# A Comprehensive Survey of Isocontouring Methods: Applications, Limitations and Perspectives
## 3 Tessellation-Based Algorithms
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Authors** & **Year** & **Structured** & **Unstructured** & **Quality** & **2D** & **3D** \\ \hline \hline [PERSON] et al. & [2] & 1987 & \(\bullet\) & & visual inspection & \(\bullet\) \\ [PERSON] et al. & [4] & 1990 & \(\bullet\) & & visual inspection & \(\bullet\) \\ [PERSON] et al. & [5] & 1991 & \(\bullet\) & & & \(\bullet\) \\ [PERSON] & [7] & 1994 & \(\bullet\) & & comparison with MC and Greedy & \(\bullet\) \\ [PERSON] & [6] & 1995 & \(\bullet\) & & & \(\bullet\) \\ [PERSON] et al. & [3] & 1996 & \(\bullet\) & & visual inspection & \(\bullet\) \\ [PERSON] & [10] & 2004 & \(\bullet\) & & visual inspection & \(\bullet\) \\ [PERSON] et al. & [15] & 2008 & \(\bullet\) & & calculation time and memory requirement & \(\bullet\) \\ [PERSON], [PERSON] et al. & [9] & 2010 & \(\bullet\) & & calculation time and memory requirement & \(\bullet\) \\ [PERSON] et al. & [13] & 2013 & \(\bullet\) & & comparison with original MC & \(\bullet\) \\ [PERSON] et al. & [18] & 2015 & \(\bullet\) & & visual inspection & \(\bullet\) \\ [PERSON], [PERSON] et al. & [11] & 2016 & \(\bullet\) & & comparison with original MC & \(\bullet\) \\ [PERSON], [PERSON] et al. & [17] & 2017 & \(\bullet\) & & comparison with original MC & \(\bullet\) \\ [PERSON], [PERSON] et al. & [14] & 2018 & \(\bullet\) & & comparison with original MC & \(\bullet\) \\ [PERSON], [PERSON] et al. & [16] & 2019 & \(\bullet\) & & comparison with original MC & \(\bullet\) \\ [PERSON], [PERSON] et al. & [8] & 2021 & \(\bullet\) & & calculation time and memory requirement & \(\bullet\) \\ [PERSON], [PERSON] et al. & [12] & 2021 & \(\bullet\) & & calculation time and deviation & \(\bullet\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of marching cubes algorithms. single tetrahedron. Thus, the entire surface is wrapped with triangles. [PERSON] and [PERSON] have extended the algorithm in [23; 24] in terms of surface simplification and the necessary memory capacity. [PERSON] et al. clarify in [25] that, despite the decomposition of the cubes into tetrahedra, the ambiguity problem has not yet been solved. Therefore, the authors propose a method of determining the intersection points and a criterion to test the intersection between the isosurface and the edge of the tetrahedron, which are based on the assumption that the function is linear along the edge of a cube. However, the authors deduce in [25] that there is a quadratic variation in the function value \(f(t)\) along the triangle edge \(t\), as can be seen in Equation (4). \[f(t)=c_{0}+c_{1}t+c_{2}t^{2},\quad t\in[0,1] \tag{4}\]  
The values for \(c_{0}\), \(c_{1}\) and \(c_{2}\) are determined by the function values at the vertices. If one or both zeros of the function  
\[c_{2}t^{2}+c_{1}t+c_{0}-C=0\]  
are an element of the interval \([0,1]\), the isocontour with the isovalue \(C\) intersects the triangle edge. In addition, the connection of intersection points in the tetrahedra to construct the polygons and the triangulation of the polygons are discussed. They show that, with the proposed method, the isosurfaces are independent of the exact distribution of the tetrahedra resulting from the decomposition of the cubes. In [26], [PERSON] and [PERSON] present an algorithm that computes a triangulated surface that separates a collection of data points segmented into several different classes instead of only two. The algorithm can be used for both structured, rectilinear and scattered data sets. For tetrahedralization, each cell is decomposed into five or six tetrahedra for structured data sets, while tetrahedralization is assumed for scattered data sets. However, the algorithm assumes that the data are already segmented. Authors [PERSON] et al. describe, in [27], a new algorithm for the reconstruction of material interfaces from data sets with volume fractions.