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Equation of Egg Shaped Curve VI

Nobuo YAMAMOTO

You can copy and use all the figures in this page freely.

    We will treat the following equation which may be thought to give the nearest shape of egg to an actual one as a result of transformation of the equation from the concave circle (referred in convex circle) into an egg shaped curve.

                     .                      (1)
If we solve Eq.(1), the following equation is derived.

                     .                       (2)
The condition that Eq.(2) exists is

                     .                       (3)

    In the result of calculation of Eq.(2) with the use of computer in order to pursuit the nearest shape to an actual egg, it is recognized that the egg shaped curves exist only in the narrow range of small values of the parameter a including a=0.   An example in this case is displayed in Fig.1.   The pink colored curve in this figure indicates the previously found egg shaped curve (referred in egg shaped curve) as a reference.
    The value of a becomes large, the curve is transformed from the egg shaped curve into that shown in Fig.2 gradually.   When a becomes larger, Eq.(3) is not satisfied, and any curve does not exist.   Furthermore, as the value of a exceeds some value, a figure dispersing to the y direction is displayed as shown in Fig.3.

Fig.1 Comparison between the newly obtained curve by Eq.(5)
(The type in this article) and the previously founded one
Fig.2

Fig.3

    Comparison between the curve in the case that a=0, b=2.0, c=2.65 and d=k1=k2=k3=1 and the shape of an actual egg is shown in Fig.4, where the value of the parameter c is a little changed from that in Fig.1.   As seen in Fig.4, the egg shaped curve made from the equation of the concave distorted circle may approximately coincide to the shape of an actual egg.


Fig.4 Comparison between the egg shaped curve in
the case that a=0, b=2.0, c=2.65 and d=k1=k2=k3=1

(pink colored curve) and the shape of an actual egg

    In purpose to calculate the numerical coordinates data of a single curve as shown in Figs.1, 2 and 3, a C++ program originated from Eq.(2) 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.

    In the next, the transformation of the equation from the convex circle to an egg shaped curve leads to

                     .                      (4)
If we solve Eq.(4), the following equation is derived.

                     .                       (5)
The condition that Eq.(5) exists is

                     .                       (6)
    When we pursuit the nearest shape to an actual egg as calculating Eq.(5) with the use of computer, it is recognized that the egg shaped curves are found in the wide range of parameters.   Three cases of ,     and the other one are displayed with the blue colored curves in Figs.5, 6 and 7 respectively.   The pink colored curves in these figures indicate the previously found egg shaped curve as a reference.

   It must be paid attention to that Eq.(4) itself is invalid as when .


Fig.5 Comparison between the newly obtained curve by Eq.(2)
(The type in this article) and the previousely founded one
Fig.6

Fig.7

    As an example, comparison between the curve in the case that a=0, b=1.5, c=2.2 and d=k1=k2=k3=1 and the shape of an actual egg is shown in Fig.8, where the values of the parameters a and b are a little changed from those in Fig.5.   As seen in Fig.8, the egg shaped curve made from the equation of the convex distorded circle do not coincide to the shape of an actual egg so well.



Fig.8 Comparison between the egg shaped curve in
the case that a=0, b=1.5, c=2.2 and d=k1=k2=k3=1

(pink colored curve) and the shape of an actual egg

    In purpose to calculate the numerical coordinates data of a single curve as shown in Figs.5, 6 and 7, a C++ program originated from Eq.(5) is given by C++_program_single_curve.



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Updated: 2009.11.26, edited by N. Yamamoto.
Revised on Mar. 16, 2015, Jul. 22, 2016, May 05, 2020, Jan. 17, 2021 and May 08, 2021.