{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T05:08:54Z","timestamp":1778562534272,"version":"3.51.4"},"reference-count":128,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2026,4,27]],"date-time":"2026-04-27T00:00:00Z","timestamp":1777248000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>Deriving from the Latin \u201covum\u201d (egg), the oval is a commonly used term, but does not have the status of a standard geometric figure like a circle or ellipse. Consequently, the oval lacks both a mathematical descriptive basis to attribute a set of key geometric parameters and an elegant formula to describe its contours. Herein, we consider the basis for deriving the formula of an oval for typical egg profiles. Specifically, these are round, ellipsoid, classic oval, pyriform (conical) and biconical shapes. To do this, we adhered to four basic postulates: (i) the ability to describe all possible egg shapes; (ii) a minimum set of measurable geometric parameters; (iii) the application of some universal indices (ratios of key geometric dimensions) to describe mathematical models; (iv) conformity with the \u201cMain Axiom of the Mathematical Formula of the Bird\u2019s Egg.\u201d Additionally, we sought to comply with the principles of mathematical elegance. Following these theoretical assumptions and practical verification, we obtained a mathematically supported, elegant formula for this well-known but non-standardized geometric figure. The derived oval geometry equation will find use in applied problems of biology, construction, engineering and school curricula, alongside the classical figures of the circle and ellipse.<\/jats:p>","DOI":"10.3390\/computation14050101","type":"journal-article","created":{"date-parts":[[2026,4,27]],"date-time":"2026-04-27T19:24:51Z","timestamp":1777317891000},"page":"101","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["What Is an Oval, Officially and Overall? Old and New Mathematical Descriptions"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6799-6605","authenticated-orcid":false,"given":"Valeriy G.","family":"Narushin","sequence":"first","affiliation":[{"name":"Independent Researcher, Soborny 145-B, 69035 Zaporizhya, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3176-4911","authenticated-orcid":false,"given":"Stefan T.","family":"Orszulik","sequence":"additional","affiliation":[{"name":"Independent Researcher, Grove, Wantage, Oxfordshire OX12 0QA, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3584-4644","authenticated-orcid":false,"given":"Michael N.","family":"Romanov","sequence":"additional","affiliation":[{"name":"School of Natural Sciences, University of Kent, Canterbury, Kent CT2 7NJ, UK"},{"name":"Animal Genomics and Bioresource Research Unit (AGB Research Unit), Faculty of Science, Kasetsart University, Chatuchak, Bangkok 10900, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7595-3226","authenticated-orcid":false,"given":"Darren K.","family":"Griffin","sequence":"additional","affiliation":[{"name":"School of Natural Sciences, University of Kent, Canterbury, Kent CT2 7NJ, UK"},{"name":"Animal Genomics and Bioresource Research Unit (AGB Research Unit), Faculty of Science, Kasetsart University, Chatuchak, Bangkok 10900, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,4,27]]},"reference":[{"key":"ref_1","unstructured":"Clement, P.A., and Hoffleit, H.B. (1969). Moralia, Quaestiones Convivales (Table Talk), Book 8, Question 2, \u201cWhat is Plato\u2019s Meaning, When He Says That God Always Plays the Geometer?\u201d. Moralia, Harvard University Press. Loeb Classical Library 425."},{"key":"ref_2","first-page":"87","article-title":"What Plato meant by saying that God is always doing geometry: Responses to the question posed by Plutarch in Table-Talk 8.2","volume":"Volume 74","author":"Rochberg","year":"2026","journal-title":"Studies in Premodern Sciences in Memory of Noel M. Swerdlow"},{"key":"ref_3","first-page":"252","article-title":"\u201cThe egg reopened\u201d again","volume":"56","author":"Gilbert","year":"1974","journal-title":"Art Bull."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1080\/0025570X.1990.11977491","article-title":"D\u00fcrer\u2019s paradox or why an ellipse is not egg-shaped","volume":"63","year":"1990","journal-title":"Math. Mag."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Sarhangi, R. (2000). On Growth and Form in Nature and Art: The Projective Geometry of Plant Buds and Greek Vases. Bridges: Mathematical Connections in Art, Music, and Science, 3rd Annual BRIDGES Conference Proceedings, Winfield, KS, USA, 28\u201330 July 2000, Bridges Conference. Available online: https:\/\/archive.bridgesmathart.org\/2000\/bridges2000-267.pdf.","DOI":"10.1007\/s00004-000-0018-x"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1057\/s41599-020-00541-8","article-title":"Eggstraordinary artefacts: Decorated ostrich eggs in the ancient Mediterranean world","volume":"7","author":"Hodos","year":"2020","journal-title":"Humanit. Soc. Sci. Commun."