## 4 Proposed Geometric Similarity Evaluation Methodology
The Frechet distance is often illustrated through the analogy of finding the shortest leash length needed to connect a person and a dog as they each walk along their respective curves [56]. This metric captures both the ordering of points along the curves and the overall path flow; hence, it is well suited for comparing time-series data or trajectory-based shapes. Specifically, the discrete Frechet distance approximation enables efficient computation of this metric in \(O(pq)\) time via a dynamic programming-based algorithm. \[f_{7}=\underset{\tau\in\mathcal{C}}{min}\ \underset{(i,j)\in\tau}{max}\big{\|}y_{1,i }-y_{2,j}\big{\|}. \tag{38}\]  
In Equation (38), \(f_{7}\) is the Frechet distance, which quantifies the overall path-flow similarity between two point sets \(y_{1},y_{2}\in\mathbb{R}^{n\times m}\). Each point set is treated as a trajectory of \(n\) sequentially connected points, where \(y_{1,i}\) and \(y_{2,j}\) denote the \(i\)-th and \(j\)-th points on the corresponding trajectories, respectively. The correspondence between the two sets is defined by an order-preserving path \(\tau=\big{(}(i_{1},j_{1}),(i_{2},j_{2}),\ldots\big{)}\) within the collection \(\mathcal{C}\) of such paths. The distances are computed based on the standard Euclidean norm. The Frechet distance then evaluates the similarity between the trajectories by choosing among all possible paths \(\tau\) the one that minimizes the maximum distance between matched points. Hence,  
Figure 4: Overall process of proposed methodology for evaluating geometric similarity. a smaller \(f_{7}\) value implies that the two trajectories maintain the same ordering and share similar shapes. The Hausdorff distance is a metric that measures the distance of every point in Set A from its nearest neighbor in Set B and then finds the maximum value among these nearest-point-pair distances in both directions. Thus, this term quantifies the largest local dissimilarity between two shapes [57]. Because it does not require explicit point-to-point correspondences, it robustly captures shape mismatches even in the presence of small positional errors. Therefore, it is widely used in image comparison and pattern recognition. Additionally, algorithms that approximate the Hausdorff distance on a binary raster grid have been proposed to efficiently compute the minimum over all possible translations. \[f_{8}=max\left\{ \begin{matrix}sup\limits_{i\in y_{1}p\in y_{2}}\|y_{1,i}-y_{2, j}\|,&sup\limits_{i\in y_{2}p\in y_{1}}\|y_{2,i}-y_{1,j}\|\end{matrix} \right\}. \tag{39}\]  
In Equation (39), \(f_{8}\) represents the Hausdorff distance, which quantifies the maximum local discrepancy between two shapes, i.e., the degree of structural mismatch. The first term reflects the extent to which \(y_{1}\) deviates from \(y_{2}\), while the second term measures the discrepancy in the opposite direction. The maximum of these two values represents the largest bidirectional distance. A smaller \(f_{8}\) indicates that the shapes overlap consistently, whereas a larger value suggests a significant local mismatch (i.e., an outlier). All distances are calculated based on the Euclidean norm. \[y_{5}(y_{1},y_{2})=\eta_{6}f_{6}+\eta_{7}f_{7}+\eta_{8}f_{8}. \tag{40}\]  
As shown in Equation (40), the geometric similarity \((y_{s})\) between two shapes in this study was calculated as a weighted sum of the distance-based metrics from the three perspectives mentioned earlier. This metric was then used to evaluate geometric similarity, as explained in Sections 4.1 and 4.2