Mathematical modelling of hen s egg shape by rotation curve

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1 Annals of Warsaw Universiy of Life Sciences SGGW Agriculure No 6 (Agriculural and Fores Engineering) 204: (Ann. Warsaw Univ. Life Sci. SGGW, Agricul. 6, 204) Mahemaical modelling of hen s egg shape by roaion curve LESZEK MIESZKALSKI Deparmen of Producion Managemen and Engineering, Warsaw Universiy of Life Sciences SGGW Absrac: Mahemaical modelling of hen s egg shape by roaion curve. The proposed mehod concerns a mahemaical modelling of he hen s eggs. To describe he hen s egg shape here was used he Bézier curve, roaed in relaions o z axis; he obained surface was scaled. The Bézier curve described a half of conour of egg s longiudinal secion. The developed mahemaical model can be used in generaing D solids ha are similar o hen s eggs in respec of shape and basic dimensions. The mahemaical model allows for he change in egg s lengh, widh and hickness as well as in coordinaes of nodal and conrol poins of Bézier curve ha describes a half of conour of he egg s longiudinal secion. Key words: hen s egg, shape, mahemaical modelling, Bézier curve INTRODUCTION Eggs are valuable componen of human die in respec of heir nuriional value. They belong o producs of animal origin and are valuable raw maerial in food indusry. They conain almos all viamins and polysauraed fay acids as well as a balanced composiion of amino acids and mineral componens [Niewiarowicz e al. 99, Trziszka 2000, Siepka e al. 200]. The shape of eggs is considered in designing of machines and equipmen used in heir producion and processing; i is also one of qualiy parameers assessed by cusomers [Rashidi and Keshavarzpour 20]. Geomerical properies of eggs are imporan in biological invesigaions and poulry indusry [Narushin 2005]. The mehods for sock managemen used in producion and rade enerprises [Buliski e al. 202, 20], poulry indusry and climae conrolling in poulry- -houses [Lysenko e al. 20], producion roboics [Shvorov e al. 202], as well as applicaion of he mehods for produc classicaion and diagnosics in respec o signicanly differen properies [Janaszek and Trajer 200] enable o arrange daa and idenify hese properies, working parameers and design decisions ha inuence he produc qualiy. In descripion of he eggs shape here can be used he mehods ha are applied in descripion of raw maerial of agriculural origin proposed by many researchers [Kska and Feder 997, Mabille and Abecassis 200, Donev e al. 2004, Frczek and Wróbel 2006, 2009, Mieszkalski 20]. Informaion needed for modelling D objecs of various origing are given by compuer graphics [Kiciak 2000, Foley e al. 200]. This work aimed a developmen of mahemaical model describing an exernal shape of he hen s eggs by roaion of Bézier curve.

2 60 L. Mieszkalski MATERIAL AND METHODS The research maerial were he hen s eggs originaed from a lier keeping sysem and disribued by P.H. Czachorowski ( Three hen s eggs of differen shape were seleced and phoographed. Their lengh, widh and hickness were measured wih a slide caliper wih accuracy 0. mm; he maximal diameers of eggs were measured along heir lengh every 0, 25, 5, 45 and 55 mm. To describe he shape of half of conour of he egg s longiudinal secion here was used he Bézier curve; by is roaion in respec of z axis he roaional surface was obained. The creaed roaional surfaces of solids were scaled o obain heir shapes similar o shape of seleced hen s eggs. Visualizaion of menioned D solid models was performed wih he use of compuer program Mahcad. The hen s egg dimensions are presened in Table, exemplary phoographs of he hen s eggs are presened in Figure. TABLE. Lengh (a) and diameer dimensions relaed o secions of seleced hen s eggs Disance beween measured diameers of eggs [mm] Diameers of eggs [mm] I a = 60 mm II a = 57 mm III a = 56 mm FIGURE. Hen s eggs designaed as I, II and III (dimensions in Table ) DESCRIPTION OF MODEL Marix equaions of coordinaes xa, za of Bézier curve poins for egg I have he form: xi xi xi 2 N N 2 2 xi N N xi 4 N N zi zi zi 2 N N 2 2 zi N N zi 4 N N () (2) Marix equaions for eggs II and III have analogical noaion. Coordinaes of nodal and conrol poins ha occur in he equaions (, 2) for Bézier curve of eggs I, II and III are wrien down in marixes, 4 and 5:

