What are an Oval and an Egg Curve?
There is no clear definion. Mostly you define:
||An oval is a closed plane line, which is like an ellipse or like the
shape of the egg of a hen.
An egg curve only is the border line of a hen egg.
The hen egg is smaller at one end and has only one symmetry axis.
The oval and the egg shaped curve are convex curves, differentiate twice
and has a positive curvature.
||You distinguish between the oval, the ovoid and the oval shape in the
same way as between the circle, the figure of the circle and the sphere.
Ellipses and its changings
The parameters a and b are called lengths of axis.
All points P, for which the distances of two fixed points or foci F1
and F2 have a constant sum, form an ellipse.
The ellipse in the centre position has the following cartesian equation.
The ellipse is the formula of a relation.
The ellipse on the left has the equation
The constant sum is 2a=6.
||You can add two halves of different ellipses to form a chicken egg.
A Gardener's Construction
You can draw an egg curve, if you wrap a rope (green) around an isosceles
triangle and draw with taut rope a closed line (1). The rope must be a
little bit longer than the circumference of the triangle. Elllipse arcs
develop, which together form an egg shaped curve (2).
The three main ellipse are totally drawn in a computer simulation (2,black,
red, blue, book 9). You are exacter, if you draw three more ellipses in
the sector of the vertical angles of the triangle angles to the sides AB,
AC und BC (3,4) .
The Danish author and scientist Piet Hein (1905-1996) dealt with the super
ellipse in great detail (book 4). In particular that the shape made by
rotation around the x-axis can stand on the top, if it is made from wood.
You don't have to use power in contrast to the Columbus' egg.
||If you take the exponent 2.5 instead of 2 in the equation (x/a)²+(y/b)²=1,
you get the equation of a super ellipse:
The modulus | | makes sure that the roots are defined.
In the drawing there is a=3 and b=2.
The super ellipse belongs to the Lamé curves. They have the
||In the drawing there is a=3, b=2 and you substitute n with 1(parallelogram,
blue), 1.5(green), 2(Ellipse, bright red), 2.5 (super ellipse, red), and
From the Oval to
the Egg Shape
You can develop the shape of a hen egg, if you change the equation
of a oval a little. You multiply y or y² by a suitable term t(x),
so that y becomes larger on the right side of the y-axis and smaller on
the left side. y(x=0) must not be changed.
The equation of the ellipse e.g. x²/9+y²/4=1 change to x²/9+y²/4*t(x)=1.
Here you multiply y² with t(x).
To the red egg shaped curve:
The ellipse is black. The egg curve is red. It lies under the ellipse
on the right side of the y-axis. The term there is larger than 1. The number
4 (=b²) becomes smaller by multiplication of y²/4. So the curves
belongs to ellipses with smaller minor axes. It is under the black ellipse.
Corresponding you explain, why the red curve lies above the black ellipse
on the left of the y-axis. (You multiply with a number smaller than 1...)
To the blue and green egg curves:
They have about the same shape, though the equations are different
at first glance.
t2(x)=1/(1-0,2x) can be written as geometric series.
Generally there ist 1/(1-q) = 1+q+q²+..., here is 1/(1-0,2x) =
t3(x)=exp(0.2x) kann be developped as Taylor's series
Generally there is f(x) = f(0)+x*f'(0)+x²*f''(0)+..., here is
exp(0.2x) = 1+0,2x+0,02x²+...
To compare t1(x)=1+0,2*x+0*x².
The three terms t1, t2 und t3 differ in the series not until in
the square term.
|Further there is t1<t3(x)<t2(x).
If you draw the three accompanying egg curves, the red curves is outside,
the green one in the middle and the blue one inside.
Why is the blue egg shaped curve inside the red one?
Smaller minor axes belong to t2(x) compared to t1(x).
