﻿ Equation of Egg Shaped Curve V

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Equation of Egg Shaped Curve V

Nobuo YAMAMOTO

You can copy and use all the figures in this page freely.
 This site is introduced in   ПРИЛОЖЕНИЕ НА ТЕХНИКИ ЗА АНАЛИЗ НА ИЗОБРАЖЕНИЯ ПРИ ОЦЕНКА НА ВЪНШНИ ХАРАКТЕРИСТИКИ НА ЯЙЦА ( APPLICATION OF TECHNIQUES FOR IMAGE ANALYSIS IN ASSESSING THE EXTERNAL CHARACTERISTICS OF EGGS) , Introduction to Scratch for Programming Beginners-Pen Edition (7) .     As seen by looking at each site or treatise, the author's name written in it is "Yamamoto N.".     On the otherhand, the address of this site has passed "http://www.geocities.jp/nyjp07/index_egg5_E.html" and becomes the current "http://nyjp07.com/index_egg5_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.

 Newly found equation of egg shaped curve is written in the following;                      .                       (1)     As seen in the composition of Eq.(1), this equation is regarded to be a transformation of the equation of an ellipsoid.   If we calculate Eq.(1) with the use of computer, we obtain Fig.1, where we fix the value of constant as without loss of the generality.

 Fig.1

 Comparison between the newly obtained curve by Eq.(1) (Novel type) and the previously found one (referred in ) is shown in Fig.2 as we choose the nearest shape of an actual egg.   Approximately, both curves are identical.   For the comparison, the latter curve has been displaced by some values and has been reversed in the x direction.   The shape in the case that a=1, b=0.72 and c=0.08 gives the shape of an actual egg most.   The comparison between the curve in this case and the shape of an actual egg is shown in Fig.3.

 Fig.2　Comparison between the newly obtained curve by Eq.(1) (Novel type) and the previously founded one Fig.3 Comparison between the egg shaped curve in the case that a=1, b=0.72 and c=0.08 (pink colored curve) and 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 .    Another C++ program treating a single curve is given by .    In the latter program, we can vary the form of the curve as the constants and are changed.     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, Eq.(1) is replaced into the follows as the expression of the constants in Eq.(1) are changed.

.                      (2)

If we calculate Eq.(2) with the use of computer, we obtain Figs.4 and 5.    As seen in these figures, we can obtain strange curves by some values of c.

 Fig.4

 Fig.5

 In purpose to calculate the numerical coordinates data of five species of curves as shown in Fig.3, a C++ program originated from Eq.(2) is given by .

In the last, the expression of Eq.(1) in the orthogonal coordinates is led to the follows.

.                      (3)

[General expression]
It is reported from Mr. Yasuyuki ASAI that the above equation is rewritten from the following general expression.

.                      (4)

If the constants in Eq.(4) is replaced as , Eq.(4) leads to Eq.(1) in this page.   On the other hand, if the constants in Eq.(4) is replaced as , Eq.(4) leads to the first equation in the page of Mr. Yasuyuki ASAI and leads to Eq.(1) in ouranother page.   In another speaking, it is recognized that the egg shaped curve in this page is the same as that in the page of Mr. Suugaku-Dokusyuu-Juku Jukuchou and the analyses have been performed in the different ways each other.   Here, we appreciate his suggestion.

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Updated: 2009.11.08, edited by N. Yamamoto.
Revised on Aut. 30, 2013, Sep. 02, 2013, Mar. 16, 2015, Jul. 22, 2016, May 05, 2020, Jan. 17, 2021, May 08, 2021, Sep. 23, 2021 and Mar. 06, 2022.