2.3.1 Core concepts and terminology.

Parametric modeling: The parametric modeling method utilizes parametric technology to automatically control and generate 3D models of entities based on their morphological parameters. Parametric technology is a computer-aided design technique that allows flexible control of the geometric features of entities by adjusting parameter values [56,57].

Level of detail (LOD): LOD is used to describe the complexity of geographic object representations. Its purpose is to reduce the geometric complexity of objects during visualization to improve performance. By creating multiple versions of a model with varying levels of detail, the appropriate version is dynamically selected for rendering based on the user’s distance from the model or the viewpoint.

Dynamic lightweight Processing: This refers to dynamically adjusting the level of detail and rendering strategy of a model during runtime based on real-time requirements and environmental changes. By integrating mesh simplification, LOD technology, and view frustum culling, dynamic lightweight processing of BIM models can be achieved. This significantly improves model loading and rendering speed while maintaining visual quality and reducing hardware resource demands.

Digital elevation model (DEM): A DEM is a raster model that digitally represents surface elevation using limited elevation data. Each cell in the grid stores the absolute elevation value of its corresponding location [58].

2.3.2 Design of geometric parameters for algorithms.

In parametric modeling, the design of characteristic parameters is paramount to determining the quality of the final outcome. During the modeling process, parameters are thoroughly decomposed based on topological relationships and object characteristics, yielding detailed modeling design parameters. This parameter design involves the design and constraints of parameters that drive the generation algorithms. Road design primarily encompasses the design of components such as road pavement, bridges, bridge piers, tunnels, and slopes. Depending on their topological structures, these various components can be further broken down into more intricate sub-components. Each component contains common parameters such as texture materials, geometric appearance parameters, starting and ending positions, among others. Detailed road modeling parameters can be found in S1 Fig. Prior to the generation of road components, this paper divides road planning points into segments such as road pavement, slopes, bridges, and tunnels based on topographical and geomorphological characteristics. Each component is then generated according to its corresponding algorithm. This paper developed and integrated algorithms for road pavement generation, slope generation, tunnel generation, and bridge generation.

(1) Algorithm for road pavement generation

1) Continuous road pavement generation algorithm

The continuous pavement generation algorithm necessitates consideration of road continuity, particularly in the generation of ramps, turns, and other complex sections. Since the parameters of each planned point are relatively independent, the algorithm must be locally independent. In other words, the road pavement generation should be achievable by solely considering the two adjacent planned points, ensuring that it does not disrupt previous design achievements and minimizes computational time and space costs.

1. (a)  Generation of curved road pavement coordinates

In 3D scenarios, road design often encounters curves with varying heights, sharp turns, and other complexities. These turns can broadly be categorized into horizontal turns, uphill/downhill turns, and folded ramps with height differences (as shown in Fig 3). This paper proposes a curve interpolation algorithm to accomplish the generation of curved pavement.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Typical scenarios in curve design.

(A) horizontal turns; (B) uphill/downhill turns; (C) folded ramps with height differences.

https://doi.org/10.1371/journal.pone.0340062.g003

When calculating the curve, it requires the assistance of the two points before and after the current point, making a total of three points involved; these points are sequentially labeled as Points A, B, and C; these points are then projected into the local coordinate space with Point B serving as the reference point.

This algorithm is symmetric, meaning the same algorithmic approach is employed on both sides of the road. Taking the algorithm for the left side of the road as an example, the flow of this algorithm is outlined below (Fig 4).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Algorithm process for generating coordinates of curved pavement.

https://doi.org/10.1371/journal.pone.0340062.g004

PS and PE denote the interpolation start and end points, respectively. Based on the interpolation resolution parameter, spherical interpolation is performed with point P3 as the center, starting from PS and ending at PE, to calculate the interpolated points along the curve. Starting from point Ax, points are sequentially pushed onto the stack until point Cx is reached, thereby obtaining the complete set of curve coordinates. The same algorithm can be applied to generate the coordinate points on the right side of the curve. Detailed pseudocode and algorithm implementation are provided in S1 Text.