},{"key":"ref_7","unstructured":"Grady, K. (2026, March 23). Why Are Artists and Designers so Obsessed with Eggs? House & Garden, Cond\u00e9 Nast Britain. Available online: https:\/\/www.houseandgarden.co.uk\/article\/interiors-egg-obsession?utm_source=chatgpt.com."},{"key":"ref_8","unstructured":"Freiberger, M. (2026, March 23). Perfect Buildings: The Maths of Modern Architecture. Plus Magazine. Available online: https:\/\/plus.maths.org\/content\/perfect-buildings-maths-modern-architecture."},{"key":"ref_9","first-page":"115","article-title":"Suitability analysis of Hugelschaffer\u2019s egg curve application in architectural and structures\u2019 geometry","volume":"57","year":"2011","journal-title":"Bul. Institutului Politeh. Din Iasi"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Feleki, A., and Nagy, Z. (2016). Challenges in Structural Designing of Egg-shaped Steel Structure. Challenges in Design and Construction of an Innovative and Sustainable Built Environment, 19th IABSE Congress, Stockholm, Sweden, 21\u201323 September 2016, IABSE Congress.","DOI":"10.2749\/stockholm.2016.2242"},{"key":"ref_11","unstructured":"Baj\u0161anski, I., and Jovanovi\u0107, M. (2023). A Method for Adjusting the Shape of Semioval Arches Using H\u00fcgelsch\u00e4ffer\u2019s Construction. Proceedings of the 9th International Scientific Conference on Geometry, Graphics and Design in the Digital Age \u201cMongeometrija 2023\u201d, Novi Sad, Serbia, 7\u201310 June 2023, Faculty of Technical Sciences, University of Novi Sad, Serbian Society for Geometry and Graphics. Available online: https:\/\/grafar.grf.bg.ac.rs\/handle\/123456789\/3354."},{"key":"ref_12","unstructured":"Archiproducts (2025). Egg Design: The Design Inspired By the Egg, Archiproducts.com. Available online: https:\/\/www.archiproducts.com\/en\/focus\/egg-design-the-design-inspired-by-the-egg_593638."},{"key":"ref_13","unstructured":"Weisstein, E.W. (2025). Oval. Wolfram MathWorld, Wolfram Research, Inc.. Available online: https:\/\/mathworld.wolfram.com\/Oval.html."},{"key":"ref_14","unstructured":"Cambridge Dictionary (2026). Oval, Cambridge University Press & Assessment. Available online: https:\/\/dictionary.cambridge.org\/us\/dictionary\/english\/oval."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1007\/s00283-021-10149-8","article-title":"The Cartesian ovals","volume":"44","author":"Farouki","year":"2022","journal-title":"Math. Intell."},{"key":"ref_16","first-page":"231","article-title":"A multi foci closed curve: Cassini oval, its properties and applications","volume":"14","year":"2013","journal-title":"Do\u011fu\u015f Univ. Derg."},{"key":"ref_17","unstructured":"Dixon, R. (1991). Mathographics, Dover Publications."},{"key":"ref_18","unstructured":"Wikimedia Commons (2026, March 23). Pied Kingfisher Ceryle rudis, Egg, Coll. Museum Wiesbaden, CC BY-SA 3.0 License. Klaus Rassinger and Gerhard Cammerer, Museum Wiesbaden. Available online: https:\/\/commons.wikimedia.org\/wiki\/File:Ceryle_rudis_MWNH_1249.JPG."},{"key":"ref_19","unstructured":"Wikimedia Commons (2026, March 23). Blue-Spotted Wood Dove Turtur afer, Egg, Coll. Museum Wiesbaden, CC BY-SA 3.0 License. Klaus Rassinger and Gerhard Cammerer, Museum Wiesbaden. Available online: https:\/\/commons.wikimedia.org\/wiki\/File:Turtur_afer_MWNH_0569.JPG."},{"key":"ref_20","unstructured":"Wikimedia Commons (2026, March 23). Harlequin Duck Histrionicus histrionicus, Egg, Coll. Museum Wiesbaden, CC BY-SA 3.0 License. Klaus Rassinger and Gerhard Cammerer, Museum Wiesbaden. Available online: https:\/\/commons.wikimedia.org\/wiki\/File:Histrionicus_histrionicus_MWNH_1024.JPG."},{"key":"ref_21","unstructured":"Wikimedia Commons (2026, March 23). Common redshank Tringa totanus, Egg, Coll. Museum Wiesbaden, CC BY-SA 3.0 License. Klaus Rassinger and Gerhard Cammerer, Museum Wiesbaden. Available online: https:\/\/commons.wikimedia.org\/wiki\/File:Tringa_totanus_MWNH_0210.JPG."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"9728","DOI":"10.1002\/ece3.4412","article-title":"Accurately quantifying the shape of birds\u2019 eggs","volume":"19","author":"Biggins","year":"2018","journal-title":"Ecol. Evol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"264","DOI":"10.15421\/2017_78","article-title":"Geometrical standards in shapes of avian eggs","volume":"7","author":"Mytiai","year":"2017","journal-title":"Ukr. J. Ecol."