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

Mr. Tadao ITOU   (Ikoma city, Nara prefecture, Japan)    and   Nobuo YAMAMOTO

Novel type expression of egg shaped curves has been proposed by Mr. ITOU, and contributed to this page in March, 2008.
This thesis is described by YAMAMOTO as based on fragmentary manuscripts sent from Mr. ITOU and advanced calculations are performed by YAMAMOTO.
Surprisingly, as the equation of egg shaped curve is not found so easily, this thesis will also become valuable.
An evolution of this method can be found in The Egg Equations .
    Also, about this method, The equation that represents the egg of the Twitter default icon is found (?),     same (archive version) has been announced.
    In addition, this site is introduced in GENERATING AVIAN EGG USING RATIONAL BEZIER QUADRATIC CURVES , A new oval curve inspired by Mr. Ito , and this site is also cited in papers such as Natural Aesthetic Curves for Design .

    As seen by looking at each site or treatise, the author's name written in it is "Tadao Itou", "Tadao Itou & Nobuo Yamamoto" or "Itou & Yamamoto".
    On the otherhand, the address of this site has passed "Http://", "", "", and becomes the current "".
    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. Equation of Egg-shaped-curve II   (expressed by the intermediate valuable of phase angle )

Fig.1 Egg shaped curve [Eq.(1)] found by Mr. Itou.

    The proposed equation is expressed with the use of the intermediate valuable of the phase angle of the plane coordinate as in the following;
                     .                        (1)
     In the case of , if we calculate Eq.(1) as varying the several values of with the use of computer, the each egg-like shaped curve is drawn as shown in Fig.1.

    Comparison between its curve (Itou's) and Yamamoto's one (refered in Yamamoto's egg shaped curve) is shown in Fig.2.     Both curves nearly coincide to each other.     For the comparison, Itou's curve has been displaced by the value of in the direction.

    If we remove the definition range of in Fig.1 and we treat all the range of , the displayed curves become to be shown in Fig.3, where and .     Actually, only a finite range of is sufficient to give the full curves in Fig.3.     The outline curve (connected curve a and d) is as the same as the shape of the egg shaped curve of in Fig.1.

    By the way, there is an interesting report (in Japanese) about this egg-shaped curve. (Refer to reference

Fig.2 Comparison between Itou's egg-shaped curve and Yamamoto's.
Here, Ttou's is displaced by the value of a(=0.5)
in the direction for the comparison.

Fig.3 Egg shaped curve [Eq.(1)] found by Mr. Itou
as shown over all the range of .
Each curve a, b, c, d should be referred in Eq.2.

Fig.4 Comparison between the egg shaped curve
   (pink colored curve)
and the shape of an actual egg

    The curve in the case that and (in general, the case of ) 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.4.

    The each curve a, b, c and d shown in Fig.3 is corresponding to the each range of discribed in the following equations;

                  ,           (2)
where n is an integer.

    As Eq.(2) is a little complicated to recognize the each range, we will show the typical case of n=0 in Eq.(2) as the followings;
                  .           (2)'

    However, as an area of a, b, c or d as shown in Fig.3 is duplicated as seen in the above equation while each area intersects complicatedly, so the distinction of the each area is not understood well yet.     Therefore, when only the middle part of the range of in the above equation is picked out so as not to be duplicated, we can show a simple description in respect to the range of as follows.

                  .           (2)''

    In purpose to calculate the numerical coordinates data of four species of egg shaped curves as shown in Fig.1, a C++ program originated from Eq. (1) is given by C++_program_four_eggs.     Another C++ program treating a single egg shaped curve is given by C++_program_single_egg.    In the latter program, the constants are settled as =0.5 and =0.37.     In addition, C++ program treating an original egg shaped curve displayed in Fig.3 is given by C++_program_original_egg_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.
    In order to obtain the curves shown in Fig.3, the area of should be expanded into the region from =-2 to +2 in the above mentioned C++ programs.

[Reference] Relation between and indicating the curves shown in Fig.3

    If we take the square of the both sides of the each equation in Eq.(1),
                       (3)       and             .           (4)
If we apply the sinusoidal formula      and       into Eq.(4),
Substituting Eq.(3) into the above equation, we obtain
If we arrange the above equation more clearly,
If we take the square of the both sides of the above equation,
Applying the sinusoidal fomula       into the above equation, we obtain that
By substituting Eq.(1) in the above equation, Eq.(1) is led to
If we arrange the above equation more clearly, we obtain the 5th order equation as the following;
            .              (5)
    The order number of such the obtained equation is more than that of Yamamoto's egg equation (refered in Yamamoto's egg shaped curve) by 'a unit'.

2. Egg shaped curve as a section made by cutting a Pseudo-sphere by means of inclined plane  (Written by Mr. ITOU)

    It is well known that a section made by cutting a cone by means of inclined plane does not reveal an egg shaped curve but an ellipse.
    On the other hand, a section made by cutting a pseudo-sphere reveals an egg shaped curve as shown in Fig.5.

Fig.5 Egg-shaped-curve as a section made by cutting a Pseudo-sphere by means of inclined plane

    In Fig.5, the equations of the inclined plane and the are severally given in the followings;
       Plane;                   ,
       Pseudo-sphere;       ,
    Under the consideration that the common points of both the plane and the pseudo-sphere is precisely plotted into the obtained egg-shaped-curve, we have drawn this curve by varying the value of    with the use of the following equations.

                               (the same as the above),
    Problem is to obtain the equation displaying this egg shaped curve as in the form of   .

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Updated: 2008.03.29, edited by N. Yamamoto.
Revised on Mar. 16, 2015, Aug. 15, 2019, Aug. 17, 2019, May 05, 2020, Jan. 17, 2021, May 08, 2021, Sep. 11, 2021 and Sep. 23, 2021.