Semi-automated Generation of Geometric Digital Twin for Bridge Based on Terrestrial Laser Scanning Data

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Introduction
Bridge safety has become a common research topic because bridges are an important part of highway infrastructure and are exposed to many dangers during their lifecycle [1].After a bridge is built and used, knowing how to manage and maintain them is a long and costly task.Previous studies have proven that operation and maintenance costs account for half of the total costs of bridges in their lifecycle [2].
An essential problem in the operation and maintenance of bridges is efcient data storage.Nevertheless, the emergence of digital twins (DTs) has provided an efective approach to address this problem.DTs are digital copies of real-world assets [3], consisting of a 3D geometric digital twin (gDT) and semantic, material, and mechanical information.Among them, gDT is the key component.In the bridge design stage, engineers build 3D models of a bridge design according to 2D drawings.However, diferences exist between the designed 3D model and gDT due to deviations in construction [4].To obtain gDTs, the appearance and process of a bridge and the corresponding data should be tested upon bridge completion and during its operation [5].Building information modeling (BIM) is an efective approach of realizing parametric modeling, and it is particularly suitable for generating gDTs.
Terrestrial laser scanning (TLS) is an advanced technology used to obtain building information quickly.TLS has surpassed some traditional measurement methods [6].Furthermore, TLS can accurately and comprehensively obtain the surface points (i.e., point clouds) of a detected object.However, TLS is mainly used to obtain and process the information of simple components, such as buildings, pipelines, and simple supported beams [7][8][9][10].Most of these objects have simple or single components; thus, some methods are unsuitable for directly analyzing bridge components.In recent years, relevant research on bridge measurement and modeling using TLS has been conducted.Given its simple structure and wide distribution, simply supported beam bridges have become a main research object of bridge point clouds.Lu and Brilakis [11] delivered a slicing-based object ftting method that can generate the gDT of an existing reinforced concrete bridge from four types of labeled point clusters.An average modeling distance of 7.05 cm of gDT was obtained.Qin et al. [12] proposed an automatic method of reconstructing parameterized BIM by using point clouds to target simply supported beam bridges, and the accuracy was within 2 cm.For arch bridges with a complex structure, point clouds have more advantages than traditional total station measurement.Yang et al. [13] used an iterative polynomial algorithm to ft the arch structure alignment in the laboratory and obtained data of thirteen epochs.Results showed the reliability of 3D laser scanning and its advantages over traditional methods.Riveiro et al. [14] presented a new method for fully automated point cloud segmentation of masonry arch bridges.Te method efciently created segmented, spatially related, and organized point clouds, each containing relevant geometric data for a particular component (pier, arch, spandrel wall, etc.) of the structure.Occlusions and areas of low point density can cause elements to be missed or classifed as noise.To address the problems that most existing methods for creating asbuilt BIMs from laser scanning data involve plenty of manual work, Yang et al. [15] conducted semi-automated generation of parametric BIM based on TLS data for complex steel structures.Te authors believed that future research is needed to extend the developed technology to other common types of structural components, such as L-shaped components, T-shaped components, and other components formed by combining basic primitives.
TLS has great potential in the research of long-span arch bridges.First, with the arch ring as the main force-bearing object of arch bridges, the existing methods are not accurate enough for its local details.Furthermore, for large bridges, such as arch, cable-stayed, and suspension bridges, comprehensive data collection stations are difcult to set up because of their large span, narrow site, and limited feld of vision.Terefore, complete point clouds are also difcult to obtain.Knowing how to process data and generate gDT in the absence point clouds is an urgent problem.Tus, this study proposes a method of processing the absence point clouds and gDT generation.Te main contributions of this method are as follows: (1) an algorithm based on the combination of the iterative polynomial ftting curve and the sliding window is developed to extract the arch ring accurately.(2) An improved random sample consensus (RAN-SAC) algorithm based on distribution density is adopted to extract the cross sections of arch bridge components except the arch ring.(3) Finally, for the T-beam, a model alignment method is proposed to best match the characteristic intersections extracted by the improved RANSAC algorithm and the points corresponding to the design model.Te quality of the generated models is gauged using a point cloud deviation chromatogram.In addition, the stressed component piers are compared with its design parameters to verify the accuracy of the proposed method.Tree algorithms are used to extract the geometric information of the arch bridge components, and the arch bridge gDT is parameterized using BIM software.
Te remainder of this paper is organized as follows: Section 2 provides the research background with respect to the three aspects of bridge gDT generation: shape representation, geometric information extraction, and bridge gDT generation.Te developed approach to extract geometry information from laser scanning data and generate gDT is described in Section 3. Section 4 presents illustrative examples to validate the proposed method and discusses the experimental results.Section 5 provides a summary and the conclusion of this study.

