from plane to space and proposed a numerical calculation method by establishing the spatial analysis model, and with this method, the main cable curves in both the cable completion stage and the bridge completion stage can be analyzed. At present, the segmented catenary theory has formed a relatively mature calculation system. The constraints of the mechanical balance state and the geometric relationship are key to deducing the recursive relationship between line parameters. On this basis, many achievements have been accumulated in the boundary conditions of cable saddle, in the calculation efficiency of nonlinear equations, in theoretical optimization, etc. [7; 8; 9; 10]. However, the iterative calculation of linear deviation and internal force correction is tedious and sensitive to the selection of the initial value, and the iterative convergence speed is slow as well.