**1. Introduction**

Three-dimensional geological modeling and visualization is one of the hot topics in the field of geosciences [1]. As a geological body with high economic value, orebody modeling is also one of the current research hot topics and is of great significance for promoting mine digitization and improving mining efficiency [2]. In 2002, Mallet [3] proposed a new method called discrete smooth interpolation (DSI), based on which a number of orebody modeling works have been published [4–7]. Additionally, some commercial software [8] has been developed using different technologies including triangulate surfaces, radial functions, and parametric surfaces [9]. However, due to the complex shape of the orebody, much research work still needs to be done in orebody modeling. Our research is dedicated to developing the software for modeling orebody using small triangles based on the Coons surface [10].

We can often obtain the cross-contour polylines of planes and sections of orebodies through geological logging in the geological exploration and production exploration of mines, especially of some precious metal mines with flat orebody shapes. These contour polylines are interlaced with complex shapes. Therefore, if the explicit modeling method based on the contour stitching method is used, the model will be constructed with poor quality and low efficiency [11]. Modeling orebodies with high speed and great quality is of great significance to improving the production efficiency of mines [12]. The orebody

**Citation:** Wu, Z.; Bi, L.; Zhong, D.; Zhang, J.; Tang, Q.; Jia, M. Orebody Modeling Method Based on the Coons Surface Interpolation. *Minerals* **2022**, *12*, 997. https://doi.org/ 10.3390/min12080997

Academic Editor: Behnam Sadeghi

Received: 1 July 2022 Accepted: 3 August 2022 Published: 6 August 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

modeling method we propose based on Coons surface interpolation can automatically divide the contour polylines into closed loops and interpolate and model the formed n-sided regions; then, the constructed sub-meshes will be combined without manual intervention. Based on the above steps, the complete orebody model can be quickly constructed.

### *1.1. Related Research Work*

Based on the above analysis, we transform the 3D modeling of the orebody into the Coons surface interpolation of the interpreted contour polylines. Therefore, we will briefly summarize the research work on free-form surface reconstruction, contour interpolation modeling, and orebody modeling.

#### 1.1.1. Free-Form Surface Reconstruction

Free-form surface reconstruction is a key technology of orebody 3D modeling [13–15]. In practical application, the commonly used surface reconstruction methods include implicit function surface, triangular Bezier surface, B-spline surface, NURBS surface, Coons surface, polygon model, and so on.

The implicit function surface [16–18] refers to the surface represented by equation *F*(*x*, *y*, *z*) = 0, which can represent quadric surfaces such as spherical surfaces, cylindrical surfaces, and more complex surfaces. When reconstructing the surface, the implicit function equations are constructed using the existing data, and then the surface model can be generated by obtaining isosurfaces through the Marching Cube method or other methods. Though using the implicit function to reconstruct a surface has the advantages of small calculation and easy solution, there are also some problems such as unclear geometric meaning. In the future, a great amount of research work needs to be carried out to make extensive use of implicit modeling.

In 1982, Farin [19] proposed the method of constructing the triangular Bezier surface based on the triangulation of scattered data. This surface is explicit and *C* 1 continuous with flexible construction. Using the triangular Bezier surface for surface reconstruction requires less manual intervention and is easy to automate. However, it also has some disadvantages, such as the insufficient ability for surface modification and poor controllability.

The B-spline curve and surface [20–22] have many excellent properties, such as geometric invariance, convex hull, convexity preservation, and local support, which lead to the powerful function of representing and designing free-form curves and surfaces. It is a commonly used free-form surface suitable for engineering shape design, but it cannot be well applied to represent and design elementary curves and surfaces. Therefore, researchers proposed the NURBS surface method [23–25] for surface reconstruction, which both has the powerful function of representing and designing free-form curves and surfaces like B-spline and can accurately represent quadric arcs and quadric surfaces. The unification of analytical geometry and free-form curves and surfaces is well realized by this method. Due to the introduction of a weight factor and a manipulation control vertex, it provides sufficient flexibility for various shape designs. However, the parameterization effect will become very poor with an inappropriate weight factor. In addition, there are still many problems to be further solved in the calculation of NURBS surface intersection.

In 1964, Professor S. A. Coons of the Massachusetts Institute of Technology (MIT) proposed a general method to describe surfaces, the Coons surfaces method [10], which has unique advantages in engineering applications. Given the boundary conditions, Coons patches of the corresponding degree can be constructed. Therefore, in theory, the Coons method can be used to construct arbitrary order patches through boundary constraints [26,27], which makes Coons surfaces show strong advantages in representing and designing surfaces. However, in the Coons surface method, it is difficult to control the surface shape. When the boundary curve is determined, the shape can only be modified by changing the torsion vector [28].

The most commonly used polygon model method is the triangle model method based on triangulation. The triangulation technology in the plane has matured [29], and great research progress has been made in the triangulation of surface data and tetrahedrons of spatial scattered data [30,31]. The characteristics of the polygon model are: (1) adjacent surface patches can only be *C* 0 continuous and (2) data amounts are large. When there needs high accuracy of a model, a large number of polygons need to be used to represent the surface approximately, which has a huge amount of data. As a result, this method is not suitable for constructing complex models with high accuracy. Therefore, in this paper, the polygon model method is used only in the modeling of simple single-sided regions.

## 1.1.2. Contour Interpolation Modeling

Visualization based on contour interpolation refers to the modeling of three-dimensional entities using a sequence of two-dimensional contours, which is an important research direction in scientific computing visualization. According to the number of contour polylines of two adjacent layers, it can be divided into single contour modeling and multi-contour modeling.

Single contour modeling refers to the three-dimensional model reconstruction of the contour polylines when there is only one contour polyline in each layer on two adjacent planes, which is relatively simple. Researchers have proposed many optimization methods for single contour modeling, such as the minimum surface product method [32], the maximum volume method [33] based on the global search strategy, the shortest diagonal method [34], and the synchronous advance method of adjacent contours [35] based on local calculation and decision.

Multi-contour modeling refers to 3D model reconstruction when there are multiple contour polylines on two adjacent planes, which difficulty is the correspondence and branch processing between contour polylines. In terms of the correspondence of contour polylines, Meyers et al. [36] proposed the minimum spanning tree method, but this method still has some limitations. In some special cases, the contour polylines cannot correctly correspond, and subsequent manual processing is required. For the branching problem between contour lines, Ekoule et al. [37] proposed a method for constructing the middle contour polyline in the middle of the two adjacent planes of contour polylines. This method transforms the branching problem into the connection problem between a series of single contour polylines. However, in some cases, this method will leave holes on the constructed model surface.

Jones et al. [38] proposed an isosurface construction method using volume data, which requires that contour polylines have simple shapes and be completely closed. There is no need to judge the corresponding relationship and branch relationship when using this method for surface reconstruction, and it is suitable for both convex and non-convex contour polylines. However, it also has the disadvantage of a large amount of calculation. Additionally, Zhong et al. [39] proposed a method of reconstructing the 3D orebody by cross-section contour polylines, which allows for adding geometric constraints and can easily control the shape of the model, but many discontinuous parts may be generated after reconstruction if the data are relatively sparse. Moreover, Wu et al. [40] proposed a 3D orebody modeling method based on the normal estimation of cross-contour polylines. First, the normals of the cross-contour polylines will be estimated, based on which the radial basis function will be used for interpolating and modeling. This method improves the modeling automation of contour polylines, but it requires strict intersection between contour polylines, and they can be completely broken at intersections.