Related Work Background
Te two kinds of point-cloud reverse-generation models are the 3D grid models [16] and geometric structures [17], and they are defned in accordance with curve shapes, vertical elevations, and cross sections.Te former reconstruction method is simple, but it cannot be used with gDT to collect other information.Te latter is an essential approach to gDT, and it generates data for logical objects that can be used to model the visualization and modifcation of these objects.In general, gDT generation for arch bridges consists of the following steps: (1) shape representation, (2) geometric information extraction, and (3) bridge gDT generation.

Shape Representation Method.
No universal solution can describe 3D objects, and diferent representations have their own advantages and disadvantages.Te choice of a shape representation method depends on the preferred modeling technology and the characteristics of the target object.Te most commonly used shape representation methods can be divided into four categories: implicit representation, boundary representation, constructed solid geometry, and swept solid representation (SSR).Implicit representation requires mathematical formulas to represent 3D shapes, and it has the advantage of accurately describing the 3D shape of the object, i.e., whether it is a plane [18], a sphere, a ring [19], and so on.However, implicit representation can hardly 2 Advances in Civil Engineering represent edges and vertices.Boundary representation provides information about each vertex, edge, and cycle, and it describes shapes by using their limits, thus overcoming the weakness of implicit representation.However, the results of boundary representation may be highly complicated due to the required high-resolution detail level [20], which is detrimental to operations and maintenance.Constructive solid geometry is a set of basic entity primitives that follow a certain "logic."A basic body can be in the form of a cuboid, cylinder, sphere, cone, and so on.Te random sampling method of Schnabel et al. [19] can be used to model objects comprising fve basic shapes: plane, sphere, cylinder, cone, and torus.Nguyen et al. [9] proposed the ftting of a segmented cylinder on the basis of normal region growth to segment and identify pipeline components.However, the shapes of bridge components are generally complex.Te method of constructed solid geometry can be described as an idealized or simplifed topology design, but bridge components, such as arch ring, cable, T-beam, are difcult to describe accurately.As a representation model, SSR creates a 3D shape by sweeping a 2D cross section along a specifc path.Arch ring alignment, pier verticality, and cable sag are important parameters of bridges, and they can be described by a 3D path.In addition, most bridge components have a uniform section; thus, the cross section is fxed.Terefore, SSR is a suitable shape representation method to describe bridges [11,12,15].

Geometric Information Extraction
. SSR includes two aspects: 2D cross section and 3D sweep path.Many studies have attempted to obtain the 2D cross section of point clouds.Ramamurthy et al. [21] extracted geometric features, such as line segments related to cross sections, from point clouds that contain noise and rough surfaces.Moreira et al. [22] used the concave packet algorithm to extract the concave hulls of a local XY plane of a slice.Laefer and Hong [23] proposed a kernel density estimation method to reconstruct the point cloud of standard steel beams in a BIMcompatible format.Zhou et al. [24] developed a parameter extraction method based on grid points to extract quickly and efectively bolt hole features.As for obtaining a 3D sweep path of point clouds, the line shape of the curve component with respect to a straight line is difcult to extract.Bauer and Polthier [25] proposed an automatic method of parametric reconstruction of a curved surface by using unorganized point sets.Tey applied the principle of using the moving least square method to calculate the spine curve of the pipe surface and approximate the polygon curve via the continuous arc spline.To obtain the deformation of each stage of an arch ring accurately, Yang et al. [13] defned the best-ftting surface and changed the usual order to move the polynomial surface closer to the actual point cloud.
For large bridges, such as arch, cable-stayed, and suspension bridges, comprehensive point-cloud collection stations are difcult to set up because of their large span, narrow site, and limited feld of vision.Consequently, the complete point clouds of these bridges are also difcult to obtain.Knowing how to process absence point clouds and generate gDT is an urgent problem.Yang et al. [15] extracted data from 39 pillars of a bridge for a large-scale bridge-like steel structure and completed an as-built BIM.In achieving a complete model, the modeling parameters of the two missing struts in their work and the corresponding connection plates were manually generated.Te arch ring is the main bearing component of an arch bridge, and thus, ensuring its high-precision alignment is important.Knowing how to process absence point clouds is another urgent problem.Tus, an improved RANSAC algorithm based on distribution density is adopted in this study to extract the sections of arch bridge components except the arch ring.For sections that lack point clouds, the translation strategy is used to supplement the unknown line segment.For the T-beam, a model-matching algorithm is used to best align the design model and gDT.

