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Equation of Egg Shaped Curve IV (Egg Shaped Curve II proposed by Mr. Asai)

Mr. Yasuyuki ASAI  (Machida city, Tokyo, Japan)   and   Nobuo YAMAMOTO

Mr. Yasuyuki ASAI has informed more novel types of egg shaped curves to Yamamoto in September, 2009.
This thesis is described by YAMAMOTO in account of these novel two types of the equations expressing the egg shaped curves.


Fig.1 The first type of egg shaped curves
1.   The first type of equation

    One of two novel types of the equations expressing egg shaped curve proposed by Mr. Yasuyuki ASAI is written in the following;

                     .                      (1)

   Eq.(1) is transformed into the following equation.

                    .                      (2)

    If we calculate Eq.(2) as varying several values of b with the use of computer, we obtain Fig.1.    Thus, we may find egg shaped curves as seen in Fig.1.

   Comparison between the curve in the case that a=1.5, b=1.05 and c=11 and the shape of an actual egg is shown in Fig.2.    As seen in Fig.2, the egg shaped curve may approximately coincide to the shape of an actual egg.




Fig.2 Comparison between the egg shaped curve
in the case that a=1.5, b=1.05, c=11
(pink colored
curve) and the shape of an actual egg

   In purpose to calculate the numerical coordinates data of five species of curves as shown in Fig.1, a C++ program originated from Eq. (1) is given by C++_program_five_curves.    Another C++ program treating a single curve is given by C++_program_single_curve.
    By executing the either C++ program, a common text file named "egg_shaped_curve.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 an egg shaped curve with the use of a graph wizard attached on the Excel file.


Fig.3 The second type of egg shaped curves


2.   The second type of equation


   Another one of two novel types of the equations expressing egg shaped curve proposed by Mr. Yasuyuki ASAI is written in the following;

                    .                      (3)

    If we calculate Eq.(3) as varying the several values of b with the use of computer, we obtain Fig.3.

   Thus, we may find egg shaped curves as seen in Fig.3.

    Comparison between the curve in the case that a=1.35, b=0.9 and c=0.5 and the shape of an actual egg is shown in Fig.4.    As seen in Fig.4, the egg shaped curve may approximately coincide to the shape of an actual egg.





Fig.4 Comparison between the egg shaped curve
in the case that a=1.35, b=0.9 and c=0.5
(pink
colored curve) and the shape of an actual egg

   In purpose to calculate the numerical coordinates data of five species of curves as shown in Fig.1, a C++ program originated from Eq. (1) is given by C++_program_five_curves3b.    Another C++ program treating a single curve is given by C++_program_single_curve3.
    By executing the either C++ program, a common text file named "egg_shaped_curve.txt" including the calculated data is produced.    Each interval of these data is devided by 'comma'.    After moving these calculated data into an Excel file, we obtain an egg shaped curve with the use of a graph wizard attached on the Excel file.



3.   A conventional method to connect two ellipsoids into an egg shaped curve
Fig.5 Egg shaped curves into the each of which two ellipsoids are connected


   A simple method to connect two ellipsoids into an egg shaped curve is also proposed by Mr. Yasuyuki ASAI.   The equation for expressing such an egg shaped curve is written in the following;

                     .                      (4)
In this equation, sgn(x) is defined as

                     .                      (5)

    In the case of c=a, Eq.(4) gives an ellipsoid.    If we calculate Eq.(4) as varying the several values of c with the use of computer, we obtain Fig.5.
    Comparison between the curve in the case that a=1, b=0.72 and c=0.9 and the shape of an actual egg is shown in Fig.6.    As seen in Fig.6, the egg shaped curve may approximately coincide to the shape of an actual egg.




Fig.6 Comparison between the egg shaped curve
in the case that a=1, b=0.72 and c=0.9
(pink
colored curve) and the shape of an actual egg

   In purpose to calculate the numerical coordinates data of five species of curves as shown in Fig.5, a C++ program originated from Eq. (4) is given by C++_program_five_curves3b.    Another C++ program treating a single curve is given by C++_program_single_curve3.
    By executing thte either C++ program, a common text file named "egg_shaped_curve.txt" including the calculated data is produced.    Each interval of these data is devided by 'comma'.    After moving these calculated data into an Excel file, we obtain an egg shaped curve with the use of a graph wizard attached on the Excel file.


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Updated: 2009.09.18, edited by N. Yamamoto.
Revised on Mar. 03, 2020, May 05, 2020, Jan. 17, 2021 and May 08, 2021.