## 3 Methods
### Construction of the Shape Complexity Model  
[PERSON] et al. initially selected 27 shape variables, and 9 indexes were retained by a stepwise multiple linear regression method for use in constructing the shape complexity model. In this way, the relationship between the shape complexity of the polygons and the computational performance of a spatial overlay analysis algorithm was established. The model shows that the number of polygon vertices is the most important factor affecting the overlay analysis, with its coefficient in the complexity model reaching 0.97 [29]. In multiple regression analysis, as the number of independent variables increases, the model performance may improve, but this change is accompanied by greater computational overhead and obvious variable redundancy. Therefore, while ensuring the accuracy of the model, minimizing the redundancy of the variables can effectively reduce the time needed to calculate the complexity index. This paper is based on reference [29], retains the index of the number of vertices with the greatest impact, and selects 5 indexes that take less time to calculate. The final indexes are as follows: Number of vertices (NOV), Amplitude of the vibration (Ampl), Number of holes (NOH), Average nearest neighbor (ANN), Concavity (Conv), and Equivalent rectangular index (ER). Detailed descriptions of these shape indexes are shown in Table 3.  
In this paper, 100,000 polygons are randomly selected from the land use patch data for China and the 2009 land use data for several counties in Ningxia for use as target polygons, and a typical complex polygon is selected as a clipping polygon. For each target polygon, an overlay analysis operation is performed with the selected clipping polygon, with the above six indicators as independent variables and the running time of the polygon overlay analysis algorithm as the dependent variable. The stepwise multiple linear regression method in the IBM SPSS Statistics 27.0.1 software is used to construct the shape complexity models for the intersection operator and difference operator of the Vatti algorithm. The model equations are (1) and (2):  
\[C_{1}=0.971x_{1}+0.059x_{4}+0.018x_{5}+0.034x_{6} \tag{1}\]  
\[C_{2}=0.97x_{1}-0.004x_{2}+0.037x_{3}+0.06x_{4}+0.024x_{5}+0.037x_{6} \tag{2}\]  
Figure 2: Flow chart of the Vatti algorithm. where \(C_{1}\) is the shape complexity of the intersection operator of the Vatti algorithm, \(C2\) is the shape complexity of the difference operator, \(x_{1}\) is the Number of vertices, \(x_{2}\) is the Amplitude of the vibration, \(x_{3}\) is the Number of holes, \(x_{4}\) is the Average nearest neighbor, \(x_{5}\) is the Concavity, and \(x_{6}\) is the Equivalent rectangular index. Regression analysis is widely used in the fields of econometrics, psychology, and sociology. The coefficient of determination (\(R^{2}\)) is mainly used to evaluate the quality of a model. The closer \(R^{2}\) is to 1, the better the model fitting effect and the higher the consistency. In this paper, shape complexity models are established based on the intersection operator and difference operator of the Vatti algorithm. The coefficients of determination of both shape complexity models based on the intersection operator and the difference operator of the Vatti algorithm are greater than 0.89, indicating that the shape complexity of the input polygon is highly correlated with the calculation time. Specifically, the shape characteristics of the input polygon can explain more than 89% of the calculation time of the algorithm. The model summary is shown in Table 4.  
By means of the constructed shape complexity models, the shape complexity of each polygon subjected to clipping can be calculated. In Figure 3, the calculation times for 100,000 polygons are plotted versus their shape complexities, as evaluated based on the intersection operator and difference operator of the Vatti algorithm in the form of scatter plots. Linear fits to these data are also shown.