Comparison of two simplified versions of the Gielis equation for describing the shape of bamboo leaves

Weihao Yao, Ülo Niinemets, Wenjing Yao*, Johan Gielis, Julian Schrader, Kexin Yu, Peijian Shi*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)
13 Downloads (Pure)


Bamboo is an important component in subtropical and tropical forest communities. The plant has characteristic long lanceolate leaves with parallel venation. Prior studies have shown that the leaf shapes of this plant group can be well described by a simplified version (referred to as SGE-1) of the Gielis equation, a polar coordinate equation extended from the superellipse equation. SGE-1 with only two model parameters is less complex than the original Gielis equation with six parameters. Previous studies have seldom tested whether other simplified versions of the Gielis equation are superior to SGE-1 in fitting empirical leaf shape data. In the present study, we compared a three-parameter Gielis equation (referred to as SGE-2) with the two-parameter SGE-1 using the leaf boundary coordinate data of six bamboo species within the same genus that have representative long lanceolate leaves, with >300 leaves for each species. We sampled 2000 data points at approximately equidistant locations on the boundary of each leaf, and estimated the parameters for the two models. The root–mean–square error (RMSE) between the observed and predicted radii from the polar point to data points on the boundary of each leaf was used as a measure of the model goodness of fit, and the mean percent error between the RMSEs from fitting SGE-1 and SGE-2 was used to examine whether the introduction of an additional parameter in SGE-1 remarkably improves the model’s fitting. We found that the RMSE value of SGE-2 was always smaller than that of SGE-1. The mean percent errors among the two models ranged from 7.5% to 20% across the six species. These results indicate that SGE-2 is superior to SGE-1 and should be used in fitting leaf shapes. We argue that the results of the current study can be potentially extended to other lanceolate leaf shapes.
Original languageEnglish
Article number3058
Pages (from-to)1-11
Number of pages11
Issue number22
Early online date11 Nov 2022
Publication statusPublished - Nov 2022

Bibliographical note

Copyright the Author(s) 2022. Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.


  • leaf shape
  • percent error
  • Pleioblastus
  • polar angle
  • polar radius


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