## 2 Flattening-Based Morphing Method  
Before explaining the flattening-based morphing method, the symbols and terminologies used in the current study are described. Let us consider a baseline model whose surface is represented as triangular mesh and define this triangular mesh as \(M\). When the mesh \(M\) is mapped to the 2D parametric space by a mapping function \(\zeta\), the parameterized mesh of \(M\) is denoted by \(U\). Then, the locations of the \(i\)-th vertex of \(M\) and \(U\) can be defined as \(\mathbf{x}_{i}\in\mathbb{R}^{3}\) and \(\mathbf{u}_{i}\in\mathbb{R}^{2}\), respectively. In addition, the one-ring neighborhood of the \(i\)-th vertex, meaning the vertices that are connected to the vertex \(i\), is defined as \(N(i)\) and the edge between the vertices \(i\) and \(j\) is defined as \(e_{ij}\). When baseline models are morphed, if their meshes are topologically the same (meaning the number of vertices and their connectivity are identical to each other), establishing one-to-one correspondences among their vertices becomes straightforward, and morphing can be easily implemented by linear interpolations among corresponding sets of baseline meshes' vertices. However, there is no guarantee that the topologically equivalent baseline meshes will be provided. Rather, it is more reasonable to claim that the process to match the topology of the baseline meshes is required to perform morphing. In the flattening-based morphing method, this task is implemented in the parametric space because creating the topology of the baseline models is much easier in the planar space than in 3D. For a more detailed explanation, suppose two meshes (which will be referred to as Baseline models 1 and 2) are given and we want to morph them to create a new design. The first step of the proposed method is to distribute the anchor points on the baseline models. The purpose of defining anchor points is to obtain proper connections among the baseline models in the morphing process. To create a reasonably shaped new design, the geometric features of the baseline models should be matched during the morphing process. For example, when two aircraft are morphed, the leading edge of Baseline model 1 should be morphed with that of Baseline model 2. Similarly, the cockpit of Baseline model 1 should be morphed with the corresponding part of Baseline model 2. These proper connections can be secured by linking the corresponding anchor points between the baseline models. The locations of the anchor points are manually determined by a user, but it is recommended to place the anchor points where the baseline objects' geometry features are varied, such as the ends of leading edge line and trailing edge line of the BWBs. Once the anchor points have been defined, the baseline meshes are mapped to the parametric space for the remeshing. In the proposed method, a conformal mapping conceptually indicating the minimization of angle distortion between \(M\) and \(U\) is used. With a triangular mesh, the conformal mapping can be constituted by using the cotangent Laplace matrix, which can be derived by minimizing the energy over \(N(i)\)[42; 43; 44; 45]. If the energy saved in the mesh \(U\) over \(N(i)\) is defined as \(E\), it can be computed as  
\[E=\sum_{j\in N(i)}(cot\alpha_{ij}+cot\beta_{ij})|\textbf{u}_{i}-\textbf{u}_{j}| ^{2}, \tag{1}\]  
where \(\alpha_{ij}\) and \(\beta_{ij}\) are the opposite angles of the edge \(e_{ij}\) of \(M\) and \(|\textbf{u}_{i}-\textbf{u}_{j}|\) is the length of \(e_{ij}\) of \(U\) (see Figure 1). Minimizing \(E\) with respect to \(\textbf{u}_{i}\) is required to find a proper mapping between \(M\) and \(U\), which leads to the following equation:  
\[\sum_{j\in N(i)}(cot\alpha_{ij}+cot\beta_{ij})(\textbf{u}_{i}-\textbf{u}_{j} )=0. \tag{2}\]  
This equation can be expressed as  
\[\textbf{L}\textbf{b}=\textbf{c} \tag{3}\]  
in a matrix form where **L** is the cotangent Laplace matrix of \(M\) computed by the left side of Equation (2), **b** represents the 2D coordinate of \(U\), and \(c\) is the zero column vector.