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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.

    This site is introduced in   GENERATING AVIAN EGG USING RATIONAL BEZIER QUADRATIC CURVES   and    Make An Egg-Shaped Model With Origami.
    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-y/Egg_by_SuudokuJuku2/index_egg_by_SuudokuJuku2_E.html" , "http://www.geocities.jp/nyjp07/Egg_by_SuudokuJuku2/index_egg_by_SuudokuJuku2_E.html" , "http://www.geocities.jp/nyjp07/index_egg_by_SuudokuJuku2_E.html" and becomes the current "http://nyjp07.com/index_egg_by_SuudokuJuku2_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.


1.   The first type of equation (extend the ellipse non-uniformly along the y-axis)
Fig.1　The first type of egg shaped curves

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

                     .                      (1)


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


   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.




   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 to calculate the numerical coordinates data of 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.


2.   The second type of equation
Fig.3　The second type of egg shaped curves

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

                    .                      (3)


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


    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.








   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 to calculate the numerical coordinates data of 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


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


                     .                      (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.





   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 to calculate the numerical coordinates data of 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, May 08, 2021, Sep. 23, 2021, Oct. 17, 2021, Oct. 28, 2021, Nov. 05, 2021, Nov. 08, 2021, Nov. 19, 2021, Dec. 01, 2021, Jan. 12, 2022 and Mar. 06, 2022.