Bridge gDT Generation.
In contrast to the component model generation of bridges, that of buildings is relatively simple.Jung et al. [26] built the inner and outer walls of buildings through point cloud segmentation and feature recognition.Te object of their study only included a single type of a component.Danielle et al. [27] suggested the use of a data-processing algorithm as provided by a point cloud library to create walls and foors.For the common existence of simple and complex shapes, Barazzetti et al. [28] proposed a parametric BIM generation method that can preliminarily separate two aspects.For simple shape modeling, commercial BIM software is used.For complex shape modeling, nonuniform rational basis splines are utilized.Commercial BIM software provides powerful tools for modelers, and it is the platform of choice for most modelers.Quattrini et al. [29] used Autodesk Recap and Revit to create 3D models directly from point cloud data and subsequently modeled them using parameterized elements in Revit (i.e., built-in family library and custom family).However, current experience suggests that no software can perform all geometric modeling [30].
Most software platforms support only a few standard components, such as wall and pipe modeling [31].In addition, information may be lost due to manual modeling operations [32].Tus, modeling efciency is difcult to ensure [33].In this study, the CATIA platform, which is known for its powerful parameterization ability, is adopted to generate arch bridge gDT.

Methods
3.1.Overview.Tis section describes the semi-automated generation of gDT for arch bridge based on TLS data.For large bridges, such as arch, cable-stayed, and suspension bridges, comprehensive point cloud collection stations are difcult to set up because of their large span, narrow site, and limited feld of vision.Terefore, the object of this study is the absence point cloud.
Figure 1 shows an overview of our method that includes three steps: (1) data segmentation, (2) extraction of the bridge components' geometric information, and (3) Advances in Civil Engineering parametric bridge gDT generation.Section 3.2 introduces that the point cloud processing software Geomagic is used to separate diferent components of the arch bridge and classify them into diferent categories.Section 3.3 discusses the algorithms that have been developed and implemented in MATLAB for various components to extract the bridge components' geometric information.Among them, each object class has its own workfow.Section 3.4 presents the process of how to generate the parametric bridge gDT in CATIA.In this study, all solutions of bridge component generation are in SSR.
Te assumptions for the shape of components are described as follows: (1) an arch ring section is an equal section, and any section is the design section size.(2) Te assumption of pier column, tie beam, and so on section is that one side cannot be ftted efectively.If the corresponding parallel edge has been successfully ftted, then the translation strategy is adopted; otherwise, the design information is used for supplementation.(3) Te T-beam section is consistent with the design size because the T-beam is modeled by aligning the design model with extract feature intersection points to the greatest extent using the ICP algorithm.

Data Segmentation.
After pretreating the point clouds, the arch bridge point clouds are manually divided into four categories of components in Geomagic.Te manual division time of the entire bridge is approximately 15 min.Te frst category is the arch ring.Te second category is the pier column and tie beam.Te third category is the pier, pier seat, and bent cap.Te fourth category is the T-beam.Te arch ring is the main stress structure of the arch bridge, as depicted by the rectangular stretching along the arch axis in Figure 2(a).Te quality of several piers is crucial in identifying the stress of an arch bridge.Te pier column and tie beam of the arch bridge are usually rectangular section components, as shown in Figure 2(b).Te pier, pier seat, and bent cap of the arch bridge can be regarded as a standard contour along a line; hence, they are divided into the same category, as shown in Figure 2(c).Te T-beam is the main component of the arch bridge deck, and it has a complex shape.However, in this study, the T-beam adopts the prefabrication method and has a unifed size in the bridge, as shown in Figure 2(d).