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"67","DOI":"10.37828\/em.2020.38.9","article-title":"Mathematical interpretation of avian egg shapes","volume":"38","author":"Mytiai","year":"2020","journal-title":"Ecol. Monten."},{"key":"ref_25","first-page":"679","article-title":"The mathematics of egg shape","volume":"78","author":"Nishiyama","year":"2012","journal-title":"Int. J. Pure Appl. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1111\/nyas.14680","article-title":"Egg and math: Introducing a universal formula for egg shape","volume":"1505","author":"Narushin","year":"2021","journal-title":"Ann. N. Y. Acad. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1111\/nyas.15059","article-title":"A novel approach to egg and math: Improved geometrical standardization of any avian egg profile","volume":"1529","author":"Narushin","year":"2023","journal-title":"Ann. N. Y. Acad. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"454","DOI":"10.2307\/4083294","article-title":"The shapes of birds\u2019 eggs: Mathematical aspects","volume":"85","author":"Preston","year":"1968","journal-title":"Auk"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"246","DOI":"10.2307\/4083498","article-title":"Shapes of birds\u2019 eggs: Extant North American families","volume":"86","author":"Preston","year":"1969","journal-title":"Auk"},{"key":"ref_30","unstructured":"Wikimedia Commons (2026, March 23). Isabelline Wheatear Oenanthe isabellina, Egg, Coll. Museum Wiesbaden, CC BY-SA 3.0 License. Klaus Rassinger and Gerhard Cammerer, Museum Wiesbaden. Available online: https:\/\/commons.wikimedia.org\/wiki\/File:Oenanthe_isabellina_MWNH_1866.JPG."},{"key":"ref_31","unstructured":"Wikimedia Commons (2026, March 23). Rook Corvus frugilegus, Egg, Coll. Museum Wiesbaden, CC BY-SA 3.0 License. Klaus Rassinger and Gerhard Cammerer, Museum Wiesbaden. Available online: https:\/\/commons.wikimedia.org\/wiki\/File:Corvus_frugilegus_MWNH_1322.JPG."},{"key":"ref_32","unstructured":"Wikimedia Commons (2026, March 23). Red-Necked Grebe Podiceps grisegena, Egg, Coll. Museum Wiesbaden, CC BY-SA 3.0 License. Klaus Rassinger and Gerhard Cammerer, Museum Wiesbaden. Available online: https:\/\/commons.wikimedia.org\/wiki\/File:Podiceps_grisegena_MWNH_0120.JPG."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/j.biosystemseng.2020.06.008","article-title":"Digital imaging assisted geometry of chicken eggs using H\u00fcgelsch\u00e4ffer\u2019s model","volume":"197","author":"Narushin","year":"2020","journal-title":"Biosyst. Eng."},{"key":"ref_34","first-page":"507","article-title":"The method of transformed co-ordinates applied to the deformations produced by the walls of a tubular viscus on a contained body: The avian egg as a model system","volume":"104","author":"Smart","year":"1969","journal-title":"J. Anat."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"160","DOI":"10.2307\/4081145","article-title":"The shapes of birds\u2019 eggs","volume":"70","author":"Preston","year":"1953","journal-title":"Auk"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"ukac028","DOI":"10.1093\/ornithology\/ukac028","article-title":"Preston\u2019s universal formula for avian egg shape","volume":"139","author":"Biggins","year":"2022","journal-title":"Ornithology"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1080\/00071666808415706","article-title":"The hen\u2019s egg: A mathematical model with three parameters","volume":"9","author":"Carter","year":"1968","journal-title":"Br. Poult. Sci."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1006\/jaer.1997.0188","article-title":"The avian egg: Geometrical description and calculation of parameters","volume":"68","author":"Narushin","year":"1997","journal-title":"J. Agric. Eng. Res."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1006\/jaer.2001.0721","article-title":"Shape geometry of the avian egg","volume":"79","author":"Narushin","year":"2001","journal-title":"J. Agric. Eng. Res."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1179","DOI":"10.1093\/auk\/119.4.1179","article-title":"A geometric method for determining shape of bird eggs","volume":"119","author":"Baker","year":"2002","journal-title":"Auk"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Narushin, V.G., Volkova, N.A., Dzhagaev, A.Y., Griffin, D.K., Romanov, M.N., and Zinovieva, N.A. (2025). Coupling artificial intelligence with proper mathematical algorithms to gain deeper insight into the biology of birds\u2019 eggs. Animals, 15.","DOI":"10.3390\/ani15030292"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1098","DOI":"10.1111\/1749-4877.12936","article-title":"Pear-shaped eggs evolved to maximize the surface area-to-volume ratio, increase metabolism, and shorten incubation time in birds","volume":"20","author":"Narushin","year":"2025","journal-title":"Integr. Zool."