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Equation of Egg Shaped Curve of the Actual Egg is Found

Nobuo YAMAMOTO

You can copy and use all the figures except for Fig.2b in this page freely,
where Fig.2b is not created by the author of this page.

 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".     This starts from the short article that Yamamoto has published in Library Newsletter No. 63 (2006) of Ibaraki National College of Technology.     Succeedingly, "Equation of Egg Shaped Curve of the Actual Egg is Found" is released in this page on Apr. 27, 2007,   "Equation of Egg Shaped Curve II (Egg Shaped Curve I proposed by Mr. Itou" is released on Mar. 29, 2008,   "Equation of Egg Shaped Curve III (Egg Shaped Curve I proposed by Mr. Asai)" is released on Aug. 02, 2009,   "Equation of Egg Shaped Curve IV (Egg Shaped Curve II proposed by Mr. Asai)" is released on Sep. 18, 2009,   "Equation of Egg Shaped Curve V" is released on Nov. 08, 2009,   "Equation of Egg Shaped Curve VI" is released on Nov. 26, 2009,   and "Egg Shaped Curve VII (Egg Shaped Curve III proposed by Mr. Asai)" is released on May. 29, 2011.     For a while after these, there was no publication of an oval curve close to the shape of a real chicken egg.     However, as of March 2020, "Equation to fit an egg", "HÜGELSCHÄFFER EGG". "Egg Curve", "Egg-curve", "Let's display an egg curve" and "The Python Egg Curve" have been issued.

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 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 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.

 b=a b=0.8a b=0.7a b=0.6a

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.

Fig.1 and the culcuration process from Eqs.(1) to (6) may suggest the generation process of an actual chicken egg.

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 = nfinitive.

Mr. Akira NAGASHIMA indicated Fig.2 as an animation clearly in March in 2011.    I'll be thankful.    It is shown in Fig.2b.

 (If clicked, you can see by expansion.)
Fig.2a Comparison between the egg shaped curve
of
(pink colored curve)
and the shape of an actual egg
Fig.2b Egg animation by Mr. Akira NAGASHIMA

 In purpose to calculate the numerical coordinates data of five species 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 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.    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 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 of the egg shaped curve]

 Fig.2c  The major and minor axes and , 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

.          (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)

 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 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).
More detail analyses are described in "How can I find the surface area of a normal chicken egg?".

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

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.

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

[Remark 1]   The minor axis is not though =.   As described in the above, the value of cannot be obtained analytically.
So, its value will be obtained by computer calculation with C program.
[Remark 2]   The nearest shape of egg to an actual egg exists arround as painted with pink color.

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

 (If clicked, to be seen by expansion.) Fig.3 Relationship between the ratio of the minor to the major axes and the values of volume and surface area of egg shaped solid figure.

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.

 In purpose to calculate the coordinates data for displaying the relationship as shown in Fig.3, a C++ program using Eqs.(15b) and (16) etc.is composed as given by .
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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 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.

 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 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.    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 species 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.4a or Fig.4b is obtained.

 Fig.4a　a=1, p=1.5, Red curve (b=0]) gives the ellipse to spread in the side.

 Fig.4b　a=1, p=0.7, Red curve (b=0]) gives the ellipse to spread lengthwise.
 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.
 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 < 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 .

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

Accepting a suggestion from Dr. Hiroyasu OKUYAMA 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.
This follows the basic formula proposed by Dr. Hiroyasu OKUYAMA to analyze the light distribution curve of a solar lamp by the least squares method, referring to Yamamoto's earlier short paper.

(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.

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.

 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.

 In purpose to calculate the numerical coordinates data of egg-shaped curves within 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.

 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.