1. (b)  Generation of 3D road model

Based on the above algorithms, generate all key point data to complete the coordinate generation for the continuous road pavement. Further interpolation and refinement of these coordinate points are necessary for the generation of the road pavement. Due to the symmetry of the algorithm, the number of points on both edges of the road will be the same after calculation, making it easier to complete the interpolation of coordinates based on the interpolation resolution parameter. The calculation of the triangulated irregular network (TIN) and texture coordinates for the road is then performed based on parameters such as road pavement thickness. The detailed process is as follows:

1. Step 1: Based on the interpolation resolution parameter r, calculate the interpolated coordinates for the continuous road pavement along the straight segments on both the left and right sides of segment ABC.
2. Step 2: Based on the road pavement thickness parameter d and the interpolated road pavement coordinates, calculate the triangulated irregular network (TIN) model of the road.
3. Step 3: Calculation of road pavement Texture Coordinates:
   1. ① The texture coordinates are calculated based on the road sidelines obtained from the above-mentioned curve generation algorithm, denoted as M.
   2. ② Calculate to obtain the texture coordinates (s, t1), where s is designated as 0 for the left side of the road and 1 for the right side.
4. Step 4: Based on the texture coordinates, the road pavement texture is mapped to each coordinate.
5. Step 5: Based on the length of the road sidelines, obtain the anchor point positions for the guardrail at fixed intervals.
6. Step 6: Based on the guardrail coordinates, generate the guardrail model point by point. The detailed process of road pavement generation is provided in S2 Fig.

After applying the above algorithms and texturing, the basic appearance of the road is formed. Since the road pavement is considered as a single entity, the subsequent generation of models for bridges, tunnels, and other structures will no longer require separate consideration for the corresponding road pavement portions.

2) Slope generation algorithm

The generation of slopes primarily adopts a terrain meshing and partitioning approach. Based on the design parameters for slope generation, the triangular mesh of the slope is calculated according to the algorithmic relationships between different levels and the intersection lines between the slope and the excavated terrain. The slope is then generated by applying texture mapping based on the texture style specified in the slope parameters. By treating the entire algorithm as a function, the inputs are the design parameters and the terrain model, and the output is a 3D slope model. The core algorithmic formula is as follows:

(1)

Here, represents a complete 3D slope model, denotes the base elevation of the foundation, is the slope angle, represents the width of each platform level, indicates the number of slope levels, is the terrain model, and represents the texture style collection. The detailed process of slope generation is provided in S2 Text.

(2) Bridge algorithm design

Bridge parameters encompass three major modules: bridge approach parameters, bridge deck parameters, and pier parameters. The semantic information is expressed in detail, encompassing components such as piers, guardrails, and bridge bodies. Customized parameter settings can be applied to each planning point based on the characteristics of the project area, including parameters like pier type, pier texture, pier base, interpolation resolution, guardrail type, and bridge deck type. In the design of bridge algorithms, particular attention must be paid to low-lying areas along the planned route, the length of these areas, bridge approach locations, pier construction, and texture types to ensure the adaptability and flexibility of 3D model generation.The detailed construction process for bridges is outlined as follows:

① Calculation of depressed areas

The road planning route is first mapped out, and the road is subsequently divided into n segments based on the intersections between the planning route and the terrain. The depressed areas, which refer to sections where the terrain height is lower than the planned route, are then identified. Assuming the average height of the i-th segment of the planned route is denoted as , and the terrain height function is represented as (where x, y, z are the coordinates of the terrain), the depressed areas meet the following condition:

(2)

Wherein, represents a point on the terrain, and can be obtained through interpolation or retrieved from the planned route data. The above steps are reiterated until all n segments of the road are traversed, and road segments where the formula result is true are designated as Rm and input into the next stage for calculation.

② Determination of the necessity for bridge construction

The traversed road segments from the previous step are further examined. Only when there exists a continuous depressed road segment of a certain length within the planned route, will that segment be designated for bridge construction. represents the length of the depressed segment in road segment Rm. If (where is a threshold value), a bridge is required and designated as Ri. Conversely, if , bridge construction is not necessary, and the construction of an ordinary road should be considered.

