The leaf is the primary photosynthetic organ in most plants (Wright et al., 2004), and its morphology significantly influences plant growth and nutrient transport (Daas-Ghrib et al., 2010). As a result, an increasing number of mathematical models, such as the superellipse equation, have been developed to capture the geometric shape of leaves (Gielis, 2003). Bamboo, an essential component of ecosystems, is widely distributed in tropical and subtropical regions. Most bamboo leaves exhibit a long lanceolate shape, posing challenges in finding suitable mathematical models for accurate shape quantification. Fortunately, a polar coordinate equation was proposed by Gielis (2003), referred to as the Gielis equation hereinafter, which can effectively describe the two-dimensional contours of bamboo leaves, offering a promising approach for geometric modeling (Lin et al., 2016; Shi et al., 2018; Yao et al., 2022).

Gielis (2003) extended the superellipse equation to model a wide range of geometric shapes found in plants, which is usually reparameterized in the following form (Shi et al., 2020; Tian et al., 2020):

where r and θ represent the polar radius and polar angle, respectively; a, k, n ₁, n ₂ and n ₃ are parameters to be fitted; and m is a positive integer that determines the number of angles of the Gielis curve within the interval [0, 2π) (Wang et al., 2022a). In recent years, various studies have demonstrated the validity of the Gielis equation for describing actual biological geometries, e.g., leaf shapes (Lin et al., 2016; Shi et al., 2018, 2019; Yao et al., 2022), planar projections of seed and fruit (Tian et al., 2020; Yao et al., 2024), geometries of the outer rims of corolla tubes (Wang et al., 2022b), and shapes of some sea stars (Shi et al., 2020), as well as egg shapes of birds (Shi et al., 2022a). Particularly, bamboo leaves—with their easy accessibility and clear morphological boundaries—have become a preferred material for validating the Gielis equation in studies of natural geometries (Shi et al., 2015a; Yao et al., 2022). Shi et al. (2015a) employed a simplified Gielis equation with two parameters to describe the shape of bamboo leaves, in which one parameter represented the overall ratio of leaf width to leaf length. Yao et al. (2022) compared a three-parameter Gielis equation with a two-parameter version using leaf boundary coordinate data from six bamboo species within the same genus, all characterized by distinct long lanceolate leaves. Their study aimed to determine whether the three-parameter Gielis equation could enhance the model’s fitting accuracy for bamboo leaf shapes.

However, previous researches on model evaluation have primarily focused on assessing goodness of fit (e.g., the coefficient of determination) or examining the trade-off between goodness of fit and model complexity (e.g., the Akaike information criterion) (Shi et al., 2015a; Lin et al., 2016; Yao et al., 2022). Despite its potential to offer valuable insights into plant leaf formation mechanisms, the nonlinearity of the Gielis equation has remained largely unexplored in terms of quantification and comparison using relative curvature measures of nonlinearity. In fact, relative curvature measures of nonlinearity offer a more comprehensive evaluation of the nonlinear behavior of models such as the Gielis equation, providing insights beyond traditional criteria like the coefficient of determination or the Akaike information criterion. While conventional metrics primarily assess how well a model fits the data, they fail to capture the intrinsic nonlinearity of the model and its potential impact across varying datasets or conditions. However, relative curvature measures quantify how closely a nonlinear regression model approximates linear behavior (Bates and Watts, 1980), which is particularly relevant for Gielis curves derived from the Gielis equation, where nonlinearity is an inherent characteristic. By incorporating these nonlinear measures, studies can ensure that selected models not only achieve better fit to the data but also offer a deeper understanding of their structural properties. This makes them particularly valuable for comparing nonlinear models, especially in ecological and biological research, where nonlinearity is a fundamental feature.

To systematically investigate the intrinsic nonlinearity of the Gielis equation, we employed three simplified versions (i.e., the four-parameter, three-parameter, and two-parameter Gielis equations) to model leaf contours of two bamboo species within the same genus, both exhibiting characteristic lanceolate morphology. Each species had a dataset of more than 250 leaves, ensuring robust statistical analysis. Model performance was evaluated using root-mean-square error, the Akaike information criterion, and relative curvature measures of nonlinearity to identify which of the three nonlinear models best captured the geometric properties of bamboo leaves. This study aims to validate the effectiveness of relative curvature measures of nonlinearity in nonlinear regression analysis and to introduce a novel approach for assessing the Gielis equation’s suitability in describing the geometries of natural plants. Our work establishes curvature analysis as a vital complement to conventional model selection criteria in plant morphometrics.