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"108143","DOI":"10.1016\/j.jfca.2025.108143","article-title":"Egg Geometrical Index: Encompassing a wide range of avian egg profiles with potential for novel AI applications in research and industry","volume":"148","author":"Narushin","year":"2025","journal-title":"J. Food Compos. Anal."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"20","DOI":"10.25162\/sl-2013-0003","article-title":"Fermat\u2019s analysis of extreme values and tangents","volume":"45","author":"Breger","year":"2013","journal-title":"Stud. Leibnitiana"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"242","DOI":"10.1016\/j.cognition.2019.04.008","article-title":"Intuitions about mathematical beauty: A case study in the aesthetic experience of ideas","volume":"189","author":"Johnson","year":"2019","journal-title":"Cognition"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"121","DOI":"10.36253\/Aisthesis-14790","article-title":"Mathematical beauty: On the aesthetic qualities of formal language","volume":"16","year":"2024","journal-title":"Aisthesis"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1016\/j.cagd.2015.10.004","article-title":"Uniform approximation of a circle by a parametric polynomial curve","volume":"41","year":"2016","journal-title":"Comput. Aided Geom. Des."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1007\/s11075-017-0322-0","article-title":"On parametric polynomial circle approximation","volume":"77","author":"Kozak","year":"2018","journal-title":"Numer. Algor."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"100038","DOI":"10.1016\/j.nls.2025.100038","article-title":"The pros and cons of the Preston\u2013Biggins egg shape model: A reconsideration case based on mathematical modeling and simulation","volume":"4","author":"Narushin","year":"2025","journal-title":"Nonlinear Sci."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1007\/s12064-025-00447-6","article-title":"Smart and smarter: Improving on a classic egg shape model","volume":"144","author":"Narushin","year":"2025","journal-title":"Theory Biosci."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"437","DOI":"10.17537\/2025.20.437","article-title":"Eggs from the Baker: Simplifying and improving on an under-appreciated oological model","volume":"20","author":"Narushin","year":"2025","journal-title":"Mat. Biol. Bioinform."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"333","DOI":"10.3732\/ajb.90.3.333","article-title":"A generic geometric transformation that unifies a wide range of natural and abstract shapes","volume":"90","author":"Gielis","year":"2003","journal-title":"Am. J. Bot."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1111\/nyas.14799","article-title":"Comparison of a universal (but complex) model for avian egg shape with a simpler model","volume":"1514","author":"Shi","year":"2022","journal-title":"Ann. N. Y. Acad. Sci."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1080\/00071667008415805","article-title":"The hen\u2019s egg: Shell shape and size parameters and their interrelations","volume":"11","author":"Carter","year":"1970","journal-title":"Br. Poult. Sci."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"104196","DOI":"10.1016\/j.psj.2024.104196","article-title":"Comparison of two polar equations in describing the geometries of domestic pigeon (Columba livia domestica) eggs","volume":"103","author":"Wang","year":"2024","journal-title":"Poult. Sci."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"482","DOI":"10.1093\/ps\/84.3.482","article-title":"Egg geometry calculation using the measurements of length and breadth","volume":"84","author":"Narushin","year":"2005","journal-title":"Poult. Sci."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/j.fbp.2024.04.007","article-title":"\u2018Eggology\u2019 and mathematics of a quail egg: An innovative non-destructive technology for evaluating egg parameters in Japanese quail","volume":"146","author":"Narushin","year":"2024","journal-title":"Food Bioprod. Process."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1016\/j.fbp.2024.07.013","article-title":"Reimagining Archimedes: An innovative and accurate calculation of volumes and asserting another standard method for defining the surface area of quail and any avian eggs","volume":"147","author":"Narushin","year":"2024","journal-title":"Food Bioprod. Process."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"e7681","DOI":"10.56083\/RCV5N3-047","article-title":"Morphometry and classification of chicken white eggs using computational vision and artificial neural network techniques","volume":"5","author":"Moreira","year":"2025","journal-title":"Rev. Contemp."