# Inspiring Designers' Innovative Thinking: An Evolutionary Design Method for Product Forms
## 2 Modeling of Product Form Evolution Design
### Extraction of Product Form Features
#### 2.1.2 Quantification of Product Form Features  
In the cognitive study of curves, changes in the curve's curvatures are perceivable. The product form contour, as a kind of curve, contains rich information. In information theory, information entropy is commonly used to quantify information. Therefore, [PERSON] proposed a curve information quantization method [28]. This method considers a curve as a source of information and the change in the curvature of the curve as a source symbol, and uses curvature entropy to characterize the shape information of the curved contour. This indicates that humans are capable of discerning alterations in curvature. The curvature change of a node in a curve is influenced by the curvature of the preceding node. To address this, [PERSON] et al. enhanced the curvature entropy through the introduction of the Markov process, defining the quadratic curvature entropy and delineating its application methodology [29]. In this study [29], the quadratic curvature entropy was found to be more closely aligned with human cognitive evaluation than curvature entropy, as evidenced by experimental analysis. Furthermore, the applicability of quadratic curvature entropy in product forms was substantiated by a case study examining the side view of an automobile. Therefore, quadratic curvature entropy has been gradually applied to quantitative research on product form information. For example, quadratic curvature entropy was adopted in [30] to construct a macroscopic form information evaluation system to evaluate products. Quadratic curvature entropy was adopted in [31] to quantify the complexity of information about product forms for esthetic form generation. Therefore, this study applied quadratic curvature entropy to quantify the amount of information in product form curves to provide data support for studying the evolution of product form features.  
Firstly, the curve extraction method was used to obtain the curve contour of the design element \(T_{e}\) (\(w\)) in Table 1. Then, as in [29], this study quantified the abovementioned form design element features by considering the curve contour as the information source and the quadratic curvature entropy value as the information of the product form feature.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Design Element** & \multicolumn{4}{c}{**Element of Form Design through the Generations**} \\ \hline \(T_{1}\) & \(T_{1}\) (1) & \(T_{1}\) (2) & \(\cdots\) & \(T_{1}\) (\(w\)) \\ \(T_{2}\) & \(T_{2}\) (1) & \(T_{2}\) (2) & \(\cdots\) & \(T_{2}\) (\(w\)) \\ \(T_{3}\) & \(T_{3}\) (1) & \(T_{3}\) (2) & \(\cdots\) & \(T_{3}\) (\(w\)) \\ \(\cdots\) & & & & \\ \(T_{e}\) & \(T_{e}\) (1) & \(T_{e}\) (2) & \(\cdots\) & \(T_{e}\) (\(w\)) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Deconstruction of product-form feature elements.  
First, the curve contour was divided into \(N\) equivalent curve cells using sampling points (Figure 1a), and the curvature \(\rho_{n}\) was calculated for each sampling point (Figure 1b). Second, the value \(O=\rho_{n}/\sigma\) of the curvature and its standard deviation were calculated, where \(\sigma\) is the standard deviation of the curvature \(\rho_{n}\), and the set of source symbols was constructed with the value of \(O\) (Figure 1c). Then, the range of values of \(O\) was determined, the number of source symbols \(V\) and the number of source symbols \(d\) that form a state, and construct the state \(S_{i}\). Finally, \(q_{i}\) (the probability of the occurrence of state \(S_{i}\)) and \(q_{ij}\) (the probability of the transfer of the source symbol \(O\) from state \(S_{j}\) to state \(S_{i}\)) were calculated (Figure 1d). According to information theory, the above parameters can be used to calculate the quadratic curvature entropy \(H_{\text{QC}}\) of the curve, as illustrated in Equation (1).  
\[H_{\text{QC}}=-\frac{1}{\log_{2}V}\sum_{i=1}^{V^{d}}\sum_{j}^{V}q_{i}q_{ij} \log_{2}q_{ij} \tag{1}\]