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3D Displays of Egg and Apple

Mr. Svein Daniel Solvenus (in Norwey) and Nobuo YAMAMOTO
 The request is informed from Mr. Svein Daniel Solvenus to Yamamoto in 2011 how we can make 3D display of an egg with the use of the equation described in Yamamoto's website.   Therefore, Yamamoto make an equation of egg surface and C program for its calculation.   Then, the coordinates data of the egg surface written in an Excel file by calculation are sent to Mr. Daniel.   Mr. Daniel create 3D figure of egg using its Excel file, and moreover, create that of apple. 1. Apple Shape     The first model is an egg shape made as like that the egg shaped curve given in Eq.(9b) in the page of Yamamoto's Website is made revolve around x axis.   If we introduce the phase angle in (y, z) plane, the egg surface is formulated as ,                      (1) where indicates the phase angle in (y, z) plane and is refined in 0 < < 2 .     With the use of the coordinates data calculated by Eq.(1), Svein Daniel Solvenus creates the 3D displays as seen in Figs.1 and 2. Fig.1 3D display of an egg in the first modelcreated by Mr. Svein Daniel Solvenus Fig.2 the same as the left created by Mr. Svein Daniel Solvenus

 However, as seen in Fig.3, this model has the demerit that the holes exist in both sides and the mesh intervals are extremely not uniform.Moreover, it is much difficult that its demerit is removed in this first model. Fig.3　The hole existance and non-uniformity of mesh

 Therefore, as the second model, we treat an egg shape which is made rotating the egg curve described in Eq.(1) given in the webpage of "Egg Shaped Curve II" around x axis.   If we introduce the phase angle in (y, z) plane, the egg surface is formulated from Eq.(1) in this webpage in the same manner as described in the above as ,                      (2) where 0 < < 2 .     Then, with the use of the coordinates data calculated by Eq.(2) with computer, we obtain the front view and plant of an egg shape as seen in Fig.4.   It seems to moderate the demerit mentioned above.  Fig.4 The front and side views of egg shape by the second model [Eq.(2)]

 In purpose to calculate the numerical coordinates data of an egg surface as shown in Fig.4, a C++ program originated from Eq.(2) is given by .    By executing the C++ program, a text file named "egg_surface.txt" including the calculated 3D coordinates data of egg surface is produced.    Each interval of these data is divided by 'comma'.    After moving these calculated data into an Excel file, we obtain a spiral with the use of a graph wizard attached on the Excel file.

2. Apple shape
Though the equation of apple shaped curve in (x, y) plane is given in the Eq.(3) in the page of "apple shaped curve", it is written in the following equation to give 3D display of an apple in accordance to the same process described above. ,                      (3)
where 0 < < 2 .

With the use of the coordinates data calculated by Eq.(3) with computer, Svein Daniel Solvenus creates the 3D displays as seen in Figs.5 and 6.   The 3D picture in Fig.6 is very beautiful.   Thanks to Mr. Svein Daniel Solvenus. Fig.5 3D display of an apple in the second model created by Mr. Svein Daniel Solvenus Fig.6 3D display of an apple created and colord by Mr. Svein Daniel Solvenus

 In purpose to calculate the numerical coordinates data of an apple surface as shown in Figs.5 and 6, a C++ program originated from Eq.(3) is given by .    By executing the C++ program, a text file named "apple_surface.txt" including the calculated data is produced.    Each interval of these data is divided by 'comma'.    After moving these calculated data into an Excel file, we obtain a spiral with the use of a graph wizard attached on the Excel file.

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Updated: 2011.01.22, edited by N. Yamamoto.
Revised on Mar. 16, 2015, Jul. 24, 2016, May 05, 2020, Jan. 30, 2021, May 05, 2021 and Mar. 22, 2022.