③ Calculation of bridge approach location

For road segments Ri requiring bridge construction, the bridge approach is positioned by extending a certain length B from the intersection point of the bridge and the terrain. The latitude and longitude coordinates of the bridge approach are calculated accordingly. Assuming that the elevation of the bridge approach and the bridge end remains essentially constant, the latitude and longitude of the intersection point between the bridge and the terrain are denoted as (, ). By simplifying the extension in a certain direction, the latitude and longitude of the bridge approach can be calculated as follows:

(3)

(4)

Where and are the components of the direction vector in latitude and longitude, respectively, and and represent the actual distances corresponding to each degree of latitude and longitude, respectively.

④ Pier construction

Firstly, it is necessary to determine whether piers are required for the bridge. Assuming the total length of the bridge is L_bridge, if (where C is a threshold value), then piers are necessary; otherwise, they are not required. The number of piers, A, can be calculated by dividing the bridge length by the span of a single pier (rounding down or up). When , the following equation holds:

(5)

Then, based on the number of piers to be constructed, A, the locations of the piers are calculated. The pier locations can be evenly distributed along the bridge, and the position of the i-th pier, , is calculated as follows:

(6)

Here, represents the starting position of the bridge.

Subsequently, the height of each pier is calculated by determining its terrain elevation. The height of the pier, , is determined by the terrain elevation and the deck height:

(7)

Where is the designed height of the bridge deck.

Finally, based on the pier heights and positions, the pier geometries are modeled, the texture mapping coordinates are calculated, and the texture mapping is applied.

⑤ 3D Modeling and texture mapping

Based on the bridge approach locations and the bridge deck width, the geometric modeling of the bridge deck is conducted, and its dimensions are calculated. Simultaneously, texture calculations and mapping are performed. For elevated terrain at the bridge approach, terrain modifications are also required, and slope modeling and texture mapping are conducted using the methods for slope construction described earlier in this paper. The detailed algorithm for texture mapping is described in S3 Text.

(3) Tunnel algorithm design

Tunnel parameters primarily encompass three components: portal model, tunnel model, and portal slope parameters. The semantic information is expressed in detail, including portal length, portal width, interpolation accuracy, tunnel radius, texture material parameters, etc. In the algorithm design process, since tunnels are primarily generated when the road is located at a significant height from the mountain surface, the generation of tunnels requires calculations from the points of entry and exit from the mountain. By utilizing the tunnel radius, all coordinate points for the tunnel’s generation are computed, and a 3D tunnel is created by applying textures. At the entrance and exit of the tunnel, excavations need to be made in the terrain to ensure the tunnel’s connectivity.

The detailed construction process of the tunnel is as follows:

① Calculation of highland areas

The roadway is divided into n segments based on the intersection points between the planned route and the terrain, allowing for the identification of low-lying areas, where the terrain height is below the planned route. Similarly, the highland areas, where the terrain height exceeds the planned route, are calculated using the intersection points. Assuming the average height of the k-th segment of the planned route is denoted as (where are the coordinates of the terrain), and the terrain height function is hterrain (x,y,z), the highland areas satisfy the following condition:

(8)

represents points on the terrain, and the road segments for which the formula result is true are designated as Rt and fed into the next stage for further calculation.

② Determination of tunnel construction necessity

The road segments passed through in the previous step are further iterated. Only when a continuous highland segment of a certain length exists within the planned route, will that segment be considered for tunnel construction. Rt(L) denotes the length of the highland segment within the road segment Rt. If (where B is a threshold value), then a tunnel construction is required, designated as Rk. Conversely, if , the segment is considered for construction as an ordinary road.

③ Calculation of tunnel portal locations

For road segments Rk that require tunnel construction, based on the starting intersection point between the tunnel and the terrain, let the intersection be denoted as (, , ). The tunnel portal is determined as the point where the terrain height exceeds the threshold B above the planned route height. Assuming the tunnel entrance and exit elevations for this segment are approximately the same, B represents an increment relative to the planned route height at that point, with its geographical coordinates being (, , ). Thus, the following condition holds true when:

(9)

(10)

(11)

Otherwise, the current road segment is designed as an ordinary road with slopes, and the algorithm automatically proceeds to the next road segment for calculation. The algorithm for entering and exiting the tunnel portal is consistent.

