## 1 Introduction and Related Work  
In the field of science and especially in engineering, the formulation of mathematical models for the treatment of concrete problems plays a key role. Often, however, the models are characterized by a high degree of complexity, which is why an analytical solution is only possible in rare cases. For this reason, one often has to rely on numerical methods to approximate the solution. A common application is the determination of roots, since these often represent characteristic points in the course of the function. For the numerical approximation of the roots of one-dimensional functions, the bisection method ([1], p. 250), [PERSON]'s method ([2], p. 243) or the method by [PERSON], who combines the secant method ([1], p. 254) with the bisection method in [3], are prominent representatives. On the other hand, the determination of roots of two-dimensional functions is much more complex. Here, the roots form a level set, which is represented as a contour line. In the context of our work, we focus on two-dimensional functions (scalar fields) of the form \(f:\mathbb{R}^{2}\rightarrow\mathbb{R}\) and assume that these are given as a point cloud \(\mathcal{P}\ \in\ \mathbb{R}^{n\times 3}\), \(n\ \geq\ 3\). This has the advantage that even unstructured point clouds, which arise, for example, during the recording of measuring points, can be handled. In the following, we develop a simple and robust function for approximating the zeros of a point cloud with a plane oriented arbitrarily in space in Matlab\({}^{\copyright}\). The literature reveals that different methods are used to determine isolines or isosurfaces, such as the Marching Squares (2D) or Marching Cubes (3D) algorithm developed by [PERSON] and [PERSON] [4]. It is mainly used in computer graphics and image processing. The core of the method is a divide-and-conquer approach, in which the data set at hand is divided into square grid cells and each cell is assigned a defined isovalue. The next step is to check whether the vertices are above or below the isovalue. By assigning a binary value, it is possible to determine which combination of corners of the cell corresponds to the isovalue, whereupon the polyline or polygon can be created. This is performed for all cells, from which the isoline or isosurface can be constructed. The method has been widely studied in research. Thus, [PERSON] presents, in [5], an extension of the method for approximating the enclosed area or volume for use in room planning. [PERSON] and [PERSON] use the Marching Cubes algorithm in [6] to generate planetary terrain in a spherical environment. In [7], [PERSON] et. al extend the Marching Cubes algorithm to ensure topological correctness. Another application can be found in [8]. Here, [PERSON] et al. develop an approach to implement the Marching Cubes algorithm based on edge growth. Closely related to the Marching Cubes algorithm is the Asymptotic Decider algorithm developed by [PERSON] and [PERSON], which generates isosurfaces from a given scalar field and provides topologically correct contours [9]. A modified approach to robust and topologically correct triangulation is developed by [PERSON] in [10]. In this context, [PERSON] and [PERSON] provide a framework based on the Asymptotic Decider algorithm for analyzing uncertainties related to isocontour generation [11]. Other approaches regarding the determination of contour lines include the Octree-based algorithm [12] and the Span Space algorithm [13]. We note that the described algorithms basically aim to divide the considered scalar field into cells or cubes, to which a specific indicator is assigned in order to reconstruct the contour lines based on this information. These algorithms are primarily used in image processing, computer visualization, or medical imaging. In our research, we want to deviate from this approach and instead take an unstructured point cloud into account, using the information provided by the individual points directly.