## 2 Configuration Topography
### Similarity Point Discrimination  
The point degree indicates the number of edges or neighboring vertices connected to it in the topology. As presented in Figure 3, point 1 is a four-degree point, point 2 is a two-degree point, and point 3 is a three-degree point. A path or chain in a topology is an alternating, non-repeating sequence of vertices, and the adjacency matrix expresses the  
Figure 1: Synchronized deployment requirement of three folding wings: (**a**) initial folded state, (**b**) fully deployed state.  
Figure 2: Topography of the particularized eight-bar mechanism: (**a**) Topography \(\alpha\); (**b**) Topography b; (**c**) Topography c; (**d**) Topography d; (**e**) Topography e; (**f**) Topography f; (**g**) Topography g.  
point-to-point connection relationship. The elements of this matrix can be calculated using Equation (1).  
\[a_{ij}=\begin{cases}1&\text{$i$ is adjacent to $j$}\\ 0&\text{others}\end{cases} \tag{1}\]  
In Figure 3, the Arabic numerals (1, 2, 3, etc.) represent the point numbers in the configuration topology diagram.  
Although the shapes and labeling of topologies differ, their point-edge connection relationships remain identical, often resulting in duplicate functional topologies during function point selection. To address the issue of isomorphism in the synthesis process, similarity point identification is performed using the topological quadruple power matrix and distance matrix eigenmodes. This enables accurate selection of function points by analyzing similarity through multiple matrices.  
**(1) Weighting matrix.** In topology, numerical values can be assigned to lines corresponding to joints to represent their characteristics, known as assignments [28]. These values are applied to the edges of the lines, resulting in a weighting matrix. The elements of this matrix must satisfy the conditions outlined in Equation (2).  
\[a_{ij}^{q}=\left\{\begin{array}{ll}1/d&\text{$(i\text{ is adjacent to $j$})$}\\ 0&\text{$(i\text{ and $j$ are not neighbours})$}\end{array}\right. \tag{2}\]  
where \(d\) is the largest point degree of the point \(i,j\).  
**(2) Quadruple power matrix and minimum distance matrix.** The weighting matrix is calculated using quadruple power to obtain the weighted quadruple power matrix. Before calculating the minimum distance matrix, the weighting matrix must first be transformed. The Warshall-Floyd algorithm [29] can then be applied to compute the minimum distance matrix from the processed weighting matrix. The details of the algorithm are not introduced in this paper.  
The resulting quadruple power matrix and minimum distance matrix are sorted in descending order to produce two types of eigenmodes for each vertex. These eigenmodes represent the fixed position of the vertex within the topology and its connection relationships. If two vertices in the same topology represent the same quadruple power and minimum distance eigenmodes, they are considered similar; otherwise, they are distinct.  
Table 1 presents the corresponding quadruple power eigencodes and minimum distance eigencodes for each point in the topology depicted in Figure 3.  
Figure 3: Topography of an eight-bar mechanism with adjacency matrix.  
By comparing the feature codes of each vertex, the similar point groups in the configuration topology in Figure 3 are identified as {[1]; [2; 7]; [3]; [4]; [5]; [6]; [8]}. Identifying these similar points is crucial for selecting functional points in the topology. Similarly, Table 2 presents the results of similarity point identification for the seven eight-bar topologies.