④ Calculation of tunnel model starting point

A fixed length C is extended outward from the tunnel portal location to determine the starting point of the tunnel model. Assuming that the elevation of the tunnel model’s starting point remains unchanged from that of the tunnel portal, let the coordinates of the model’s starting point be (, , ). Then:

(12)

(13)

(14)

Where is the angle between the direction of tunnel extension and a reference direction (e.g., due north).

⑤ Tunnel portal modeling, texture mapping, and terrain clipping

The extended portions of the tunnel entrances and exits are constructed similar to ordinary roads. To ensure the safe operation of both the ordinary roads and the tunnel, it is necessary to construct slopes and modify the terrain. The slope levels on both sides are calculated separately. Following the slope construction methods outlined in previous sections, slope modeling and texture mapping are performed. This ensures that the transitions between the tunnel portals and the surrounding terrain are visually coherent and meet safety requirements. The detailed algorithm for texture mapping is described in S3 Text.Utilizing the locations of the tunnel entrances and exits, as well as the tunnel radius R, a semi-cylindrical tunnel is modeled. The starting and ending points of the tunnel body coincide with the positions of the entrance and exit portals, respectively, with a radius of R. Based on the locations, widths, and thicknesses of the tunnel portals, terrain clipping is performed to excavate the area occupied by the tunnel from the surrounding terrain. Finally, texture mapping is applied to the modeled tunnel to generate a visually realistic representation.

2.3.3 Model lightweighting processing.

To enhance the loading and rendering efficiency of BIM models in GIS platforms, model lightweighting is essential. This process reduces model information complexity, compresses data dimensions, and increases the level of abstraction, while preserving model accuracy and functional completeness.

(1) Triangular mesh information simplification

The geometric information of the model is stored in a triangular mesh, where complex BIM models often require millions of triangular meshes to depict model features. Such a vast amount of data poses significant challenges for the transmission and rendering of the model on the Web. Therefore, this paper employs the Garland edge collapse algorithm based on QEM to simplify the triangular mesh [59,60].

The edge collapse algorithm initially computes the cost of folding each edge in the model, sorts these costs in ascending order, and iteratively removes and folds the edge with the lowest cost from the front of the queue. After folding an edge, all associated edges are removed from the priority queue, their folding costs are recalculated, and edges that meet the folding criteria are reinserted into the queue in priority order. This process continues until no edges meet the folding criteria. The Garland algorithm, specifically, is an edge collapse algorithm based on the quadric error metric, which utilizes the sum of squares of distances from a vertex to its associated triangular planes as the error measure. This algorithm is known for its speed, high quality of simplification, and overall efficiency as an edge collapse method. The detailed implementation process is outlined below.

The error for folding a vertex to a new vertex is defined as:

(15)

Where: , with , and ;

represents the set of planes associated with ; denotes the fundamental error of which is:

(16)

We define as the quadratic error measurement matrix for vertex , as shown below:

(17)

During the initialization process, since each initial vertex is the intersection point of its associated triangles, the initial error for the original vertex is 0. When performing an edge collapse, the edge collapse cost and optimal position of the new vertex are given as follows:

(18)

(19)

Where: and are the quadratic error measurement matrices for and respectively. After the collapse, the quadratic error measurement matrix for the new vertex is:

(20)

The edge collapse algorithm simplifies the model grid evenly, allowing for adjustable LOD to achieve varying model resolutions (Fig 5).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5. Simplification of triangular mesh information.

https://doi.org/10.1371/journal.pone.0340062.g005

(2) Dynamic LOD hierarchy generation

The original road BIM generated experiences severe lag and slow rendering speeds during browsing in 3D engines. Due to the limited capacity of the engines, spatial partitioning of BIM models with complex components and large data volumes is necessary.

Based on the fundamental process of the Garland edge collapse algorithm, which performs well in terms of model simplification effects, there is still room for improvement in efficiency. Therefore, this paper leverages the characteristics of the edge collapse algorithm to partition the railway model into multiple regions and then applies edge collapse operations to each region individually.

