Mathematical progression of avian egg shape with associated area and volume determinations

Abstract

Development of nondestructive techniques for estimating egg parameters requires a comprehensive approach based on mathematical theory. Basic properties used to solve theoretical and applied problems in this respect are volume (V) and surface area (S). There are respective formulae for calculating V and S of spherical, ellipsoidal, and ovoid eggs in classical egg geometry; however, the mathematical description and calculation of these parameters for pyriform eggs have remained elusive. In the present study, we derived the appropriate formulae and established that this would be not only applicable and valid for the category of pyriform eggs, but also universal and explicit for all other naturally occurring avian egg shapes. Thus, we have demonstrated “mathematical progression” of this natural object, considering the egg as a sequence of geometric figures that transform from one to another in the following sequence of shapes: sphere → ellipsoid → ovoid (whose profile corresponds to Hügelschäffer's model) → pyriform ovoid.

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

“…Accordingly, at $${y_{L/4}} = \sqrt 3 B/4$$, the ovoid will match a sphere, and if y L /4 is less than that calculated by Equation (2), the egg falls into the pyriform (pear‐shaped) category 4 . If the measured value of y L /4 is equal to the value of y – L /4 , then the shape of such an egg will correspond to an ellipsoidal one.…”

Hügelschäffer egg curve and surface

“…The reason for the use of the Hügelschäffer egg-shaped curve model is its presentation as a non-destructive oomorphological model when calculating the area of a planar curve A and the volume V and the surface area S of a 3D egg surface [26], [30], [31]. Note that "oomorphological models" is used as a substitute for "egg-shaped models" [18].…”

Hugelschaffer egg curve and surface

“…Analogous to the previous example, a Hügelschäffer egg-shaped curve model can be generated according to the profiles of egg-shaped pipe sewers obtained through composition of circular arcs with the shape index B/L = 2/3 or B/L = 3/4 as given in [47]. In recent years, the application of Hügelschäffer models, as well as other mathematical models for describing the shape of eggs -ovoid forms (curves and surfaces) in the poultry industry and food engineering is on the rise, as evident from papers [2], [13], [14], [16], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28].…”

“…The reason for the use of the Hügelschäffer egg-shaped curve model is its presentation as a non-destructive oomorphological model when calculating the area of a planar curve A and the volume V and the surface area S of a 3D egg surface [23], [27], [28]. The authors Narushin et al used 2D digital images of egg contours to obtain concrete values for the parameters L = 2a, B = 2b and w, [22], [23].…”