Many natural objects exhibit radial or axial symmetry in a single plane. However, a universal tool for simulating and fitting the shapes of such objects is lacking. Herein, we present an R package called ‘biogeom’ that simulates and fits many shapes found in nature. The package incorporates novel universal parametric equations that generate the profiles of bird eggs, flowers, linear and lanceolate leaves, seeds, starfish, and tree‐rings, and three growth‐rate equations that generate the profiles of ovate leaves and the ontogenetic growth curves of animals and plants. ‘biogeom’ includes several empirical datasets comprising the boundary coordinates of bird eggs, fruits, lanceolate and ovate leaves, tree rings, seeds, and sea stars. The package can also be applied to other kinds of natural shapes similar to those in the datasets. In addition, the package includes sigmoid curves derived from the three growth‐rate equations, which can be used to model animal and plant growth trajectories and predict the times associated with maximum growth rate. ‘biogeom’ can quantify the intra‐ or interspecific similarity of natural outlines, and it provides quantitative information of shape and ontogenetic modification of shape with important ecological and evolutionary implications for the growth and form of the living world.

Leaf lamina mass and area are closely correlated with the photosynthetic capacity and competitive ability of plants, whereas leaf age has been demonstrated to affect physiological processes such as photosynthesis. However, it remains unknown whether the lamina mass vs. area scaling relationship is influenced by leaf age, which is important for understanding plant adaptive strategies and, more broadly, resource utilization and growth. We measured the leaf functional traits of five leaf-age groups of Photinia × fraseri for a total of 1,736 leaves. ANOVA followed by Tukey’s honestly significant difference test was used to compare the functional traits among the five leaf-age groups. Reduced major axis regression protocols were used to fit the scaling relationship between lamina mass and area, and the bootstrap percentile method was used to compare the lamina mass vs. area scaling relationships among the leaf-age groups. Lamina area, and the ratio of lamina dry mass to lamina fresh mass increased with increasing leaf age. Lamina fresh mass per unit area, and lamina dry mass per unit area both exhibited a parabolic-like trend as leaf age increased, i.e., at the leaf maturation stage, it showed a slight but significant decline. The phenomenon called diminishing returns were confirmed by each of the five leaf-age groups, i.e., all of the numerical values of the scaling exponents of lamina mass vs. area were significantly greater than 1. There were significant differences in the scaling exponents of lamina mass vs. area for the leaves across different sampling times. The scaling exponents were lower at the early rapid growth stage, indicating a lower cost for increasing leaf area compared to the leaf maturation stage. These data are consistent with leaves undergoing a transition from resource acquisition to resource conservation in the process of their development and growth.

Preston’s equation is a general model describing the egg shape of birds. The parameters of Preston’s equation are usually estimated after re-expressing it as the Todd-Smart equation and scaling the egg’s actual length to two. This method assumes that the straight line through the two points on an egg’s profile separated by the maximum distance (i.e., the longest axis of an egg’s profile) is the mid-line. It hypothesizes that the photographed egg’s profile is perfectly bilaterally symmetrical, which seldom holds true because of photographic errors and placement errors. The existing parameter estimation method for Preston’s equation considers an angle of deviation for the longest axis of an egg’s profile from the mid-line, which decreases prediction errors to a certain degree. Nevertheless, this method cannot provide an accurate estimate of the coordinates of the egg’s center, and it leads to sub-optimal parameter estimation. Thus, it is better to account for the possible asymmetry between the two sides of an egg’s profile along its mid-line when fitting egg-shape data. In this paper, we propose a method based on the optimization algorithm (optimPE) to fit egg-shape data and better estimate the parameters of Preston’s equation by automatically searching for the optimal mid-line of an egg’s profile and testing its validity using profiles of 59 bird eggs spanning a wide range of existing egg shapes. We further compared this method with the existing one based on multiple linear regression (lmPE). This study demonstrated the ability of the optimPE method to estimate numerical values of the parameters of Preston’s equation and provide the theoretical egg length (i.e., the distance between two ends of the mid-line of an egg’s profile) and the egg’s maximum breadth. This provides a valuable approach for comparing egg shapes among conspecifics or across different species, or even different classes (e.g., birds and reptiles), in future investigations.

The Montgomery-Koyama-Smith (MKS) equation predicts that total leaf area per shoot is proportional to the product of the sum of individual leaf widths and maximum individual leaf length, which has been validated for some herbaceous and woody plants. The equation is also predicted to be valid in describing the relationship between the total stomatal area per micrograph (A_T) and the product of the sum of individual stomatal widths (denoted as L_KS) and maximum individual stomatal length (denoted by W_KS) in any particular micrograph. To test the validity of the MKS equation, 69,931 stomata (from 720 stomatal micrographs from 12 Magnoliaceae species) were examined. The area of each stoma was calculated using empirical measurements of stomatal length and width multiplied by a constant. Six equations describing the relationships among A_T, L_KS, and W_KS were compared. The root-mean-square (RMSE) and the Akaike information criterion (AIC) were used to measure the goodness of fit, and the trade-off between the goodness of fit and the structural complexity of each model, respectively. Analyses supported the validity of the MKS equation and the power-law equation A_T ∝ (L_KS∙W_KS)^α, where a is a scaling exponent. The estimated values of α at the species level and for the pooled data were all statistically smaller than unity, which did not support the hypothesis that A_T ∝ L_TS∙W_TS. The power-law equation had smaller RMSE and AIC values than the MKS equation for the data from the 12 individual species and the pooled data. These results indicate that A_T tends to allometrically scale with L_KS∙W_KS, and that increases in A_T do not keep pace with increases in L_TS∙W_TS. In addition, using the product of L_KS and W_KS is better than using only one of the two variables.

The Lamé curve is an extension of an ellipse, the latter being a special case. Dr. Johan Gielis further extended the Lamé curve in the polar coordinate system by introducing additional parameters (n₁, n₂, n₃; m): $r\left(\mathsf{\phi }\right)={\left({\left|\frac{1}{A}\cos \left(\frac{m}{4}\mathsf{\phi }\right)\right|}^{n_2}+{\left|\frac{1}{B}\sin \left(\frac{m}{4}\mathsf{\phi }\right)\right|}^{n_3}\right)}^{-1/n_1}$, which can be applied to model natural geometries. Here, r is the polar radius corresponding to the polar angle φ; A, B, n₁, n₂ and n₃ are parameters to be estimated; m is the positive real number that determines the number of angles of the Gielis curve. Most prior studies on the Gielis equation focused mainly on its applications. However, the Gielis equation can also generate a large number of shapes that are rotationally symmetric and axisymmetric when A = B and n₂ = n₃, interrelated with the parameter m, with the parameters n₁ and n₂ determining the shapes of the curves. In this paper, we prove the relationship between m and the rotational symmetry and axial symmetry of the Gielis curve from a theoretical point of view with the condition A = B, n₂ = n₃. We also set n₁ and n₂ to take negative real numbers rather than only taking positive real numbers, then classify the curves based on extremal properties of r(φ) at φ = 0, π/m when n₁ and n₂ are in different intervals, and analyze how n₁, n₂ precisely affect the shapes of Gielis curves.