Hügelschäffer egg curve and surface

Petrović, Maja; Malešević, Branko

doi:10.2298/aadm220526027p

Cited by 3 publications

(6 citation statements)

References 30 publications

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“…Since pyriform eggs were the stumbling block in the mathematical description of egg contours, 28,55 our research on the possibility of egg shape description using Equation ( 17) and its conformity with the geometric standard was focused on this particular category. Hereby, a pyriform profile meant not a purely visual assessment of how pear-shaped their pointed end is, but a distinct mathematical calculation.…”

Section: Methodsmentioning

confidence: 99%

“…In the course of our previous studies, 6,28,29,34,41,56 we accumulated a large number of photographic and digital images of various eggs obtained directly by us or available as open source. Thanks to this, we selected the egg images that can be classified as pyriform from 32 bird species.…”

Section: Methodsmentioning

confidence: 99%

“…These were subject to subsequent processing and examination. A detailed listing and description of the sources from which the appropriate images were taken was given in Narushin et al 56 Digital and photographic images of chicken eggs were also used; their detailed description, along with a method for obtaining them, was laid out in Narushin et al 41 Since chicken eggs relate to the category of classical ovoids and, in accordance with our previous studies, 28,41 are described by the respective Hügelschäffer model for classical ovoids (Equation 1), three characteristic egg shapes were chosen for these experiments: elongated, standard, and round. This sample, in our opinion, was sufficient to conclude that the derived egg profile formula (Equation 17) is universal.…”

Section: Methodsmentioning

confidence: 99%

“…These have included (i) a comparative analysis of the universal formula with others developed before or after its publication [20][21][22][23][24][25][26][27][28] ; (ii) its use for deriving applied calculation formulae, for example, for computing egg volume or surface area that can be successfully used in biological, zootechnical, and/or other studies [29][30][31][32] ; (iii) technological developments 33,34 ; (iv) space research, for example, for building egg-F I G U R E 1 Combination of two Hügelschäffer's models conforming to (i) a classical ovoid with parameters B, L, and w (blue line) and (ii) a parabolic ovoid whose axial shift corresponds to half the egg length, that is, w p = L/2. shaped lunar habitat structures 35 and modeling broad-line regions when studying supermassive black holes 36 ; (v) computer games 37 ; and even (vi) interest in its inclusion in the school curriculum in mathematics as an adapted version of our research.…”

Section: Introductionmentioning

confidence: 99%

“…As the vast majority of avian eggs are those of domestic fowl (chicken, duck, goose, quail, etc.) and most other birds’ eggs are similarly ovoid (literally “egg shaped”), then Hügelschäffer's model largely serves as the basis for their description 1–4 . In its most simplistic form, Hügelschäffer's model is expressed as:

\begin{equation*}y = \pm \frac{B}{2}\sqrt {\frac{{{L}^2 - 4{x}^2}}{{{L}^2 + 8wx + 4{w}^2}}} ,\end{equation*}

where B is the egg maximum breadth, L is the egg length, and w is the distance between two vertical lines corresponding to the maximum breadth and the half length of the egg, L /2.…”

Section: Introductionmentioning

confidence: 99%

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A novel approach to egg and math: Improved geometrical standardization of any avian egg profile

Narushin,

Orszulik,

Romanov

et al. 2023

Annals of the New York Academy of Sciences

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Developing a geometric formulation of any biological object has a number of justifications and applications. Recently, we developed a universal geometric figure for describing a bird's egg in any of the possible basic shapes: spherical, ellipsoidal, ovoid, and pyriform. The formulation proved widely applicable but had a number of drawbacks, including a very obvious join between two parts of the egg. To correct this, we developed the Main Axiom of the universal mathematical formula. This essentially involved making the ordinate of the extremum of the function correspond to half the maximum egg breadth (B), and the abscissa to the reciprocal of the parameter w that reflects the shift of the vertical axis to its coincidence with B. This, in turn, helped us develop a new, simplified mathematical model without a nonbiological join. Experimental verification was performed to confirm the adequacy of the new geometric figure. It accurately described actual avian eggs of various shapes more closely than our previous model. To the best of our knowledge, our new, simplified equation can be applied as a standard for any bird egg that exists in nature. As a rather simple equation, it can be used in a broad range of applications.

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Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

\begin{equation*}y = \pm \frac{B}{2}\sqrt {\frac{{{L}^2 - 4{x}^2}}{{{L}^2 + 8wx + 4{w}^2}}} ,\end{equation*}

where B is the egg maximum breadth, L is the egg length, and w is the distance between two vertical lines corresponding to the maximum breadth and the half length of the egg, L /2.…”