From the Egg to the Triangle
The 3 lines of the triangles are described by the 3 factors in (x/a+y/b-1)(x/a-y/b-1)(x/a+1)=0
||If you substitute the term t(x)=(1+kx)/(1-kx) in the equation
x²/9+y²/4*t(x)=1, you get the curves on the left for different
black: k=0,1 red: k=0,2
green: k=0,3 blue k=1/3.
The black egg becomes a blue triangle.
The black egg is the same as those of t1(x), t2(x) oder t3(x)
above, because the geometric series (1+0,1x)/(1-0,1x)=1+0.2x+0.02x²+...
correspondend to the first terms.
You get a triangle for k=1/3. a=3 is the major axis.
The equations x²/a²+y²/b²*(1+x/a)/(1-x/a)=1 and
(x/a+y/a-1)(x/a-y/b-1)(x/a+1)=0 are equivalent. If you simplify both terms,
||Don M. Jacobs, M.D., from Daly City, USA developped a nice egg shape
by changing the circle equation x²+y²=1 a little: x² + [1.4^x*1.6y]²
The egg equation is an exponential equation of the type t3. This shows
Inversion of an Ellipse
in a Circle
An inversion is the function of the Argand plane one-one by reciprocal
radii or a reflection in a circle with the radius R. The centre of the
reflection is the origin (0|0). The equation of the function is z'=R²/z.
||If you reflect an ellipse in a straight line, you get an ellipse again
(on the left).
If you reflect an ellipse in a circle, you get an egg curve (on the
Curves as Loci of Points top
All points P, for which the distances of two fixed points or foci F1
and F2 have a constant product, form a Cassini
oval. The Cassini oval has the following Cartesian equation in the centre
position (x²+y²)² - 2e² (x²-y²) - (a²)²
2e is the distance of both fixed points, a² is the constant product.
The curve on the left has the equation
(x²+y²)² - 72(x²-y²) - 2800 = 0.
There is e=6, a=8.
||This drawing originated from fixing e=6 and substituting a =10 (blue),
8.5 (grey), 7 ( red), 6 (black) und 4 (green) in the formula.
If a>[e multiplied by the square root of 2] there is a egg figure.
If a=[e multiplied by the square root of 2] there also is an
egg figure, but the curvation is 0 on the vertical axis.
If e<a<[e multiplied by the square root of 2] there is a figure
cut into the middle.
If a=e there is a lemniskate.
If a<e there are two ovals.
||The ovals inside with a<e have interesting egg shapes if the variable
a approaches e=6.
All points, for which the simple and the double distances of
two fixed points or foci F1 and F2 have
a constant sum, form a Cartesian oval. The Cartesian oval has the
following cartesian equation.
c is the distance of the fixed points and m=2 ("double distance").
The origin of the coordinate system is the left fixed point.
This long equation is derived with the formulation s1+2*s2=a
and by using Pythagoras' formula twice.
||The distance of the fixed points is c=5 and the sum a=12.
The equation is now
2304((5-x)²+y²) - (3x²+3y²-40x+44)²=0.
||The graph from above is incomplete. Surprisingly the equation 2304((5-x)²+y²)-(3x²+3y²-40x+44)²=0
produces another curve outside the egg curve.
These egg curves go back to Renatus Cartesius alias René Descartes
(1596-1650), therefore the name.
||If you substitute m=2 with m=2.2, you produce another egg shape. You
keep c=5 and a=12.
Curves by Loops
Folium of Descartes
(Torsten Sillke's idea)
|More egg curves this way:
>Trisextrix of MacLaurin y²(1+x)+0,01=x²(3-x)
>Lemniskate of Bernoulli (x²+y²)²-(x²-y²)+0,01=0
>Conchoid of de Sluze 0,5(x+0,5)(x²+y²)-x²+0,02=0
Transfer the well known drawing of an ellipse with the help of two
concentric circles (on the left) to a two-circle-figure.
Draw in the order M1, M2
P1, P2 , and P.
a and b are the radii of the circles, d is the distance of its centres.