3 Mahemaical modelling of hen s egg shape by roaion curve 6 xi zi 0 60 xi 2 zi xi zi xi 4 zi4 0 0 xii zii 0 57 xii 2 zii xii zii xii 4 zii4 0 0 () (4) ai aii aiii bi bii biii ci cii ciii (2) To obain he given dimensions (a, b, c) of he hen s egg model one should scale equaions 7, 8 and 9. The scaled marix equaions of coordinaes XI, YI and ZI of nodal poins of he ne of surface ha describes he shape of egg I have he following form: xiii ziii 0 56 xiii 2 ziii xiii ziii 9 0 xiii 4 ziii4 0 0 (5) To obain roaional surface ha represens he egg I one should roae Bézier curve in relaions o z axis wih he use of equaions: XI,j = xi sin( j) (6) YI,j = xi cos( j ) (7) ZI,j = zi (8) where: 2 j j (9) N j = 0 N (0) = 0 N () where: N marix size (i was assumed N = 8); number of rows; j number of columns. Similarly, one should roae Bézier curves for he eggs II and III. The lengh a, widh b and hickness c of real measuremens on he eggs I, II and III are included in marix 2. XI YI ZI bi XI () max( XI) min( XI) ci YI max( YI) min( YI) aa ZI max( ZI) min( ZI) (4) (5) Similar calculaions should be made for eggs II and III. The D models for eggs I, II and III are presened in Figure 2. VERIFICATION OF MODEL The developed mahemaical model ha describes he shape of hen s eggs was iniially veried by comparison of overlapped projecions on XZ plane of he eggs models and images (Fig. ). Comparing he overlapped projecions of eggs I, II and III models and images one can nd, ha he shapes of solids represened by he eggs models (Fig. 2) are similar o real eggs presened in Figure. Basic dimensions lengh, widh and hickness are idenical for he models and images (resul of scaling).

4 62 L. Mieszkalski FIGURE 2. D models for eggs I, II and III FIGURE. Overlapped projecions on XZ plane of models and images of eggs I, II and III Small discrepancy may occur in he remaining secions (Table 2). Imaging inaccuracy occurred wihin he secion a disance 0 mm (egg II wih relaive error 5.7%, egg III wih relaive error 5.8%) and wihin he secion a disance 55 mm (egg II wih relaive error 6.7%, egg III wih relaive error 5.7%). The proposed mahemaical models can be used in generaing he D solids similar o hen s eggs in respec of heir shape and basic dimensions and can be applied in agriculural and food engineering, where here is no need for very high imaging accuracy. TABLE 2. Difference in diameer measuremens for model and egg and relaive error Disance beween measured diameers of eggs [mm] Difference in diameer measuremen Relaive error [%] for model and egg [mm] I II III I II III

