## 2 Flattening-Based Morphing Method
If the boundary vertices of \(U\) are defined as \(\tilde{\textbf{u}}\) and the arclength along the boundary of \(M\) and its total length are defined as \(l\) and \(L\), respectively, solving Equation (2) with the subsequent boundary condition  
\[\tilde{\textbf{u}}=\left(\cos(2\pi l/L),\ \sin(2\pi l/L)\right) \tag{4}\]  
allows us to parameterize the surface of \(M\) (defined in the physical space) into a unit circle. Figure 1: The vertex \(i\) and its one-ring neighborhood vertices \(N(i)\) are represented as yellow and blue points. The edge between the vertices \(i\) and \(j\) is denoted by \(e_{ij}\), and two opposite angles of \(e_{ij}\) are defined as \(\alpha_{ij}\) and \(\beta_{ij}\). However, when the baseline meshes are parameterized to the planar space, an anchor point should be located at the same position with its counterparts. To accomplish this, modification of the boundary condition is needed. When the anchor point is not located on the boundary, matching the position of the anchor point in the parametric space to its counterparts can be easily performed by replacing a row of the \(\mathbf{L}\) and column vector \(\mathbf{b}\) to the Dirichlet boundary condition [46]. However, when the anchor points are located on the boundary, a special function specifying the locations of \(\tilde{\mathbf{u}}\) is needed. For example, let us consider the case that all anchor points are located on the boundary. Then, let us define the 2D location of the \(i\)-th anchor point in the parametric space and its 3D location in the physical space as \(\mathbf{v}_{i}\) and \(\mathbf{y}_{i}\), respectively. Let us also define the angle measured in the parametric space between \(\mathbf{v}_{i}\) and the horizontal axis in the counter-clockwise direction as \(\phi_{i}\). In addition, the \(j\)-th boundary point in the parametric space and its 3D position in the physical space are defined as \(\tilde{\mathbf{u}}_{j}\) and \(\tilde{\mathbf{x}}_{j}\), respectively. Once the angle of the first anchor point (defined as \(\mathbf{v}_{1}\)) is set to zero, \(\phi_{i}\) can be computed as  
\[\phi_{i}=\frac{2\pi l_{i,1}}{L}, \tag{5}\]  
where \(l_{i,1}\) is the edge length from \(\mathbf{y}_{1}\) to \(\mathbf{y}_{i}\). Then, if the azimuthal angle of \(\tilde{\mathbf{u}}_{j}\) that is located between \(\mathbf{v}_{i}\) and \(\mathbf{v}_{i+1}\) is defined as \(\theta_{j}\), this can be computed as  
\[\theta_{j}=\phi_{i}+\frac{s_{i,j}}{l_{i,j+1}}\times(\phi_{i+1}-\phi_{i}) \tag{6}\]  
where \(s_{i,j}\) is the edge length between \(\mathbf{y}_{i}\) and \(\tilde{\mathbf{x}}_{j}\) and, \(l_{i,j+1}\) is the edge length between \(\mathbf{y}_{i}\) and \(\mathbf{y}_{i+1}\) (see Figure 2). After computing \(\theta_{j}\), the boundary points in the parametric space can be obtained as  
\[\tilde{\mathbf{u}}_{j}=\left(\text{cos}\theta_{j},\,\text{sin}\theta_{j}\right) \tag{7}\]  
Figure 2: Notations related to the boundary points. The \(j\)-th boundary point of \(U\) and \(i\)-th anchor point located on the boundary are defined as \(\tilde{\mathbf{u}}_{j}\) and \(\mathbf{v}_{i}\), respectively. Their azimuthal angles are defined as \(\theta_{j}\) and \(\phi_{i}\), respectively, in the current study. To compute \(\theta_{j}\), the values of \(\phi_{i}\) need to be determined first. For this, one of the baseline models (let us use the \(k\)-th baseline model) is mapped to the parametric space with the boundary parameterization presented in Equation (4). Based on this parameterization, the azimuthal angle of the mapped \(k\)-th baseline mesh's anchor points can be obtained from their positions. As the anchor points should be located at the same positions as their counterparts, the anchor points of other baseline models should be placed at the same locations as those of the \(k\)-th baseline model. This consequently leads to defining the values of \(\phi_{i}\) in Equation (6), allowing us to designate the positions of the boundary points of the other baseline models in the parametric space.