The parameters a, b, c are suitable to describe the egg shape. 2a is
its length, 2b its width and d shows the broadest position.
The equation of the egg shaped curve is
an equation of third degree:
x²/a² + y²/b²[1 + (2dx+d²)/a²] =
The drawn egg shaped curve has the parameters a=4, b=2 und d=1. The
equation is 4x²+16y²+2xy²+y²-64=0.
Origin: (11), page 67/68
||In this example there is a=4, b=3 and d=1.
The equation is 9x²+16y²+2xy²+y²-144=0.
Granville's Egg Curve
>There is given a line, which starts at point A and lies horizontally.
Then there is a vertical line in the distance of a and a circle with the
radius r being symmetric to the horizontal line in the distance of a+b
(drawing on the left).
>If you draw a line (red) starting at point A, it cuts the vertical
line at B and the circle at C. If you draw then a vertical line through
C and a horizontal line through B (green), they meet at P.
>If the point C moves along the circle, then the points P lies on a
egg shaped curve (animation on the right).
See more: (13), Jan Wassenaar (Granville's egg, URL below), Torsten
Sillke (Granville's egg, URL below)
Mechanical Egg Curve
See more: (12), www.museo.unimo.it/theatrum/macchine/, Jan Wassenaar (quartic
egg curve, URL unten)
||Let P be a fixed point and A a point, which moves on a circle around
P with the radius r=PA.
Link the bar a=QA at A . Its free end Q moves on a horizontal through
P forth and back. The point B on the line AQ with BQ=b describes an egg
Chains of Eggs top
A Double Egg
||The polar form r(t)=cos²t produces a double egg
A second equation is r(t)=exp(cos(2t))*cos²(t) (Hortsch 1990).
Another Double Egg
There is a wide field of experimenting.
||The Equation x4+2x²y²+4y4-x³-6x²-xy²=0
produces a double egg.
You can form and combine sinus curves in such
a way, that you get a chain of eggs.
Also polyominals can produce chains (see Torsten Sillke, URL below).
The equation y² = abs[sin(x)+0.1sin(2x)] describes
a sinus chain more elegant:
Egg Curves with Arcs top
||Two small (red) and two big (grey) quarter circles, which have a square
in common, form an oval.
(The angles of the sectors don't have to be 90°.)
||A semi-circle (green), a quarter circle (red) and two eighth circles
(grey), which have a triangle in common, form a second figure. If you cut
the egg in nine pieces, you get the tangram puzzle "The Magic Egg" or "The
Egg of Columbus".
||You can generalize the figure: Take a smaller dark grey triangle.
||Divided and reassembled again
||Divided and reassembled again.
(14), Seite 122..
Section through Rotation
If you make a sloping section through a cone
or a cylinder you often get an ellipse as a section line. If you choose
an hyperbolic funnel, you get egg curves in the form of a hen egg. Hyperbolic
funnels are figures, which develop from rotation of an hyperbola around
the symmetry axis.
||There is the hyperbolic funnel to f(x) =1/x².
The y-axis is perpendicular to the x-z-plane
in direction to the back.
The straight line shows the section plane perpendicular to the x-z-plane.
||The given plane intersects the hyperbolic funnel
three points in the x-z-plane.
If you project the section lines in the x-y-plane you get the red curves.
||You get an egg curve in the section plane.
If you make a sloping section through other figures,
you get more egg curves.
Three More Curves top
Equations of 3rd and 4th Degree
||Equations with the form y²=p(x-a)(x-b)(x-c)... produce egg curves.
There are two examples on the left:
2y²=(x-1)(x-2)(x-3) and y²=(x-1)(x-2)(x-3)(x-4)
||The polar form r(t)=cos³t produces the folium or wrong Kepler
A Crooked Egg
"Jedes legt noch schnell ein Ei und dann kommt der Tod herbei."