The three models (i.e., SGE-1, SGE-2, and SGE-3) generally provided effective representations to the boundary of bamboo leaves for both species ( Supplementary Tables S3–S5 in the Supplementary Materials ). Figure 2 illustrates the fitting results of the leaf profile using the three models for two leaf examples as intuitively shown in Figure 1 . For species of I. decorus, the results of Wilcoxon signed rank test for every two models, conducted at a significance level of 0.05, revealed that SGE-3 exhibited significantly highest RMSE values than other two models. While there was no significant difference in RMSE values derived from SGE-1 and SGE-2 ( Figure 3 ; Supplementary Table S6 ). For species of I. longiauritus, the results of Wilcoxon signed rank test for every two models indicated that SGE-3 exhibited significantly highest RMSE values and SGE-1 exhibited significantly lowest RMSE values ( Figure 3 ; Supplementary Table S6 ). These findings suggest that SGE-1 demonstrated the best goodness of fit for both species.

Furthermore, the Wilcoxon signed rank tests revealed that SGE-3 had the highest AIC values among the three models for the empirical data of both species at a significance level of 0.05 ( Figure 4 ; Supplementary Table S7 ). And for species of I. decorus, there was no significant difference in AIC values derived from SGE-1 and SGE-2; for species of I. longiauritus, SGE-1 exhibited significantly lower AIC values than SGE-2 ( Figure 4 ; Supplementary Table S7 ). These results indicate that SGE-1 outperformed the other models in nonlinear regression by achieving a favorable trade-off between model structural complexity and goodness of fit.

The overall nonlinearity of the nonlinear regression models was evaluated by the root-mean-square relative curvatures, i.e., γ RMS N , γ RMS T , and K_c . For I. decorus leaf data across the three models, 99.20% (SGE-1), 100% (SGE-2), and 97.86% (SGE-3) of the 254 leaves had γ RMS N values smaller than the corresponding K_c . Additionally, the proportions of γ RMS T values less than the corresponding K_c were 45.60% (SGE-1), 58.20% (SGE-2), and 88.03% (SGE-3) ( Figure 5A ; Supplementary Tables S3-S5 ). For I. longiauritus leaf data, all of the γ RMS N values were less than the corresponding K_c across the three models. The proportions of γ RMS T values smaller than the corresponding K_c were 63.20% (SGE-1), 68.80% (SGE-2), and 98.80% (SGE-3) ( Figure 5B ; Supplementary Tables S3–S5 ).

These results indicated that SGE-3 exhibited the best linear approximation among the three models, while the SGE-1 showed the worst performance in linear approximation. Notably, all three models demonstrated exceptional adherence to the planar assumption, with over 97% of γ RMS N values being smaller than the corresponding K_c in both species. Regarding the uniform coordinate assumption, the SGE-3 emerged as the most satisfactory among all models, as evidenced by over 88% of γ RMS T values being less than the corresponding K_c in both species.

In terms of individual parameter-level nonlinear behavior, examining the percentage bias (P_b ) of parameters reveals insightful findings. For SGE-1, the absolute values of P_b for a, n ₁, n ₂, and n ₃ were smaller than 1% in 2.40%, 29.20%, 61.60%, and 59.20% of cases, respectively, in the data of I. decorus leaves. Correspondingly, these proportions were 9.20%, 40.80%, 67.20%, and 66.40% for the data of I. longiauritus leaves ( Figure 6A ; Supplementary Table S3 ). Moving to SGE-2, 7.79%, 14.75%, and 42.62% of the absolute values of P_b for a, n ₁, and n ₂ were below 1% for I. decorus, and these figures were 18.00%, 16.80%, and 40.40% for I. longiauritus ( Figure 6B ; Supplementary Table S4 ). Regarding SGE-3, 94.44%, and 89.74% of the absolute values of P_b for a, and n ₁, respectively, were below 1% in I. decorus leaves data, and 99.60%, and 99.20% in I. longiauritus leaves data ( Figure 6C ; Supplementary Table S5 ).

These results underscored that SGE-3 exhibited the best close-to-linear behavior among the three models, with over 89% of the absolute values of P_b for each parameter being smaller than 1% in both species. SGE-1 demonstrates relatively good close-to-linear behavior, except for parameter a, where <10% of the absolute values of P_b are less than 1%. However, SGE-2 performs poorly, with three parameters (a, n ₁, and n ₂) having less than 43% of their absolute values of P_b below 1% for both datasets.

The Gielis equation, originating from the superellipse formulation (Lamé, 1818), has demonstrated remarkable versatility in modeling diverse biological morphologies. Previous applications include: (1) simplified versions for characterizing tree ring cross-sections in conifers (Shi et al., 2015a, 2015b) and leaf boundaries across 46 bamboo species (Lin et al., 2016); (2) modified twin versions showing superior performance in describing the morphologies of some sea stars (Shi et al., 2020); and (3) successful applications in quantifying seed projections in Ginkgo biloba cultivars (Tian et al., 2020) and fruit geometries in Koelreuteria paniculata (Li et al., 2022). Notably, Wang et al. (2022b) extended its utility to corolla tube geometries in Vinca major. These collective findings underscore the equation’s adaptability, though further validation across morphologically diverse specimens within single taxa remains valuable.

