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When we search for the internet about an oval curve, several sites of Egg Curves and Ovals, Oval and Cassini oval are found. But there seem to be little equations of a curve near a real egg shape. So, an equation of egg shaped curve which resembles closely to the shape of a chicken egg is pursued here apart from a mathematical definition of "oval curve".

1. Basic Derivation of Equation of Egg Shaped Curve
An equation of egg shaped curve (egg curve), which resembles to the shape of the actual egg more than Cassini oval etc. (continuing from the left), is obtained below.
In 2021, we received an inquiry about how to obtain Eq.(7). Certainly, the derivation of Eq. (7) does not seem to be so easy. Therefore, for reference, "the derivation of Eq. (7)" is shown. 
For simple expression of the above equation, we displace the trajectory by the value of in the direction, and we introduce the next two constant values; 


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

If Eq.(9) is rewritten, the following equation is led into the expression indicating the obvious relation to this enveloping circle.
The major axis is obtained as = as strictly understood from Eq.(9). However, the minor axis cannot be obtained analytically. On the other hand, one round length of an egg shaped curve is given by
In the next, the plane area of an egg shaped curve as shown in Fig.2 is given by the follows with the use of Eq.(9b).
2. The Volume and the surface area of Egg Shaped Figure in the Three Dimensional Space
and in the case of b=0, the volume of sphere having the radius a/2 is led to as . (15c) As a reference, the calculation process of Eq.(15a) is written as In the next, the surface area of an egg shaped solid figure is calculated by the following equation. , (16) where is given by Eq.(9b), and is given by Eq.(14b). More detail analyses are described in "How can I find the surface area of a normal chicken egg?".

Surface area of the egg shaped solid figure =  
 
 
egg shaped curve =  
egg shaped curve = 
Especially, the relationship between the ratio of the minor to the major axes and the values of volume and surface area of egg shaped solid figure is shown in Fig.3.
3. Expression of the Equations of the Egg Shaped Curves with the Use of the Intermediate Variable
return 4. The Case Using Ellipsoid instead of Circle as a Basic Figure



If we solve Eq.(20) with the usual method of the solution of the 2nd order equation, we obtain that return 5. The Case that the Constants "a" and/or "b" are/is out of the Defined Region





return 6. General Extension from Egg Shaped Curves to Pear Shaped Curves

where condition of constant c indicated in Eq.(10) is released for the reason of the extension, the conditions of d > 0 and f > 0 are needed for giving a closed curve, and new conditions which are explained in Eqs.(29) and (31) described below are added. Furthermore, the constant b is not used in this equation because the relation of has already been used in the process of derivation of Eq.(10) described in the first section. 
In the region of 0 < x< a in the above equation 
[Annotation] As a solution outside of 0 < x < a, the following equation also may exist besides Eq.(27). 

As seen in Fig.8, a closed curve goes out of the area of 0 < x < a when c > 2 whose condition does not satisfy Eq.(31). Moreover, as seen in Fig.9, a closed curve goes out of the area of 0 < x < a when e > 0 whose condition does not satisfy Eq.(29).
Furthermore, some interesting figures can be drawn as shown in Figs.10 and 11. 

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7. The Higher Order Equation I If we replace to x in Eq.(24), we rewrite the following equation in the eight order. Such obtained equation can also be solved analytically. Moreover, the conditions of constants does not vary. However, it must be paid attention that the region of a closed curve given by the following equation is . 
The solution of the above equation is given as 
Some examples of the curves which are given by Eq.(32) or its solution Eq.(33) are shown in Figs.12 and 13 in the region of .


As seen in Fig.13, a closed curve does not exist within when e > 0. This is as like as described in the previous section.
Furthermore, some interesting figures can be drawn as shown in Figs.14 and 15 corresponding to Figs.10 and 11 respectively. 

When the values of some constants are varied, 'spade shaped curves' are obtained as shown in Figs.16 and 17.


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8. The Higher Order Equation II If we replace to y in Eq.(32), we rewrite the following equation in the eight order. The conditions of constants and the region of closed curve to be obtained are the same as in the previous section. (34) The solution of the above equation is given as (35) Some examples of the curves which are given by Eq.(34) or its solution Eq.(35) are shown in Figs.18 and 19 in the region of . 

As seen in Fig.19, a closed curve does not exist within when e > 0. This is also as like as described in the previous section.
Furthermore, some interesting figures can be drawn as shown in Figs.20 and 21 corresponding to Figs.10 and 11, or to Figs.14 and 15 respectively. 

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9. Sites and papers in which this site is cited

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