 In purpose to calculate the numerical coordinates data of egg-shaped curves within 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.

 return 9.  Sites and papers in which this site is cited     Afterwards, this site is introduced in    How can I find the surface area of a normal chicken egg? ,    How to place random points on an egg? ,    The room of Jukchou ,    271828_slide_Log ,    WolframAlpha Computational Intelligence ,    draw egg curves from an equation ,    Egg quations ,    Creating a 3D ellipsoid from a 2D ellipse ,    Egg Curves and Ovals   [ Eilinien und Ovale  in German ] ,    Eilinien und Ovale ,    Two-dimensional Yamamoto (2017)'s Egg Curve    GENERATING AVIAN EGG USING RATIONAL BEZIER QUADRATIC CURVES ,    Creating a 3D Egg in SketchUp with U-V PolyGen ,    If anyone knows the equation of the egg curve, please let me know! ,    Graphs of Eggs ,    Negative Volume of Revolution? ,    How do you find the surface area of an egg using calculus? , etc., and this site is also cited in papers such as    Predictive Calculation Model of Vehicle Cabin Temperature in Environmental Laboratory and Comparison with Cooling Experiment ,    An Egg Volume Measurement System Based on the Microsoft Kinect ,    "DSEG": Original 7-segment and 14-segment fonts 5.Misc ,    "DSEG": Original 7-segment and 14-segment fonts ,    Distributed mesh generation for objects with shape variability ,    Negative Volume of Revolution? ,    Projecting Cartesian Coordinates onto the Surface of an Egg in Egg-bot Coordinates ,    Distributed mesh generation for objects with shape variability ,    Packing Ovals In Optimized Regular Polygons ,    KONSTRUKSI PERSAMAAN PERMUKAAN BENTUK-TELUR MENGGUNAKAN KURVA BENTUK-TELUR HUGELSCHAFFER ,    A Modified Avian Egg Shaped CPW Fed Printed Monopole for SWB Applications ,    Membrane-Wrapping Contributions to Malaria Parasite Invasion of the Human Erythrocyte ,    طراحي مفهومي و بهينهسازي کشتي هوايي ارتفاع باال بهوسيلة الگوريتم ژنتيک (Persian) (Conceptual design and optimization of high altitude aircraft by genetic algorithm) ,    Fun With SAS ODS Graphics: Easter Egg Equation Hunt ,    KONSTRUKSI PERSAMAAN PERMUKAAN BENTUK-TELUR MENGGUNAKAN KURVA BENTUK-TELUR HUGELSCHAEFER (Indonesian) ( CONSTRUCTION OF EGG-SHAPE SURFACE EQUATIONS USING HUGELSCHAEFER'S EGG-SHAPE CURVE) ,    MATHEMATICS IA -- Mathematical investigation of an egg ,    Regulatory Mechanisms in Biosystems -- Mathematical interpretation of artificial ovoids and avian egg shapes (part I) , etc.     Also, this site was also introduced in the blog of What is mathematical equation for an egg? (Archive version) , though this article in its blog is gone now.     As seen by looking at each site or treatise, the author's name written in it is "Nobuo Yamamoto",   "N. Yamamoto" or   "Yamamoto N.".     On the otherhand, the address of this site has passed "Http://www16.ocn.ne.jp/~akiko-y/Egg/index_egg.html",  "http://www.geocities.jp/nyjp07/Egg/index_egg.html",  "http://www.geocities.jp/nyjp07/index_egg.html", and becomes the current  "http://nyjp07.com/index_egg.html".     We can't use the old address, but searching for old addresses with Internet Archive search may reveal the site at the time it was saved.

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Updated: Apr. 29, 2007, edited by N. Yamamoto.
Revised on Mar. 16, 2015, Dec. 04, 2015, Sep. 26, 2016, Nov. 30, 2017, Jul. 13, 2019, Feb. 03, 2020, Feb. 12, 2020, Mar. 03, 2020, Mar. 16, 2020, Apr. 13, 2020, Apr. 21, 2020, May 05, 2020, Jun. 21, 2020, Jun. 28, 2020, Oct. 03, 2020, Jan. 17, 2021, May 06, 2021, Sep. 12, 2021, Sep. 20, 2021, Sep. 23, 2021, Sep. 28, 2021, Jan. 25, 2022, Feb. 13, 2022, Mar. 04, 2022, Apr. 21, 2022, Apr. 24, 2022, Apr. 25, 2022, Apr. 26, 2022, May 09, 2022, Jul. 10, 2022, Jul. 12, 2022, Jul. 18, 2022, Aug. 05, 2022, Aug. 08, 2022, Aug. 10, 2022, Aug. 18, 2022, Aug. 24, 2022 and Oct. 27, 2022.