Traditional model evaluation metrics (e.g., the goodness of fit and the Akaike information criterion) face limitations when assessing complex nonlinear models with multiple parameters (Ratkowsky and Reddy, 2017). Beyond mere fitting accuracy, an ideal nonlinear model should ensure parameter estimators exhibit close-to-linear behavior including unbiasedness, normality, and minimum variance (Ratkowsky, 1990; Ratkowsky and Reddy, 2017). Our comprehensive evaluation framework addresses these requirements through simultaneous consideration of both fitting performance and intrinsic nonlinearity characteristics.

Our analysis reveals distinct trade-offs among the three nonlinear models scrutinized in this study. We found that SGE-1 demonstrated superior fitting performance, as illustrated in Figure 3 . Meanwhile, SGE-1 exhibited a robust performance in adhering to the planar assumption, with almost all of γ RMS N values being smaller than the corresponding K_c in both species ( Figure 5 ). However, SGE-1 showed limitations in confirming the uniform coordinate assumption, as over 36% of γ RMS T values were greater than the corresponding K_c in both species ( Figure 5 ). Particularly, only two out of four parameters within SGE-1, namely n ₂ and n ₃, demonstrated close-to-linear behavior, as assessed by metrics such as lower percentage bias ( Figure 6 ). For the other end of the goodness of fit scale, SGE-3 was found to be poor in fitting the observed data ( Figure 3 ). Despite its shortcomings in goodness of fit, SGE-3 demonstrated commendable performance in terms of being close-to-linear. To be specific, SGE-3 had the best linear approximation ( Figure 5 ). And at the individual parameter level, all parameters of SGE-3, a and n ₁, were close-to-linear as judged by the lower percentage bias ( Figure 6 ).

After assessing the nonlinear regression models using various methods, it can be concluded that SGE-3 exhibited the poorest goodness of fit and the highest AIC values, whereas SGE-1 showcased the best goodness of fit and the lowest AIC values. However, despite SGE-3’s suboptimal goodness of fit, it demonstrated the best close-to-linear behavior among the three models, both as an overall measure and at the individual parameter level of nonlinear behavior. On the other hand, SGE-1 faced challenges related to the uniform coordinate assumption, and displayed drawbacks in the behavior of four of its parameters, with two of them, i.e., a, and n ₁, were not close-to-linear. Therefore, among the three models examined, SGE-1 emerges as the clear choice when considering mainly the goodness of fit and AIC. While if one focuses on the nonlinear behavior, SGE-3 might be considered the optimal selection. It is essential to note that future studies on different species may lead to different conclusions. The choice between models should therefore be guided by study objectives—prioritizing either morphological characterization or statistical inference.

Generally, through an appropriate nonlinear reparameterization can effectively reduce parameter-effects curvature in nonlinear models (Bates and Watts, 1980; Ratkowsky and Reddy, 2017). It has been confirmed that performing specific parameter transformations like exponential modification in nonlinear regression models can simultaneously improve the parameter-effects curvature and close-to-linear behavior of the model parameters (He et al., 2024; Wang et al., 2024b). However, such approaches risk increasing model complexity, particularly problematic for already intricate formulations like the Gielis equation. Based on this trade-off analysis, we recommend retaining the current parameterization while identifying this as a promising direction for future methodological research.

In summary, we evaluated three simplified Gielis equations for modeling leaf contours of two bamboo species. The results derived from the two datasets demonstrated a clear trade-off between model performance metrics: while SGE-3 had the best close-to-linear behavior among the three models, it exhibited the poorest goodness of fit and the highest AIC value. In contrast, although SGE-1 achieved the best fit quality and the lowest AIC value, it provided unacceptable close-to-linear least squares estimates of parameters. Consequently, the choice of model applied for capturing the geometric properties of bamboo leaves depends on the criteria used to evaluate it, e.g., goodness of fit, model structural complexity, and the close-to-linear behavior of parameters. When models exhibit comparable fit, assessing parameter nonlinearity becomes critical for optimal selection. The present work provided insights into the criteria of model selection for nonlinear regression for future researches on describing the leaf shape of bamboo and other plant species with similar lanceolate leaves. One limitation of the current work is that it does not address the ecological or functional significance of the lanceolate leaf shape in bamboo, such as its potential role in drought resistance or other environmental adaptations. Additionally, the relationship between the parameters of the simplified Gielis equations and specific ecological traits remains unclear. Future research should aim to bridge this gap by linking shape descriptors derived from geometric models with physiological and ecological functions. Such studies would deepen the understanding of how leaf morphology contributes to adaptive strategies in varying environments and enhance the applicability of geometric modeling in functional plant ecology.