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Egg Shaped Curve VII (Egg Shaped Curve III proposed by Mr. Asai)

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

Mr. Yasuyuki ASAI has informed novel type of an egg shaped curve to Yamamoto in April, 2011.
This report is described by YAMAMOTO in account of the equation of the egg shaped curve proposed by him,

    The equation expressing a novel type of egg shaped curve (obtained by the method using the difference between the first and the second order equations) proposed by Mr. Yasuyuki ASAI is written in the following;

                     ,                        (1)

or, is written in the expression of factorization as

                     .                        (2)

    In this time, we treat Eq.(2).   The solution of Eq.(2) is given as

                     .                        (3)

    Four points with which the curves displayed by Eq.(3) intersect the x axis are obtained by the following equation into which Eq.(3) is put as y=0.

                     .                        (4)

    If we solve the above equation, we obtain as

                     ,                        (5)

                     ,                        (6)


                     .                        (7)

    In the recommended case that , Eq.(3) gives an egg shaped curve in the region of and another accompanied curve in the region of .     In the other regions, any curves do not exist.     These are shown in Fig.1.

Fig.1 In the recommended case that

Fig.2 Comparison with a real egg

In purpose to calculate the numerical coordinates data of an egg-shaped curve and an accompanied curve as shown in Fig.1, a C++ program originated from Eq. (3) is given by C++_program_multi_eggs.
     By executing the C++ program, two text files named "egg-shaped curve_1.txt" and "egg-shaped curve_2.txt" including the calculated coordinates data of curves are produced.      The latter includes the coordinates data of "an egg curve" and the former includes those of "the accompanied curve".      Each interval of these data is divided by 'comma'.      After moving these calculated data into Excel file, we obtain an egg curve and an accompanied curve with the use of graph wizard attached on the Excel file.

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Updated: 2011.05.29, edited by N. Yamamoto.
Revised on Mar. 16, 2015, Aug. 15, 2019, Mar. 03, 2020, May 05, 2020, Jan. 17, 2021 and May 08, 2021.