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Equation of Egg Shaped Curve III (Egg Shaped Curve I proposed by Mr. Asai)

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

Mr. Yasuyuki ASAI has informed another novel type of an egg shaped curve to Yamamoto in July, 2009.
This thesis is described by YAMAMOTO in account of the equation of the egg shaped curve proposed by him,
and advanced calculations are performed also by YAMAMOTO.

    This site is introduced in Bounds of rotated egg? , Principal room , [Curve-12] Let's draw an egg curve , GENERATING AVIAN EGG USING RATIONAL BEZIER QUADRATIC CURVES .

    As seen by looking at each site or treatise, the author's name written in it is "jukuchou and Yamamoto", "Yamamoto".
    On the otherhand, the address of this site has passed "http://www16.ocn.ne.jp:80/~akiko-/Egg_by_SuudokuJuku/index_egg_by_SuudokuJuku_E.html" , "http://www.geocities.jp/nyjp07/Egg_by_SuudokuJuku/index_egg_by_SuudokuJuku_E.html" , "http://www.geocities.jp/nyjp07/index_egg_by_SuudokuJuku_E.html" and becomes the current "http://nyjp07.com/index_egg_by_SuudokuJuku_E.html" .
    We can't use the old address, but searching for old addresses with a free Internet Archive search may reveal the site at the time it was saved.

    The equation expressing another novel type of egg shaped curve proposed by Mr. Yasuyuki ASAI is written in the following;

                     ,                        (1)
where the constant is used instead of the variable according to the mathematical convention.

    If we calculate Eq.(1) as varying the several values of with the use of computer, we obtain three figures such as Fig.1 in the case of . Fig.2 in the case of and Fig.3 in the case of , where we fix the value of constant as without loss of the generality.


Fig.1　In the case of


Fig.2　In the case of


Fig.3　In the case of


    Thus, we may find egg shaped curves in Fig.1.
    Comparison between its curve (Mr. Yasuyuki ASAI's) and Yamamoto's one (referred in egg shaped curves) is shown in Fig.4 as we choose the nearest shape of an actual egg.     Approximately, both curves are identical.     For the comparison, Yamamoto's curve has been displaced by some values and then, has been reversed in the x direction.




Fig.4　Comparison between Mr. Yasuyuki ASAI's egg shaped curve
and Yamamoto's.
Here, Yamamoto's is displaced by some values and then, is reversed
in the x direction for the comparison.



Fig.5 Comparison between the egg
shaped curve in the case that a=1
and c=2.9 (pink colored curve)
and the shape of an actual egg

     The curve in the case that a=1 and c=2.9 gives the closest shape to an actual egg.      The comparison between the curve in this case and the shape of an actual egg is shown in Fig.5.

    In the last, Eq.(1) may be transformed into Eq.(2).

                     .                        (2)


In purpose to calculate the numerical coordinates data of five species of curves as shown in Figs.1 to 3, 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.     In the latter program, the constants are settled as =1, and =0.29.
     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.



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Updated: 2009.08.02, edited by N. Yamamoto.
Revised on Mar. 16, 2015, Mar. 03, 2020, May 05, 2020, Jan. 17, 2021, May 08, 2021, Sep. 23, 2021, Oct. 18, 2021 and Mar. 06, 2022.