Equation of Egg Shaped Curves

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Equation of Egg Shaped Curves

Nobuo YAMAMOTO

1. Basic Derivation of Equation of Egg Shaped Curve	2. The Volume of Egg Shaped Body in the Three Dimensional Space	3. Expression of the Equation with the Use of
4. The Case using Ellipsoid instead of Circle as a Base	5. The Case out of the Defined Region of Constants Values	6. General Extension from Egg to Pear Shaped Curve
7. The Higher Order Equation I	8. The Higher Order Equation II

1. Basic Derivation of Equation of Egg Shaped Curve

Fig.1 Expressing the point P
which draws an egg shaped curve

Fig.2 Egg shaped curves (in the case of a=4)

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 straight line PQ.
The point Q is supposed to move on the

-axis co-sinusoidally according to the angle

of the line PQ as the following;

. (1)
In the same time,

the length of the line 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)

b=a

b=0.8a

b=0.7a

b=0.6a

Fig.2a Comparison between the egg shaped curve
of (pink colored curve)
and the shape of an actual egg

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;

. (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 ＝∞.

     In purpose to calculate the numerical coordinates data of five spacies of egg-shaped curves as shown in Fig.2, a C++ program originated from Eq. (9) is given by .      Another C++ program treating a single egg-shaped curve is given by     In the latter program, the constants are settled as =4, and =3.2.
     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.
     A mask for the egg-shaped trimming is prepared in this button , which is edited from the above obtained curve.      Please use this mask freely.

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

, (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, one round length and the plate area of the egg shaped curve]

Fig.2b The major and
minor axes and ,
one round length ,
and the plane area

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

, (14a)
where

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).

. (14b)

     C++ program for obtaining the minor axis of an egg shaped curve is given by .
     C++ program for obtaining one round length of an egg shaped curve with the use of Eq.(14a) is given by .
     Moreover, C++ program for obtaining the plane area of the egg shaped curve with the use of Eq.(14b) is given by .

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2. The Volume of Egg Shaped Body in the Three Dimensional Space

The volume

of an egg body 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)

In the next, the surface area

of of an egg body is calculated by the following equation.

, (16)
where

is given by Eq.(9b), and the derivative of

is given as

C++ program for obtaining the volume of the egg body with the use of Eq.(15a) or (15b) is given by .
Moreover, C++ program for obtaining the suface area of the egg body with the use of Eq.(16) is given by .

The results of numerical calculations in account of the volume of an egg body by Eq.(15b) or (15c), the surface area of an egg body by Eq.(16), the major and minor axes 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.

Table 1 The volume and the surface area of an egg body, the major and the minor axes, one round length and the plane area of an egg shaped curve

Condition	b=0 (Sphere)	b=0.2	b=0.4	b=0.6	b=0.7	b=0.8	b=0.9	b=
Major axis =
Minor axis =		0.905473	0.823333	0.754247	0.724266	0.696999	0.672172	0.649519
Volume of the egg body =	=0.523599	0.428932	0.352696	0.292378	0.267402	0.245463	0.226247	0.209440
Surface area of the egg =	=3.141593	2.755236	2.429825	2.159010	2.042087	1.936446	1.841316	1.755925
One round length of egg shaped curve =	=3.1416	2.9942	2.8655	2.7544	2.7050	2.6596	2.6181	2.5803
Plane area of egg shaped curve =	=0.785398	0.710785	0.644026	0.585122	0.558614	0.534071	0.511491	0.490874

As a reference, the calculation process of Eq.(15a) is written as

. (15b)'

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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. 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 discribing 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 discribed 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.

     In purpose to calculate the numerical coordinates data of a single egg-shaped curve with the use of Eq.(19) as shown in Fig.2, a C++ program is given by .     Although the constants are settled as =1, and =0.8 in the this program, we can change the shape of the egg curve by changing the values of these two constants in this program under the restriction of    .
     By executing this C++ program, a 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.
     On the other hand, as the value of contains in Eq.(17) which gives each value of coordinate axis, a C++ program for displaying multiple spacies of egg-shaped-curves simultaneously in a single - plane as shown in Fig.2 is not simple to do programming.    Therefore, we are hesitated in showing such a program in this time.

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    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.3 or Fig.4 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 culculation 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.

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    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.

Fig.5 Fig.6

Fig.7

5.3 Reform of curves in Fig.5 into "egg shaped" ones in the case of a>0 and b<0

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

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 .

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    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 discribed 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 conversantly in Eqs.(29) and (31) discribed 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) discribed 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 conformed.    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.

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).

In purpose to calculate the numerical coordinates data of egg-shaped curves within 0 < x < a as shown in Fig.8, a C++ program is given by .

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

(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.

In purpose to calculate the numerical coordinates data of egg-shaped curves within 0 < x < a as shown in Fig.12, a C++ program is given by .

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.

In purpose to calculate the numerical coordinates data of egg-shaped curves within 0 < x < a as shown in Fig.18, a C++ program is given by .

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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updated: 2007.3.29, edited by N. Yamamoto