**4. Results and Discussion**

11, we can obtain a complete triangular mesh model.

We use C++ to implement the relevant algorithms of the modeling method based on the Coons surface interpolation and test them on a 64-bit computer with Windows 10 professional edition. This computer has 3.70 Ghz Intel (R) Core (TM) i7-8700u, 32GB ram, and NVIDIA GeForce RTX 2080 GPU. In this paper, five groups of different data are used to test the modeling method proposed in this work. These data are from the orebody of a tin mine in Yunnan, China, and are obtained in production and exploration. Based on these data, the original contour polylines can be produced through geological logging. The following experimental figures clearly show the process and the result of single closed-loop modeling and sub-mesh combination.

#### **4. Results and Discussion**  *4.1. Examples*

In this paper, five groups of different data are used to test the modeling method proposed in this work. These data are from the orebody of a tin mine in Yunnan, China, and are obtained in production and exploration. Based on these data, the original contour polylines can be produced through geological logging. The following experimental figures clearly show the process and the result of single closed-loop modeling and submesh combination. Figure 12 shows the processes of closed-loop division, single closed-loop modeling, and sub-mesh combination of the first group of cross-contour polylines. Figure 12a shows the original cross contour polylines of the orebody, each of which is in the same plane within the tolerance range. Then, the contour polylines are cut with the approximate planes. As shown in Figure 12b, each contour polyline is divided and grouped. After that, single closed-loop modeling is carried out on the contour polylines through the Coons surface interpolation, and the results are shown in Figure 12c. It can be seen that every closed loop has been modeled, but the sub-meshes are separate from each other. Finally, the sub-meshes are combined into a complete orebody model using polygon data merging technology, as shown in Figure 12d.

Figure 12 shows the processes of closed-loop division, single closed-loop modeling, and sub-mesh combination of the first group of cross-contour polylines. Figure 12a shows the original cross contour polylines of the orebody, each of which is in the same plane within the tolerance range. Then, the contour polylines are cut with the approximate Figures 13 and 14 show the modeling process on contour polylines with different shapes. It can be seen that the algorithm implemented here can still carry out closedloop division, single closed-loop modeling, and sub-mesh combination on more complex contour lines with good effects.

planes. As shown in Figure 12b, each contour polyline is divided and grouped. After that, single closed-loop modeling is carried out on the contour polylines through the Coons technology, as shown in Figure 12d.

technology, as shown in Figure 12d.

surface interpolation, and the results are shown in Figure 12c. It can be seen that every closed loop has been modeled, but the sub-meshes are separate from each other. Finally,

surface interpolation, and the results are shown in Figure 12c. It can be seen that every closed loop has been modeled, but the sub-meshes are separate from each other. Finally, the sub-meshes are combined into a complete orebody model using polygon data merging

*Minerals* **2022**, *12*, x FOR PEER REVIEW 14 of 20

**Figure 12.** Experimental results of the first set of contour polylines: (**a**) the original contour polylines; (**b**) closed-loop division; (**c**) single closed-loop modeling; (**d**) sub-mesh combination. **Figure 12.** Experimental results of the first set of contour polylines: (**a**) the original contour polylines; (**b**) closed-loop division; (**c**) single closed-loop modeling; (**d**) sub-mesh combination. division, single closed-loop modeling, and sub-mesh combination on more complex contour lines with good effects.

123,660.0 1,640.0 **Figure 13.** Experimental results of the second set of contour polylines: (**a**) the original contour polylines; (**b**) closed-loop division; (**c**) single closed-loop modeling; (**d**) sub-mesh combination. **Figure 13.** Experimental results of the second set of contour polylines: (**a**) the original contour polylines; (**b**) closed-loop division; (**c**) single closed-loop modeling; (**d**) sub-mesh combination.

2,580,020.0

(d)

123,560.0

2,580,120.0

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1,540.0

2,579,920.0

123,560.0

(c)

**Figure 14.** Experimental results of the third set of contour polylines: (**a**) the original contour polylines; (**b**) closed-loop division; (**c**) single closed-loop modeling; (**d**) sub-mesh combination. **Figure 14.** Experimental results of the third set of contour polylines: (**a**) the original contour polylines; (**b**) closed-loop division; (**c**) single closed-loop modeling; (**d**) sub-mesh combination.