},{"key":"ref_60","doi-asserted-by":"crossref","unstructured":"Kang, H., Hu, Y., Kaewunruen, S., Hu, X., and Zhang, J. (2025). Geometric and mechanical analysis of selenium-enriched eggs. J. Mar. Sci. Eng., 13.","DOI":"10.3390\/jmse13030525"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"296","DOI":"10.1016\/j.tws.2017.06.022","article-title":"Buckling of prolate egg-shaped domes under hydrostatic external pressure","volume":"119","author":"Zhang","year":"2017","journal-title":"Thin-Walled Struct."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/j.marstruc.2016.11.005","article-title":"Investigation on egg-shaped pressure hulls","volume":"52","author":"Zhang","year":"2017","journal-title":"Mar. Struct."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"103893","DOI":"10.1016\/j.ijpvp.2019.04.006","article-title":"Experimental and numerical collapse properties of externally pressurized egg-shaped shells under local geometrical imperfections","volume":"175","author":"Zhang","year":"2019","journal-title":"Int. J. Press. Vessel. Pip."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"104435","DOI":"10.1016\/j.ijpvp.2021.104435","article-title":"Buckling performance of egg-shaped shells fabricated through free hydroforming","volume":"193","author":"Zhang","year":"2021","journal-title":"Int. J. Press. Vessel. Pip."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"111057","DOI":"10.1016\/j.oceaneng.2022.111057","article-title":"Hydroforming and buckling of an egg-shaped shell based on a petal-shaped perform","volume":"250","author":"Zhang","year":"2022","journal-title":"Ocean Eng."},{"key":"ref_66","doi-asserted-by":"crossref","unstructured":"Narushin, V.G., Romanov, M.N., and Griffin, D.K. (2022). Egg-inspired engineering in the design of thin-walled shelled vessels: A theoretical approach for shell strength. Front. Bioeng. Biotechnol., 10.","DOI":"10.3389\/fbioe.2022.995817"},{"key":"ref_67","first-page":"118","article-title":"Development of laminated egg-shaped tsunami shelter structure made of steel-cushioning-steel","volume":"12","author":"Hou","year":"2024","journal-title":"Int. J. Mech. Eng. Appl."},{"key":"ref_68","unstructured":"OpenStax (2021). 6.1: Graphs of the Sine and Cosine Functions. Precalculus 1e (OpenStax), LibreTexts Mathematics, Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Available online: https:\/\/math.libretexts.org\/Bookshelves\/Precalculus\/Precalculus_1e_(OpenStax)\/06%3A_Periodic_Functions\/6.01%3A_Graphs_of_the_Sine_and_Cosine_Functions."},{"key":"ref_69","unstructured":"K\u00f6ller, J. (2026, March 23). Egg Curves and Ovals. More Plane Figures. Mathematische Basteleien. Available online: http:\/\/www.mathematische-basteleien.de\/eggcurves.htm."},{"key":"ref_70","unstructured":"Smith, J.O. (2007). Mathematics of the Discrete Fourier Transform (DFT): With Audio Applications, W3K Publishing, BookSurge Publishing. [2nd ed.]. Available online: http:\/\/ccrma.stanford.edu\/~jos\/mdft\/."},{"key":"ref_71","doi-asserted-by":"crossref","unstructured":"Bird, J. (2021). The circle and its properties. Bird\u2019s Basic Engineering Mathematics, Routledge. [8th ed.].","DOI":"10.4324\/9781003124214"},{"key":"ref_72","unstructured":"Salem, A.B.H. (2021). Geometry of the Ellipse and the Ellipsoid, Soukra Raoudha. Available online: https:\/\/www.researchgate.net\/publication\/353342312."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1038\/scientificamerican0965-222","article-title":"The superellipse: A curve that lies between the ellipse and the rectangle","volume":"213","author":"Gardner","year":"1965","journal-title":"Sci. Am."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"31","DOI":"10.2307\/3613154","article-title":"Lam\u00e9 ovals","volume":"54","author":"Gridgeman","year":"1970","journal-title":"Math. Gaz."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"95","DOI":"10.7546\/giq-25-2023-95-116","article-title":"Geometry of the ovoids: Reptilian eggs and similar symmetric forms","volume":"25","author":"Mladenova","year":"2023","journal-title":"Geom. Integr. Quantization"},{"key":"ref_76","unstructured":"Wikimedia Commons (2026, March 23). Actinemys marmorata Eggs Found in a Pool of a Permanent Creek. CC BY-SA 2.5 License. Pierre Fidenci: Sonoma County, CA, USA, 2006. Available online: https:\/\/commons.wikimedia.org\/wiki\/File:Clemmys_marmorata03.jpg."},{"key":"ref_77","unstructured":"Wikimedia Commons (2026, March 23). Apteryx australis, Egg. CC BY-SA 4.0 License. David J. Stang, Delaware Museum of Natural History. Available online: https:\/\/commons.wikimedia.org\/wiki\/File:Apteryx_australis_1zz.jpg."