|
|
|
|
|
|
|
| When we search for the internet about an oval curve, several sites of Egg Curves, 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 real 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 the x-y coordinate shown in Fig 1, there is some point Q  on the  -axis,
and there is the given segment PQ.
The point Q is supposed to move on the -axis co-sinusoidally according to the angle 
of the segment PQ as the following;
 .
(1)
In the same time,  the length of the segment PQ is supposed to vary co-sinusoidally as follows;
 , (2)
where  .
(3)
In such a way, the trajectory of the point P is tried to be found to become an egg shaped curve. In Fig.1, the point P is expressed as  , (4)
and  .
(5)
It is unexpectedly difficult to solve the above equations. Then, if we solve the equations from (1) to (5) under the assumption given in the following;  ,
(6)
we obtain as the fourth-order relation that  . (7)
|
in the direction,
and we introduce the next two constant values;
 ,
 . (8)
Then, Eq.(7) is rewritten into the following equation giving egg-like shaped curves.  , (9)
where  .
If we solve Eq.(9) with the usual solution method of the 2nd order equation, we obtain that  . (9b)
Such the solved equation may be used for programming and calculation with computer. If we introduce  , Eq.(9) is reduced into the following equation.
 .
(10)
Thus, Eq.(10) takes the simple and beautiful form like as an egg-shaped curve. It is prefer that Eq.(10) is taken as the standard equation of egg-shaped curve. However, as Eq.(9) has been treated from the beginning of this study, we intend to use Eq.(9) instead of Eq.(10) in below. In the case of  , if we calculate Eq.(9) as varying the several values of 
with the use of computer, the each egg-like shaped curve is drawn as shown in Fig.2.
In the case of  , the curve becomes a circle.
As increasing the value of ,
the more the curve is going to become near the shape of an actual egg,
In the case of  , the curve approaches the shape of an actual egg most.
In this condition, the comparison between the egg shaped curve and the shape of an actual egg is shown in Fig.2a.
The curve only in the case of  becomes pointed at x=0.
In the cases except for , the curves are never pointed,
as it is simply verified that the gradient of each curve at x=0 is infinitive, i.e., (dy/dx)x=0 ＝∞.
Mr. Akira NAGASHIMA indicated Fig.2 as an animation clearly in March in 2011. I'll be thankful. It is shown in Fig.2b. |
|
|
||
|
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.  . (11)
In the next, the equation of egg shaped surface expressed in the three-dimensional space is obtained as the following by replacing  to  .
 , (12)
where .
If Eq.(12) is rewritten, the following equation is led into the expression indicating the obvious relation to this enveloping sphere.  . (13)
[The major and miner axes  and  respectively,
one round length  and the plate area  of the egg shaped curve]
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
 , (14a)
where  . (14b)
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).
 . (14c)
return 2. The Volume  and
the surface area  of Egg Shaped Figure in the Three Dimensional Space
The volume of an egg shaped solid figure expressed by Eq.(12) or (13) is calculated by means of the following equation
with the use of rotation radius as given by Eq.(9b).
 . (15a)
The calculation result is as the followings; In the case of  ,
 , (15b)
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  . (15b)'
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).
The results of numerical calculations in account of the volume of an egg shaped solid figure by Eq.(15b) or (15c),
the surface area of an egg shaped solid figure by Eq.(16),
the major and minor axes and respectively of an egg shaped curve,
one round length of an egg shaped curve by Eq.(14a) and
the plane area of an egg shaped curve by (14b) are shown in Table 1.
|
and the surface area of an egg shaped solid figure,
the major and the minor axes and respectively,
and the plane area of an egg shaped curve
is not b
though =.
As described in the above, the value of cannot be obtained analytically.
as painted with pink color.
|
|
|
|
|
|
|
||
| = |
 =0.523599 |
|
|
|
|
|
|
|
| Surface area of the egg shaped solid figure = |
 =3.141593 |
|
|
|
|
|
|
|
| = |
|
|
|
|
|
|
|
|
| = |
|
|
|
|
|
|
|
|
egg shaped curve = |
 =3.1416 |
|
|
|
|
|
|
|
egg shaped curve = |
 =0.785398 |
|
|
|
|
|
|
|
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.
return 3. Expression of the Equations of the Egg Shaped Curves with the Use of the Intermediate Variable
In this section, the equations of the egg-shaped-curves expressed with the use of the intermediate variable will be given as pointed out by
Mr. Mr. Tadao ITOU
(another equation of an egg-shaped-curve given by him is shown in
 ).
Such an expression will be much convenient to display egg-shaped-curve graphically by computer aided calculation.
In the first, we add the constant value of to the right hand side of Eq.(4)
as according to the displacement of the trajectory by the value of
in the direction as mentioned above. Then,
 .
