Performance assessment of three simplified Gielis equations in quantifying the geometries of lanceolate bamboo leaves

Qinchao Fu^1*Jing Li¹Azuo Jimu¹Ximeng Xiao¹Lin Wang^2*

• ¹Key Laboratory of Sichuan Province for Bamboo Pests Control and Resource Development, Leshan Normal University, Leshan, China
• ²Sichuan University, Chengdu, China

Accurate quantification of bamboo leaf morphology is essential for understanding plant morphogenesis and development. However, most bamboo leaves exhibit long lanceolate shape characteristic, posing challenges in finding suitable mathematical models for accurate shape description. Previous studies indicated that the simplified versions of Gielis equation, a nonlinear polar coordinate system derived from the superellipse equation, have shown promise in describing bamboo leaf geometries. Nevertheless, selecting an optimal nonlinear equation that precisely fits empirical bamboo leaf data remains a formidable challenge in morphological studies. This persistent limitation underscores the critical need for developing systematic evaluation methods to assess the performance of such nonlinear models. In the present study, three distinct versions of simplified Gielis equation, i.e., four-parameter version (referred to as SGE-1), three-parameter version (referred to as SGE-2), and two-parameter version (referred to as SGE-3), were used to fit the two-dimensional contours of bamboo leaves with a long lanceolate shape across two species (Indocalamus decorus with 254 leaves, and Indocalamus longiauritus with 251 leaves). The root-mean-square error (RMSE) and Akaike information criterion (AIC) were employed to assess the goodness of fit and model structural complexity, and the nonlinear behavior for each model was assessed using relative curvature measures of nonlinearity. Across both datasets, SGE-1 showcased the lowest RMSE and AIC values but exhibited the poorest close-to-linear behavior based on relative curvature measures among the three models. Conversely, SGE-3 had the best close-to-linear behavior among the three models, but it exhibited the highest RMSE and AIC values. These findings provide a methodological framework for selecting nonlinear models in plant morphometrics, particularly for lanceolate-shaped leaves, while highlighting the critical balance between descriptive accuracy and statistical robustness in biological shape analysis.