Model space partitioning generally involves two methods: uniform and non-uniform grid partitioning, with octree partitioning belonging to the latter category [61,62]. The figure below compares uniform grid partitioning with octree-based non-uniform grid partitioning (Fig 6). Uniform grid partitioning divides the overall bounding box of the model into sixty-four smaller bounding boxes of equal size, then further divides the first-level bounding boxes into eight equally-sized grids, and so on, as illustrated in the diagram.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 6. Comparison between uniform grid division and octree grid division.

https://doi.org/10.1371/journal.pone.0340062.g006

The specific steps are as follows:

1. 1) Calculate the root node coordinates of the model’s maximum bounding volume (where m represents the current model). The formulas for the model’s length W, width H, and height D are as follows.

(21)

(22)

(23)

Taking the largest-sized scene as the root node, its coordinates are::

(24)

1. 2) Extend from the root node in the XYZ directions of space to divide the maximum bounding volume into eight equal-sized regions, which are referred to as child nodes. Place the face elements from the object file format (OBJ) model into the grids without child nodes, and categorize the center point coordinates of the face bounding boxes into the bounding boxes of their respective child nodes.
2. 3) Set the maximum error metric value, the child node face threshold, or the maximum recursion depth based on the number of triangular faces in the model. If the current level is less than the set threshold or has not reached the maximum recursion depth, proceed with the next level of node division. Stop dividing when the number of elements assigned to a child node is not zero and is the same as the number of elements in its parent node. The rules for octree node division are illustrated in the figure below (Fig 7).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 7. Illustration of octree node division rules.

https://doi.org/10.1371/journal.pone.0340062.g007

(3) View frustum culling technique

The BIM model of roads generated features massive data and encompasses numerous 3D models, resulting in significant storage space occupation and substantially affecting the efficiency of model rendering. When browsing through 3D models, users can only observe the portions within the field of view displayed on the Web client, and whether the remaining data are loaded has no impact on the final display outcome. Therefore, on top of LOD, introducing the view frustum culling technique can effectively reduce the amount of data loaded for the model and enhance rendering efficiency. Among these, the key components of view frustum culling are frustum culling and screen-space error algorithms.By utilizing an adaptive octree indexing structure, rapid loading of different LOD models across various hierarchical structures can be achieved. On this foundation, the frustum culling algorithm is incorporated to manage changes in camera viewpoints and angles. This algorithm determines whether a node’s 3D tile should be included in the rendering sequence based on its position and geometric error, thereby further enhancing the efficiency of scene rendering [61,62]. The design principles of this algorithm are outlined below. In the following figure (Fig 8), represents the center of the tile bounding box, are the coordinates of point , is the approximate radius of the bounding box, denotes the viewpoint, is the angle of the view frustum, and is the shortest distance between the frustum boundary and point , which can be derived from the following formula:

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 8. Illustration of frustum culling.

https://doi.org/10.1371/journal.pone.0340062.g008

(25)

The value of is compared with . If , the tile is loaded; otherwise, the tile is not loaded.For models within the view frustum, the resolution of the 3D tiles to be loaded is determined based on their distance from the viewpoint. The closer a model is to the viewpoint, the higher the resolution (i.e., LOD level) of the tiles required; conversely, the farther a model is from the viewpoint, the lower the resolution (i.e., LOD level) of the tiles required. During runtime, the screen-pace error is calculated based on the geometric error of the tiles, as illustrated in the figure below (Fig 9).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 9. Illustration of screen-space error.

https://doi.org/10.1371/journal.pone.0340062.g009

In the figure above, the distance between E and G represents the predefined spatial geometric error of the tile, is the distance from the viewpoint to the far clipping plane AE, is the screen height, and is the screen-pace error. The geometric relationship between and is as follows:

(26)

Using the above formula, the screen-pace error can be calculated. By comparing with a preset threshold α, if e > α, the next level of tiles is loaded to replace the current tiles; otherwise, the current tiles are retained.

In terms of road model management, CityGML is an international standard for extensible urban spatial data exchange and encoding, published by ISO TC211 and OGC. It follows object-oriented principles and supports hierarchical descriptions of the same geographic entity, as well as integrated storage and management of geometric and semantic features. Although the road entity models in this study possess numerous parametric features, they exhibit a stable structure and a clear relationship with terrain. Therefore, in organizing and managing road models, a hierarchical parametric approach is adopted in accordance with the CityGML standard [62].