In the proceeding, substituting Eqs.(1), (2) ,(6) and (8) into the above equation, we obtain that  . (17)
In the next, substituting Eqs.(2) and (8) into Eq.(5), we obtain the following equation;  . (18)
In such a way, we obtained two equations describing an egg-shaped-curve with the use of an intermediate variable .
The selection of the value of is not restricted
under the condition that the continued range of is taken to be 2*pai.
The above summary is described as the followings;  , (19)
where .
It is noticed that the intermediate variable in Eq.(19) is defined in Fig.1 and then,
does not indicate the phase angle of the egg curve in the polar coordinates.
return 4. The Case Using Ellipsoid instead of Circle as a Basic Figure As seen in Fig.2, the basic figure of egg shaped curves is a circle in the above sections. If we adopt ellipse instead of circle as a basic figure, many other types of egg shaped curves may appear. Expanding Eq.(9) to conform such a purpose, the following equation is considered;  . (20)
When b=0, the above equation leads to  . (21)
The equation obtained above certainly indicates that of an ellipse having two typical curvature radii of  and  .
If we show Eq.(20) to chart, a figure like as Fig.4a or Fig.4b is obtained.
If we solve Eq.(20) with the usual method of the solution of the 2nd order equation, we obtain that  . (20b)
This equation may be used for programming and calculation with computer. In the next, The volume V of the body made by rotating a figure given in Fig,3 or Fig.4 around x axis is obtained with the use of Eq.(20b) as the followings in the same manner as the calculation of Eq.(18); In the case of ,
 , (22a)
and in the case of b=0, the volume of revolving ellipsoid is led to as  . (22b)
The above analyses has been done under the condition of a>b>0. As references, the results of analyses extended to the cases of b<0 and b>a in Eq.(9) are given in the following as the condition of a>0 remains. However, the extension to the case of a<0 has not been done yet. return 5. The Case that the Constants "a" and/or "b" are/is out of the Defined Region 5.1 In the case of a>0 and b<0 ---sea urchin shaped curve? In this case, if we calculate Eq.(9) as varying the several values of with the use of computer, the curves
are drawn as shown in Fig.5.
The each curve looks like a sea urchin.
5.2 In the case of b>a>0 ---fish shaped curve? In this case, if we calculate Eq.(9) as varying the several values of with the use of computer, the curves
are drawn as shown in Fig.6.
These curves are pointed at x=0, and also run out in x<0.
The each curve looks like a fish which is drawn by a child.
|
|
|
|
|
|
If the curves displayed in Fig.5 are expanded toward x direction, the egg looking curves may appear. Therefore, when some constant value c is newly introduced as 0 < c < 1 and the variable x is displaced to cx in Eq.(9) for such an expanding, the following equation is derived.  . (23)
If we calculate Eq.(23) with the use of computer as the constants a and c are fixed in a=4 and c=0.55, a few of egg looking curves are drawn as shown in Fig.7 in respect to the several values of b . return 6. General Extension from Egg Shaped Curves to Pear Shaped Curves Accepting a suggestion from Dr. Hiroyasu OKUDA who belongs to Shimizu Institute of Technology in Simizu Corporation, we treat the case that Eq.(10) is extended as to involve the utmost numbers of new terms while leaving the capability of analytic solution as like as Eq.(9b). This case is realized in the following equation which contains not only all of the curves described above but also pear shaped curves.  (24)
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.
If we solve Eq.(24) with the usual method of the solution of the 2nd order equation, we obtain that  (25)
In the region of 0 < x< a in the above equation  (26)
is confirmed. Therefore, the negative sign of the double sign in Eq.(25) have to be eliminated. Thus, the solution of Eq.(24) is led to  . (27)
[Annotation] As a solution outside of 0 < x < a, the following equation also may exist besides Eq.(27).  (27)'
In order that Eq.(27) gives a closed curve in the region of 0< x < a, it must be y=0 at both of x=0 and x=a. In the first, if we substitute x=0 into Eq,(27) in order to investigate the case of x=0, Eq.(27) is led into  . (28)
The condition that the value of the above equation becomes to y=0 is given by the following inequality.  . (29)
In the next, if we substitute x=a into Eq,(27) in order to investigate the case of x=a, Eq.(27) is led into  . (30)
The condition that the value of the above equation becomes to y=0 is given by the following inequality.  . (31)
Coupled condition explained by Eqs.(29) and (31) is equivalent to the condition that only a sort of closed curve exists in the region of 0 < x < a. Finally, some examples of the curves which are given by Eq.(24) or its solution Eq.(27) or (27)' are shown in Figs.8 and 9. |
 
|
 
|
|
return
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  .
 (32)
The solution of the above equation is given as  (33)
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.
|
 
|
|
return
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. |
 
|
| return |
=4, and 
