The oval and the egg shaped curve are convex curves, differentiate
twice and has a positive curvature.
The ellipse is the formula of a relation.
A Gardener's Construction 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) .
Super Ellipse
The super ellipse belongs to the Lamé curves. They have the equations
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). Three examples: 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. But: 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) = 1+0,2x+0,04x²+... t3(x)=exp(0.2x) kann be developped as Taylor's series
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.
From the Egg to the Triangle
Letter
Inversion of an Ellipse in a Circle
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.
Cartesian Ovals 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.
4a²m²((c-x)²+y²)-(a²+m²c²-2cm²x+(m²-1)(x²+y²))²=0 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.
Curves by Loops
Drawing by Fritz Hügelschäffer Transfer the well known drawing of an ellipse with the help of two concentric circles (on the left) to a two-circle-figure. _{1},
M_{2
}, P_{1}, P_{2 },
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: b²x²+a²y²+2dxy²+d²y²-a²b²=0 The drawn egg shaped curve has the parameters a=4, b=2 und d=1. The equation is 4x²+16y²+2xy²+y²-64=0.
Granville's Egg Curve
>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 Construction
Another Double Egg
Chains You can form and combine sinus curves in such a way, that you get a chain of eggs. The equation y² = abs[sin(x)+0.1sin(2x)] describes a sinus chain more elegant:
Formulas:
The Folium
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)
Upright Egg
English: (1) Lockwood, E. H.: A Book of Curves.
(2) Martin Gardner: The Last Recreations,
Hydras, Eggs, and Other Math.Mystifications, Springer, New York
1997
German: (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, Berlin 1977 (6) Gellert...: Kleine Enzyklopädie - Mathematik, Leipzig 1986 (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]
Deutsch Herbert Möller
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