Arch Ring.
As this study adopts the SSR method, the following two elements are required: (1) 2D cross section and (2) 3D path.For the arch ring, the complete point clouds of the arch-ring cross section are difcult to obtain.Changing the cross-sectional size of the arch ring has a negligible efect on the stress of the arch bridge.Tus, the design size of the 2D cross section is adopted.Te 3D path of the arch ring is crucial in handling the stress of the arch bridge, further indicating that an accurate extraction algorithm is needed.First, a similar but nonidentical polynomial curve ftting method is used to describe the initial alignment of the arch ring [13].Second, the obtained polynomial is used as the initial parameter of the sliding window algorithm to ensure an accurate extraction.
Polynomial curve ftting should meet two requirements to achieve the optimal results.First, the standard deviation between all points and each order of the polynomial curve ftting should be less than a given value (i.e., 0.5 m).Second, the standard deviation of the polynomial curve ftting should be less than that of a higherorder polynomial curve ftting.From the standard deviations of all points and each order of the polynomial curve ftting, the best polynomial curve ftting can then be selected.Figure 3 shows the solution process of the optimal polynomial curve ftting.Te materials and methods section should contain sufcient detail so that all procedures can be repeated.It may be divided into headed subsections if several methods are described.
where x and z are the 2D coordinates of each point, and a 0 − a n represents the parameter to be estimated.Te unknown parameters a 0 − a n are calculated using the least square method.Te specifc steps of the sliding window algorithm are as follows: frst, the best polynomial curve ftting is used to provide the slope of the tangent line at each interpolation point tangent j � f.Te normal direction is given by normal j � −1/tangent j .Ten, the interval of the interpolation points is selected in accordance with the actual situation.Given the size of a long-span arch bridge, the interval is set to 1-2 m, as shown in Figure 4. 4

Advances in Civil Engineering
Te rotation angle φ j is derived from the angle between the normal direction of slice j and the global X-axis.In particular, the interpolation points of each slice j are used to calculate the normal direction of each slice.Te changes in coordinates of the corresponding rotation angle −φ j for each window point cloud are determined as follows: Te expected value of the window point cloud can then be solved.Finally, the expected value of each window point cloud is rotated such that it is given by −φ j .Te linear control points at the bottom of the arch ring can then be obtained.Te cubic spline is used for the connection.

Pier Column and Tie Beam.
Te axis is the key geometric feature of the pier column and tie beam because these components are subjected to axial force.Te perpendicularity of the pier column axis is taken as the key parameter in the detection.Te pier column and tie beam of an arch bridge are usually rectangular.In view of ensuring the representativeness of our method, the rectangular crosssectional pier column with two-and three-side point clouds is selected as the research object, as shown in Figure 5(a).Te point clouds of the pier are segmented via the slicing method.Te specifc method involves cutting the data along the Z-axis in a plane parallel to the XY plane to obtain multiple cross sections, as shown in Figure 5(b).During site construction, the pier height of each pouring is 0.5-1 m.A 0.5-1 m length in the Z-direction is recommended.An improved RANSAC algorithm based on distribution density is adopted to extract the sections of the arch bridge components except the arch ring.For the sections that lack point Advances in Civil Engineering clouds, the translation strategy is used to supplement the unknown line segment.Finally, the intersection of each line is obtained and then used to calculate the centroid of the intersection, as shown in Figure 5(c).Te axis of the pier column is obtained via the least square method, as shown in Figure 5(d).
Te specifc steps of the improved RANSAC algorithm based on the distribution density algorithm are in the following steps, as shown in Figure 6.First, the number of point clouds per millimeter along the X-axis and Y-axis of each cross-section point clouds is calculated as m, and the average density is calculated as ρ.In addition, the density threshold is defned as ρ t .Ten, when m ≥ ρ t , the length T corresponding to the time point cloud is used as the threshold of the algorithm.
where ρ x � N a /a, ρ y � N b /b; N a and N b represent the total number of points on the X-axis and Y-axis of the cross section, respectively; a and b represent the width and height    for i � 1to kz do Datai; //Obtain the Point clouds of each segment of point clouds after cutting.N a , a;//N a is the total number of points on the X-axis a of the cross section.a is the width of the cross section.ρ x � N a /a; //Te average density is calculated ρ Along the X axis.ρ t � 3 * ρ x ; //Defnition the density threshold.
for j � 1to kx do m x ; //Calculated the number of point clouds per millimeter along X-axis.end for Tx; //Find the number greater than ρ t in m x , and then the product of dx.Ty; //It is the same as the solution process Tx. l 1 � RANSAC (Data, Tx); //Using RANSAC algorithm to ft straight line.Data0; //Delete the point cloud in Data used in l 1 l 2 � RANSAC (Data0, Ty); //Using RANSAC algorithm to ft remaining straight line.l 3 � l 1 + (0, 0, a); //the translation strategy is used to supplement the unknown line segment.l 4 � l 2 + (0, 0, b); //like l 3 .
Advances in Civil Engineering of the cross section, respectively.Te RANSAC algorithm of threshold T is used to ft the straight line of the cross section iteratively.Te specifc algorithm for calculating the axis of the pier column is given in Algorithm 1.