Figure 15 shows the modeling process of more complex contour polylines, in which most of the contour polylines do not strictly intersect. Figure 15a shows the original contour polylines. First, search the original intersections of the contour polylines, and the result is shown in Figure 15b. The green points are the original intersections that need no further processing. It can be seen that only a few polylines possess the original intersections. Red points are isolated points that have no other contour polylines within the specified scope. Then, the contour polylines are moved to intersect within the tolerance range. Figure 15c shows the intersections after processing, and most of the contour polylines have intersected with each other. After that, the closed-loop division is carried out based on the intersected contour polylines, and the result is shown in Figure 15d. Based on the closed loops, sub-meshes can be constructed through the Coons surface Figure 15 shows the modeling process of more complex contour polylines, in which most of the contour polylines do not strictly intersect. Figure 15a shows the original contour polylines. First, search the original intersections of the contour polylines, and the result is shown in Figure 15b. The green points are the original intersections that need no further processing. It can be seen that only a few polylines possess the original intersections. Red points are isolated points that have no other contour polylines within the specified scope. Then, the contour polylines are moved to intersect within the tolerance range. Figure 15c shows the intersections after processing, and most of the contour polylines have intersected with each other. After that, the closed-loop division is carried out based on the intersected contour polylines, and the result is shown in Figure 15d. Based on the closed loops, submeshes can be constructed through the Coons surface interpolation. Figure 15e shows the modeling result of every closed loop. The sub-meshes are smooth but split from each other. Finally, the sub-meshes are combined to obtain a complete orebody model, as shown in Figure 15f.

interpolation. Figure 15e shows the modeling result of every closed loop. The sub-meshes are smooth but split from each other. Finally, the sub-meshes are combined to obtain a As shown in Figure 16, the method is applied to complex orebodies with different shapes, and good orebody models are obtained, showing that this method has strong adaptability.

#### complete orebody model, as shown in Figure 15f. *4.2. Discussion*

The above experimental results show that the proposed method can deal with the contour polylines obtained by geological logging in mine production with good effects. Firstly, the constructed model conforms to the geological trend and is smooth in every mesh. Secondly, the closed-loop division can be carried out automatically, which will greatly improve the efficiency of modeling. Thirdly, since modeling between different regions will not affect each other, it can be carried out at the same time, which will greatly reduce the overall modeling time. However, this method still has some limitations that need to be further studied and solved. Firstly, due to the inherent limitations of the Coons technology, the model constructed by Coons surface interpolation is difficult to update,

adaptability.

and it is difficult to take faulting crosscutting into account, which will affect the accuracy of the model. In mine production, many orebodies are cross-cut by faults. For these orebodies, we should choose other modeling methods. Secondly, due to the complexity of the data in actual production, when this method is applied to some complex contour polylines, as shown in Figure 15, it can not complete all closed-loop divisions well, which needs to be specified manually. Finally, before Coons surface interpolation, all regions except simple single-sided regions need to be transformed into four-sided regions, which may affect the final modeling results by introducing numerical instability issues when adding virtual edges and vertexes. Therefore, to further improve the efficiency and accuracy of the modeling of cross-contour polylines, it is necessary to develop a closed-loop division method with higher precision and an n-sided regions expansion method with less effect on interpolation modeling in the future. *Minerals* **2022**, *12*, x FOR PEER REVIEW 16 of 20

**Figure 15.** Experimental results of the fourth set of contour polylines: (**a**) the original contour polylines; (**b**) original intersections; (**c**) processed intersections; (**d**) closed-loop division; (**e**) single closed-loop modeling; (**f**) sub-mesh combination. **Figure 15.** Experimental results of the fourth set of contour polylines: (**a**) the original contour polylines; (**b**) original intersections; (**c**) processed intersections; (**d**) closed-loop division; (**e**) single closed-loop modeling; (**f**) sub-mesh combination.

As shown in Figure 16, the method is applied to complex orebodies with different

shapes, and good orebody models are obtained, showing that this method has strong

**Figure 16.** Experimental results of the fifth set of contour polylines: (**a**–**c**) the orebody models with different complex shapes. **Figure 16.** Experimental results of the fifth set of contour polylines: (**a**–**c**) the orebody models with different complex shapes.