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"39","DOI":"10.2495\/D&NE-V3-N1-39-64","article-title":"On the biological shape of the Polygonaceae Rheum petiole","volume":"3","author":"Pasini","year":"2008","journal-title":"Int. J. Des. Nat. Ecodyn."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"1333","DOI":"10.1007\/s00468-023-02425-1","article-title":"Stomatal area estimation based on stomatal length and width of four Magnoliaceae species: Even \u201ckidney\u201d-shaped stomata are not elliptical","volume":"37","author":"Zhang","year":"2023","journal-title":"Trees"},{"key":"ref_80","doi-asserted-by":"crossref","unstructured":"Huang, W., Ma, K., Tan, J., Wei, M., and Lu, Y. (2024). Superellipse equation describing the geometries of Abies alba tree rings. Plants, 13.","DOI":"10.3390\/plants13243487"},{"key":"ref_81","unstructured":"Wikimedia Commons (2026, March 23). Category: Eggs of the Natural History Collections of the Museum Wiesbaden. Available online: https:\/\/commons.wikimedia.org\/wiki\/Category:Eggs_of_the_Natural_History_Collections_of_the_Museum_Wiesbaden."},{"key":"ref_82","unstructured":"Wikimedia Commons (2026, March 23). Category: Bird Eggs of the Mus\u00e9um de Toulouse. Available online: https:\/\/commons.wikimedia.org\/wiki\/Category:Bird_eggs_of_the_Mus%C3%A9um_de_Toulouse."},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1002\/for.3980010202","article-title":"The accuracy of extrapolation (time series) methods: Results of a forecasting competition","volume":"1","author":"Makridakis","year":"1982","journal-title":"J. Forecast."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"850","DOI":"10.1109\/34.232073","article-title":"Comparing images using the Hausdorff distance","volume":"15","author":"Huttenlocher","year":"1993","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1186\/s41747-020-00200-2","article-title":"On the usage of average Hausdorff distance for segmentation performance assessment: Hidden error when used for ranking","volume":"5","author":"Aydin","year":"2021","journal-title":"Eur. Radiol. Exp."},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"79","DOI":"10.3354\/cr030079","article-title":"Advantages of the Mean Absolute Error (MAE) over the Root Mean Square Error (RMSE) in assessing average model performance","volume":"30","author":"Willmott","year":"2005","journal-title":"Clim. Res."},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"5481","DOI":"10.5194\/gmd-15-5481-2022","article-title":"Root-mean-square error (RMSE) or mean absolute error (MAE): When to use them or not","volume":"15","author":"Hodson","year":"2022","journal-title":"Geosci. Model Dev."},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"1247","DOI":"10.5194\/gmd-7-1247-2014","article-title":"Root mean square error (RMSE) or mean absolute error (MAE)?\u2014Arguments against avoiding RMSE in the literature","volume":"7","author":"Chai","year":"2014","journal-title":"Geosci. Model Dev."},{"key":"ref_89","first-page":"15","article-title":"Kurvenkonstruktionen f\u00fcr den Flugzeugentwurf","volume":"36","author":"Ursinus","year":"1944","journal-title":"Flugsport"},{"key":"ref_90","first-page":"67","article-title":"Eine exakte Eierkurvenkonstruktion mit technischen Anwendungen","volume":"3","author":"Schmidbauer","year":"1948","journal-title":"Elem. Math."},{"key":"ref_91","unstructured":"Nestorovi\u0107, M. (2010). The Complement of the Hugelschaffer\u2019s Construction of the Egg Curve. Proceedings of the 25th National and 2nd International Scientific Conference \u201cMongeometrija 2010\u201d, Belgrade, Serbia, 24\u201327 June 2010, Faculty of Architecture, University of Belgrade, Serbian Society for Geometry and Graphics. Available online: https:\/\/grafar.grf.bg.ac.rs\/handle\/123456789\/2050."},{"key":"ref_92","first-page":"110","article-title":"Generating curves of higher order using the generalisation of H\u00fcgelsch\u00e4ffer\u2019s egg curve construction","volume":"58","year":"2013","journal-title":"Sci. Bull. Politeh. Univ. Timi\u015f. Trans. Hydrotech."},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1007\/s12038-023-00361-3","article-title":"A novel model for eggs like pears: How to quantify them geometrically with two parameters?","volume":"48","author":"Narushin","year":"2023","journal-title":"J. Biosci."},{"key":"ref_94","unstructured":"Wikimedia Commons (2026, March 23). Thick-Billed Murre Uria lomvia, Egg, Coll. Museum Wiesbaden, CC BY-SA 3.0 License. Klaus Rassinger and Gerhard Cammerer, Museum Wiesbaden. Available online: https:\/\/commons.wikimedia.org\/wiki\/File:Uria_lomvia_MWNH_2182.JPG."