Equation of Egg Shaped Curve II

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Equation of Egg Shaped Curve II

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 discribed 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.

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

) 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 outside two curves (a and d) is the same as the egg-curve of

in Fig.1.


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 cver all the range of . Each curve a, b, c, d should be refered in Eq.2.

図4 Comparison between the egg shaped curve
of (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)'

     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 .      Another C++ program treating a single egg shaped curve is given by .     In the latter program, the constants are settled as =0.5 and =0.37.
     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.
     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 fomula

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

) 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.4, 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