5 Mahemaical modelling of hen s egg shape by roaion curve 6 CONCLUSIONS. 2. The developed model represened wih surface obained by roaion of Bézier curve afer scaling can be used in generaing D solids similar o he hen s eggs in respec of heir shape and basic dimensions. In he proposed mahemaical model he parameers ha conrol he shape are basic dimensions of egg (lengh, widh, hickness) and coordinaes of he nodal and conrol poins of Bézier curve ha describes he half of longiudinal conour of he egg secion. REFERENCES BULISKI J., WASZKIEWICZ C., BURA- CZEWSKI P. 202: Sock managemen as an elemen of enerprise sraegy. Annals of Warsaw Universiy of Life Sciences SGGW, Agriculure 60: BULISKI J., WASZKIEWICZ C., BURA- CZEWSKI P. 20: Uylizaion of ABC/ /XYZ analysis in sock planning in he enerprise. Annals of Warsaw Universiy of Life Sciences SGGW, Agriculure 6: DONEV A. e al. 2004: Improving he densiy of jammed disordered packings using ellipsoids. Science 0: FOLEY J.D., van DAM A., FEINER, S.K., HUGHES J.F., PHILLIPS R.L. 200: Wprowadzenie do graki kompuerowej. WNT, Warszawa. FRCZEK J., WRÓBEL M. 2006: Meodyczne aspeky oceny kszau nasion. Inynieria Rolnicza 2: FRCZEK J., WRÓBEL M. 2009: Zasosowanie graki kompuerowej w rekonsrukcji D nasion. Inynieria Rolnicza 6 (5): JANASZEK M., TRAJER J. 200: Classicaion sysem as diagnosics ool for he plan producs. Annals of Warsaw Universiy of Life Sciences SGGW, Agriculure 56: KSKA W., FEDER S. 997: Trójwymiarowa rekonsrukcja kszau elemenów rolinnych z dwuwymiarowych obrazów rasrowych. Prace Przemysowego Insyuu Maszyn Rolniczych: 5 7. KICIAK P. 2000: Podsawy modelowania krzywych i powierzchni. Zasosowanie w grace kompuerowej. WNT, Warszawa. LYSENKO V., GOLOVINSKIY B., RESHE- TYUK V., GOLUB B., SHCHERBA- TYUK V. 20: Dynamics of qualiy indexes of laying hens keeping process due o ucuaions of emperaure disurbances in an indusrial poulry house. Annals of Warsaw Universiy of Life Sciences SGGW, Agriculure 57: MABILLE F., ABECASSIS J. 200: Parameric modelling of whea grain morphology: a new perspecive. Journal of Cereal Science 7: 4 5. MIESZKALSKI L. 20: Meoda maemaycznego modelowania kszau bryy ziarna pszenicy za pomoc paramerycznej krzywej przesrzennej i czerowzowej siaki. Pospy Techniki Przewórswa Spoywczego, 2/8: NARUSHIN V.G. 2005: Producion, Modeling, and Educaion. Egg Geomery Calculaion Using he Measuremens of Lengh and Breadh. Poulry Science 84: NIEWIAROWICZ A. 99: Budowa, skad chemiczny, waciwoci zykochemiczne i funkcjonalne oraz waro odywcza jaj. W: Technologia jaj praca zbiorowa. WNT, Warszawa: 8. RASHIDI M., KESHAVARZPOUR F. 20: Classicaion of Egg Size and Shape Based on Mass and Ouer Dimensions Analysis. Libyan Agriculure Research Cener Journal Inernaional 2 (5):

6 64 L. Mieszkalski SHVOROV S., RESHETYUK V., BOL- BOT I., SHTEPA V., CHIRCHENKO D. 202: Theoreical issues cosrucion and operaion of agriculural mission roboic sysem. Annals of Warsaw Universiy of Life Sciences SGGW, Agriculure 60: SIEPKA E., BOBAK., TRZISZKA T. 200: Frakcjonowanie óka w celu pozyskiwania preparaów wzbogaconych w subsancje biologicznie akywne. ywno. Nauka. Technologia. Jako 6 (7): TRZISZKA T. (red.) 2000: Jajczarswo. Nauka, echnologia, prakyka. Wyd. AR we Wrocawiu, Wrocaw. Sreszczenie: Maemayczne modelowanie kszau jaja kurzego przez obró krzywej. Proponowana meoda doyczy maemaycznego modelowania jaj kurzych. W modelu maemaycznym do opisu kszau jaja kurzego zasosowano krzyw Béziera, kór obrócono wzgldem osi z, a uzyskan powierzchni skalowano. Opracowany model maemayczny moe by sosowany do generowania bry D podobnych pod wzgldem podsawowych wymiarów do jaj kurzych. W modelu maemaycznym isnieje moliwo zmiany waroci dugoci, szerokoci, gruboci jaja oraz wspórzdnych punków wzowych i konrolnych krzywej Béziera opisujcej lini poowy konuru wzdunego przekroju jaja. MS. received November 20 Auhor s address: Leszek Mieszkalski Wydzia Inynierii Produkcji SGGW Kaedra Organizacji i Inynierii Produkcji Warszawa, ul. Nowoursynowska 64 Poland leszek_mieszkalski@sggw.pl