},{"key":"ref_95","unstructured":"Wikimedia Commons (2026, March 23). Chukar Partridge Alectoris chukar, Egg, Coll. Museum Wiesbaden, CC BY-SA 3.0 License. Klaus Rassinger and Gerhard Cammerer, Museum Wiesbaden. Available online: https:\/\/commons.wikimedia.org\/wiki\/File:Alectoris_chukar_MWNH_1084.JPG."},{"key":"ref_96","unstructured":"Wikimedia Commons (2026, March 23). Common Swift Apus apus, Egg, Coll. Museum Wiesbaden, CC BY-SA 3.0 License. Klaus Rassinger and Gerhard Cammerer, Museum Wiesbaden. Available online: https:\/\/commons.wikimedia.org\/wiki\/File:Apus_apus_MWNH_1247.JPG."},{"key":"ref_97","unstructured":"Wikimedia Commons (2026, March 23). Greater Rhea Rhea Americana, Egg, Coll. Museum Wiesbaden, CC BY-SA 3.0 License. Klaus Rassinger and Gerhard Cammerer, Museum Wiesbaden. Available online: https:\/\/commons.wikimedia.org\/wiki\/File:Rhea_americana_MWNH_0003.JPG."},{"key":"ref_98","unstructured":"Smith, S.W. (2003). Digital Signal Processing. A Practical Guide for Engineers and Scientists, Newnes, Elsevier Science. Available online: https:\/\/books.google.com\/books?id=CdtQBAAAQBAJ."},{"key":"ref_99","first-page":"1049","article-title":"Parameter estimation of linear and quadratic chirps by employing the fractional Fourier transform and a generalized time frequency transform","volume":"40","author":"Sahay","year":"2015","journal-title":"Sadhana-Acad. Proc. Eng. Sci."},{"key":"ref_100","unstructured":"Fresnel, A. (1866\u20131870). Premier M\u00e9moire sur la Diffraction de la Lumi\u00e8re. M\u00e9moire Adress\u00e9 \u00e0 l\u2019Acad\u00e9mie des Sciences le 15 Octobre 1815. Le Texte Pr\u00e9sent\u00e9 ici a \u00e9t\u00e9 Publi\u00e9 Dans les \u0152uvres Compl\u00e8tes d\u2019Augustin Fresnel, MM. Henri de Senarmont, Emile Verdet et L\u00e9onor Fresnel, Impr. Imp\u00e9riale."},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1016\/j.optlastec.2005.08.011","article-title":"Computer simulation of Fresnel diffraction from rectangular apertures and obstacles using the Fresnel integrals approach","volume":"39","author":"Abedin","year":"2007","journal-title":"Opt. Laser Technol."},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"3370","DOI":"10.1364\/AO.51.003370","article-title":"Simple and effective method for the analytic description of important optical beams when truncated by finite apertures","volume":"51","author":"Recami","year":"2012","journal-title":"Appl. Opt."},{"key":"ref_103","first-page":"4031793","article-title":"Approximation of Fresnel integrals with applications to diffraction problems","volume":"2018","year":"2018","journal-title":"Math. Probl. Eng."},{"key":"ref_104","first-page":"2","article-title":"Allometry: The study of biological scaling","volume":"3","author":"Shingleton","year":"2010","journal-title":"Nat. Sci. Educ."},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"319","DOI":"10.2307\/1366345","article-title":"The avian egg: Surface area, volume, and density","volume":"76","author":"Paganelli","year":"1974","journal-title":"Condor"},{"key":"ref_106","doi-asserted-by":"crossref","first-page":"147","DOI":"10.2307\/1366724","article-title":"The avian egg: Incubation time and water loss","volume":"76","author":"Rahn","year":"1974","journal-title":"Condor"},{"key":"ref_107","doi-asserted-by":"crossref","unstructured":"Piiper, J. (1978). Interdependence of gas conductance, incubation length, and weight of the avian egg. Respiratory Function in Birds, Adult and Embryonic. Proceedings in Life Sciences, Springer.","DOI":"10.1007\/978-3-642-66894-4"},{"key":"ref_108","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1016\/0300-9629(88)90628-7","article-title":"Eggs of leach\u2019s storm petrel: O2 Uptake, water loss, and microclimate of the nest","volume":"91","author":"Rahn","year":"1988","journal-title":"Comp. Biochem. Physiol. Part A Physiol."},{"key":"ref_109","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1007\/BF01649755","article-title":"The initial density of avian eggs derived from the tables of Sch\u00f6nwetter","volume":"130","author":"Rahn","year":"1989","journal-title":"J. Ornithol."},{"key":"ref_110","first-page":"132","article-title":"Population growth model and avalanche","volume":"28","author":"Souza","year":"2024","journal-title":"Rev. Fisio Ter."},{"key":"ref_111","first-page":"24891","article-title":"Exponential functions through a real-world context","volume":"3","author":"Budinski","year":"2020","journal-title":"Open Sch. J. Open Sci."},{"key":"ref_112","doi-asserted-by":"crossref","first-page":"1875","DOI":"10.1128\/aem.56.6.1875-1881.1990","article-title":"Modeling of the bacterial growth curve","volume":"56","author":"Zwietering","year":"1990","journal-title":"Appl. Environ. Microbiol."