Pier, Pier
Seat, and Bent Cap.Te pier, pier seat, and bent cap of an arch bridge can be regarded as having a standard contour along one line; thus, they can be classifed in the same category.Tese components are not necessarily perfectly vertical; however, the piers are assumed to be quasivertical in this study.First, the piers and pier seat are projected onto the XZ plane, as shown in Figures 7(a) and 7(b).Te bent cap is projected onto the XY plane, as shown in Figure 7(c).Te 2D ConcaveHull α-shape [22] is used to describe the outline of the slice cross section of the point clouds.Similarly, the improved RANSAC algorithm in Section 3.3.2 is adopted to extract the sections.In particular, their point clouds are absent because of the shielding of the soil around the pier and the diminutive size of the stops, which are similar to those of the pier column and tie beam, on both sides of the bent cap.

T-Beam.
Most beam-slab bridges use precast concrete members as the main structural members [34].Te T-beam of an arch bridge is the same.Te superstructure of arch bridges is the same as that of beam-slab bridges.Usually, the point clouds on both sides of the T-beam are difcult to determine, hence the incomplete data.Terefore, the point clouds of one side of the T-beam are selected in this research.Te T-beam entails a complex modeling, further suggesting the inapplicability of the translation strategy.
Te point clouds of the pier are segmented via the slicing method.In particular, the data along the X-axis in a plane parallel to the YZ plane are cut to obtain multiple cross sections, as shown in Figure 8 (a).A 0.5-1 mm length in the X-direction is recommended.Te improved RANSAC algorithm based on distribution identity is used to obtain some characteristic intersections of the T-beams.And the calculated threshold of the improved RANSAC is shown in 8 (b).Te characteristic intersections of each T-beam section total four, as shown in Figure 8 (c).Te model alignment algorithm is proposed to best match the characteristic intersections extracted by the improved RANSAC algorithm and the points corresponding to the design model.For the T-beam, a model-aligning algorithm is proposed.In particular, the outline of some characteristic intersections of T-beams is obtained.Ten, the iterative closest point (ICP) algorithm is used to match characteristic intersections and the points corresponding to the design model.

Bridge gDT Generation.
After geometric information of the components is extracted as derived from the point clouds, it is stored in an Excel fle.By automatically assigning the coordinates stored in Excel to the corresponding PART module of CATIA, the arch bridge gDT can also be established in CATIA.Te three steps of this approach can be described as follows: Step 1 : Generation of the arch bridge skeleton.Except for the T-beam characteristic intersections, the skeleton includes the 3D path and projection plane of the other components.First, the coordinate system of the entire arch bridge is determined through an approach that is essentially the same as that of the construction coordinate system.Ten, the information is imported into CATIA's PART module, as shown in Figure 9(a).
Step 2 : Generation of the templates for each component.First, the local coordinate system is created in the PART module of CATIA.Te Z-axis is in the 3D path, which is defned as a straight line or a curve.Te X-and Y-axes should conform to the principles of the Cartesian coordinate system.Ten, the origin of the component is determined.For the arch ring, pier, and tie beam, the origin is the center point of the profle.For the pier, pier seat, bent cap, and T-beam, the origin is the corner of the profle.Finally, the corners of the cross section are sketched and then connected with the segments, as shown in

