Mathematical interpretation of artificial ovoids and avian egg shapes (part I)

Measurement of the neutral axis in avian eggshells reveals which species conform to the golden ratio

Avian eggs represent a striking evolutionary adaptation for which shell thickness is crucial. An understudied eggshell property includes the neutral axis, a line that is drawn through any bent structure and whose precise location is characterized by the k-factor. Previous studies have established that, for chicken eggs, mean k corresponds to the golden ratio (Φ = 1.618, or 0.618 in its reciprocal form). We hypothesized whether such an arrangement of the neutral axis conforms to the eggshell of any bird or only to eggshells with a certain set of geometric parameters. Implementing a suite of innovative methodological approaches, we investigated variations in k of 435 avian species, exploring which correspond to Φ. We found that mean k is highly variable among birds and does not always conform to Φ, being much lower in spherical and ellipsoid eggs and higher in pyriform eggs. While 21 species had k values within 0.618 ± 0.02 (including four falcon species) and the Falconinae subfamily (six species) revealed a mean of 0.618, it is predominantly domesticated species (chicken, ducks, and geese) that lay eggs whose neutral axis corresponds to the golden ratio. Thus, the study of the mathematical secrets of the eggshell related to the golden ratio of its neutral axis suggests its species-specific signatures in birds.

An Algorithm for Obtaining 3D Egg Models from Visual Images

Mathematical models for describing the shape of eggs find application in various fields of practice. The article proposes a method and tools for a detailed study of the shape and peripheral contours of digital images of eggs that are suitable for grouping and sorting. A scheme has been adapted to determine the morphological characteristics of eggs, on the basis of which an algorithm has been created for obtaining their 3D models, based on data from color digital images. The deviation from the dimensions of the major and minor axes measured with a caliper and the proposed algorithm is 0.5–1.5 mm. A model of a correction factor has been established by which the three-dimensional shape of eggs can be determined with sufficient accuracy. The results obtained in this work improve the assumption that the use of algorithms to determine the shape of eggs strongly depends on those of the bird species studied. It is approved with data for Mallard eggs which have a more elliptical shape and correspondingly lower values of correction coefficient ‘c’ (c = 1.55–4.96). In sparrow (c = 9.55–11.19) and quail (c = 11.71–13.11) eggs, the form tends to be ovoid. After testing the obtained model for eggs from three bird species, sparrow, mallard, and quail, the coefficient of the determination of proposed model was R2 = 0.96. The standard error was SE = 0.08. All of the results show a p-value of the model less than α = 0.05. The proposed algorithm was applied to create 3D egg shapes that were not used in the previous calculations. The resulting error was up to 9%. This shows that in the test, the algorithm had an accuracy of 91%. An advantage of the algorithm proposed here is that the human operator does not need to select points in the image, as is the case with some of the algorithms developed by other authors. The proposed methods and tools for three-dimensional transformation of egg images would be applicable not only for the needs of poultry farming, but also in ornithological research when working with different shaped varieties of eggs. Experimental results show that the proposed algorithm has sufficient accuracy.