},{"key":"ref_113","doi-asserted-by":"crossref","first-page":"12","DOI":"10.11613\/BM.2014.003","article-title":"Understanding logistic regression analysis","volume":"24","author":"Sperandei","year":"2014","journal-title":"Biochem. Med."},{"key":"ref_114","unstructured":"Clement, L. (2020). 6. Simple Linear Regression. PSLS20: Practical Statistics for the Life Sciences, Gulbenkian Training Programme in Bioinformatics (GTPB) Courses, Instituto Gulbenkian de Ci\u00eancia. Available online: https:\/\/gtpb.github.io\/PSLS20\/pages\/06-linearRegression\/06-linearRegression.html."},{"key":"ref_115","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1007\/BF01081220","article-title":"Pade approximants (review)","volume":"22","author":"Apresyan","year":"1979","journal-title":"Radiophys. Quantum Electron."},{"key":"ref_116","unstructured":"Baker, G.A., and Graves-Morris, P. (1996). Pad\u00e9 Approximants, Cambridge University Press. [2nd ed.]."},{"key":"ref_117","doi-asserted-by":"crossref","unstructured":"Kalospyros, S.A., Nikitaki, Z., Kyriakou, I., Kokkoris, M., Emfietzoglou, D., and Georgakilas, A.G. (2021). A mathematical radiobiological model (MRM) to predict complex DNA damage and cell survival for ionizing particle radiations of varying quality. Molecules, 26.","DOI":"10.3390\/molecules26040840"},{"key":"ref_118","unstructured":"Khaliduzzaman, A., Ahad, M.A.R., Kamruzzaman, M., and Aboonajmi, M. (2026). Creating a simulated virtual collection of avian egg shapes. Handbook of Egg Sensing Technology, CRC Press. Available online: https:\/\/books.google.com\/books?id=a4fc0QEACAAJ."},{"key":"ref_119","doi-asserted-by":"crossref","unstructured":"Bokulich, A. (2008). Reexamining the Quantum\u2013Classical Relation. Beyond Reductionism and Pluralism, Cambridge University Press. Available online: https:\/\/bokulich.org\/wp-content\/uploads\/2020\/06\/reexamin-q-c-book_bokulich.pdf.","DOI":"10.1017\/CBO9780511751813"},{"key":"ref_120","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1207\/s15327957pspr0804_3","article-title":"Processing fluency and aesthetic pleasure: Is beauty in the perceiver\u2019s processing experience?","volume":"8","author":"Reber","year":"2004","journal-title":"Pers. Soc. Psychol. Rev."},{"key":"ref_121","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1111\/j.1467-9264.2013.00346.x","article-title":"V\u2014Aesthetics in science: A Kantian proposal","volume":"113","author":"Breitenbach","year":"2013","journal-title":"Proc. Aristot. Soc."},{"key":"ref_122","doi-asserted-by":"crossref","unstructured":"Zeki, S., Romaya, J.P., Benincasa, D.M., and Atiyah, M.F. (2014). The experience of mathematical beauty and its neural correlates. Front. Hum. Neurosci., 8.","DOI":"10.3389\/fnhum.2014.00068"},{"key":"ref_123","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1007\/s10699-014-9378-7","article-title":"Mathematical beauty, understanding, and discovery","volume":"20","author":"Cellucci","year":"2015","journal-title":"Found. Sci."},{"key":"ref_124","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1002\/arcp.1067","article-title":"Metacognitive experiences as information: Processing fluency in consumer judgment and decision making","volume":"4","author":"Schwarz","year":"2021","journal-title":"Consum. Psychol. Rev."},{"key":"ref_125","doi-asserted-by":"crossref","unstructured":"Moskowitz, M.A. (2002). A Course in Complex Analysis in One Variable, World Scientific.","DOI":"10.1142\/4836"},{"key":"ref_126","doi-asserted-by":"crossref","unstructured":"Maasz, J., and O\u2019Donoghue, J. (2011). Mathematics and eggs. Real-World Problems for Secondary School Mathematics Students, Sense Publishers.","DOI":"10.1007\/978-94-6091-543-7"},{"key":"ref_127","first-page":"61","article-title":"Mathematics and eggs\u2014Do those two terms have something in common?","volume":"21","author":"Siller","year":"2011","journal-title":"Simul. Notes Eur."},{"key":"ref_128","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1134\/S1062359022602701","article-title":"What comes first: The egg or the mathematics? Review article","volume":"50","author":"Narushin","year":"2023","journal-title":"Biol. Bull."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/14\/5\/101\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T04:14:06Z","timestamp":1778559246000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/14\/5\/101"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,27]]},"references-count":128,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2026,5]]}},"alternative-id":["computation14050101"],"URL":"https:\/\/doi.org\/10.3390\/computation14050101","relation":{},"ISSN":["2079-3197"],"issn-type":[{"value":"2079-3197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,4,27]]}}}