Experiments
4.1.Point Cloud Acquisition and Processing.Te proposed semiautomatic method for extracting the geometric information of the bridge's components in the absence of point clouds was verifed in this study.We used a Leica 3D laser scanner to collect the point clouds of a long-span arch bridge.Te vertical and horizontal felds of view of the scanner were 300 °and 360 °, respectively.Te distance between two points at a distance of 10 m in the resolution gear was approximately 3 mm, and the data collection time of each scan was approximately 13 minutes.Similar to the stations of cable-stayed and suspension bridges, the comprehensive data collection stations in this study were difcult to set up because of the model's large span, narrow site, and limited feld of vision.In accordance with the actual situation of the bridge, three stations must be set up to obtain the required point clouds in Figure 10 (a).Te point clouds of the three stations were registered using the ICP algorithm in CloudCompare software, as shown in Figure 10 (b).Among them, the frst station was considered the target station.A total of 48,300 points in the joint station of the second and frst stations were processed, and the fnal RMS was 0.0021 m.In addition, a total of 42,380 points in the third and frst stations were processed, and the fnal RMS was 0.0018 m.To ensure the accuracy of   4.2.Results.Te method described in Section 3.4 was used to generate the gDT of the arch bridge.To check the original coordinate matching degree between the gDT and its point clouds, we converted the point clouds before segmentation and the reconstructed CATIA model into STP format and then imported them into Geomagic.We imported the bridge point clouds before segmentation, followed by the gDTof the arch bridge, into Geomagic.Figure 11 shows the coincidence of the two components and the deviation chromatogram of the arch ring, pier column, tie beam, pier, pier seat, bent cap, and T-beam.Te deviation between the point clouds of some components and their gDT was evenly distributed, and most of them were within 5 mm.
Except for the absence point cloud, most other point clouds and their gDT coincide with each other, as illustrated in Figure 11.Some deviations can be observed in some parts of each component.Tis phenomenon can be attributed to three reasons: (1) the small amount of noise in the point clouds, (2) the format conversion between models in the software, and (3) the error of the cross-sectional ftting algorithm.
Te arch ring linear is critical to the mechanical performance of the arch bridge.Taking the actual point cloud as a reference, the results of distance deviation between the polynomial curve ftting and the algorithm in this paper are compared.Among them, the root mean square error (RMSE) of the polynomial ftting curve algorithm is 0.0126 m, and the RMSE of the algorithm in this paper is 0.0055 m, which indicates that the latter is better in ftting the arch line.Among them, there are several obvious abnormalities in the distance deviation of the polynomial ftting curve algorithm, such as the midspan and quarter of the arch ring as depicted in Figure 12.Te enlarged index of these corresponding positions shows that, compared with the control points of the algorithm in this paper are closer to the point cloud, the alignment of the polynomial ftting curve algorithm is farther from the point cloud, which further illustrates the progress of the algorithm in this paper.
In addition, the quality of the pier columns is important to the arch bridge.Here, the geometric information of the extracted pier column was compared with the design information in the construction drawing in two aspects: (1) intersection of the pier column centerline and pier base horizontal line and (2) perpendicularity.Te results showed that the deviation of the intersection relative to the design intersection has a normal distribution.Moreover, the RMSE of the deviation was 0.0041 m, as depicted in Figure 13.Te deviation between the perpendicularity of the pier column and the design angle in terms of RMSE was 0.046 °, as shown in Figure 14.According to Klein et al. [35], an error of approximately 2% is acceptable in the as-built BIM model facility management.Advances in Civil Engineering

Figure 1 :
Figure 1: Overview of the proposed method for arch bridge gDT generation.

Figure 5 :Figure 6 :
Figure 5: Illustrations of axis line for pier column and tie beam: (a) original point clouds, (b) extracted cross sections along the Z-axis, (c) ftting outline, and (d) ftting axis line.

Figure 9 (
b). Step 3 : Lofting of each component.Te generation component templates are lofted on the arch bridge skeleton, as shown in Figure 9(c).

Figure 8 :Figure 9 :
Figure 8: Illustrations of characteristic intersections for T-beam: (a) original point clouds, (b) calculated threshold of RANSAC, and (c) calculation of the characteristic intersections.

Figure 10 :Figure 11 :
Figure 10: Illustration of gDT generation: (a) arch bridge scanning, (b) station processed by the ICP algorithm, and (c) point cloud used for generating gDT.

Figure 12 :
Figure 12: Accuracy analysis of arch ring linear algorithm.

Figure 13 :
Figure 13: Deviation of the intersection relative to the design intersection.

Figure 14 :
Figure 14: Deviation between the perpendicularity of the pier column and the design angle.