"Each still lays a final egg, then comes Death and out they peg." (Reclam)
||The polar equation r(t)=sin³t+cos³t produces a crooked egg
Torsten Sillke's egg of Columbus
(1) Lockwood, E. H.: A Book of Curves.
Cambridge, England: Cambridge University Press, p. 157, 1967.
(2) Martin Gardner: The Last Recreations, Hydras,
Eggs, and Other Math.Mystifications, Springer, New York 1997
(3) Sz.-Nagy, Gyula: Tschirnhaussche Eiflaechen und Eikurven. Acta Math.
Acad. Sci. Hung. 1, 36-45 (1950). Zbl 040.38402
(4) Ulrich/Hoffman: Differential- und Integralrechnung
zum Selbstunterricht, Hollfeld 1975
(5) Martin Gardner: Mathematischer Karneval, Frankfurt/M,
(6) Gellert...: Kleine Enzyklopädie - Mathematik,
(7) Wolfgang Hortsch, Alte und neue Eiformeln in der Geschichte der
Mathematik, Muenchen, Selbstverlag 1990, 30S
(8) Gebel und Seifert, Das Ei einmal anders betrachtet, (eine Schülerarbeit)
Junge Wissenschaft 7 (1992)
(9) Hans Schupp, Heinz Dabrock: Höhere
Kurven, BI Wissenschaftsverlag 1995
(10) Gardner, Martin: Geometrie mit Taxis,
die Koepfe der Hydra und andere mathematische Spielereien. Basel: Birkhaeuser
(1997), Deutsche Ausgabe von (2)
(11) Elemente der Mathematik 3 (1948)
(12) Karl Mocnik: Ellipse, Ei-Kurve und Apollonius-Kreis,
Praxis der Mathematik. (1998) v. 40(4) p. 165-167
(13)W. A. Granville: Elements of the differential
and integral calculus, Boston, (1929)
(14) Heinz Haber (Hrsg.): Mathematisches Kabinett,
München 1983 [ISBN 3-423-10121-0]
Egg Curves on the Internet
2:3-Ei - ein praktikables Eimodell (.pdf Datei)
Projekt der Universität Würzburg
rund ums Ei
CARLOS CALVIMONTES ROJAS
GEOMETRY OF THE PARABOLA
ACCORDING TO THE GOLDEN NUMBER
Eric W. Weisstein (MathWorld)
and Rollett's Egg, Mosss
an Oval (Eggs)
Chickscope project at the Beckman Institute
Nick C. Thomas
Eggs with path curves
Richard Parris (peanut Software)
Satire Software (Geometry Gallery)
The Shape of Birds Eggs (applet)
The MacTutor History of Mathematics archive (Created by John J
O'Connor and Edmund F Robertson)
|Granville's egg - quartic [Granville 1929]
Cubic curves as perturbated ellipse
Mechanical egg curve construction by a two bar linkage - a quartic
Polynomials making chains of eggs
Newton's cubic: Elliptic curve
Transforming the ellipse
|Limacon Graphics Gallery
Toric sections - hippopede of Proclus: analyzed by Perseus
The Family r = cos^p(phi) or [Münger Eggs]
Multifocal Curves - Tschirnhaussche Eikurven
Pivot transform construction of Path-curves
egg curve, Cassini
the Great (Fabergé egg), Fabergé
egg, Egg decorating,
Robert FERRÉOL (mathcurve)
DE DESCARTES, ELLIPSE,
DE GRANVILLE, COURBE
NN (published in: Pythagoras, wiskundetijdschrift voor jongeren,
Een eitje, zo'n eitje
admin @ arbuz.uz
Ellipser og æg,
of Egg Shaped Curves, Egg
Shaped Curve II, Egg
Shaped Curve III, Egg
Shaped Curve IV
A.Gärtl, Willi Jeschke, Torsten Sillke,
Gail from Oregon Coast - thanks.
Feedback: Email address on my main page
page is also available in German.
2000 Jürgen Köller