Gardener construction of an ellipse

An ellipse E is the set of all points (x,y) of the xy-plane for which the sum of the distances to two given points F_1 and F_2 is equal (=2a). Ellipses belong to the class of conic sections (see the exhibit “Conic Sections”).

Figure 1: An ellipse

The points F_1 and F_2 are called foci. The center M of their connecting line (of length 2e, eeccentricity) is called the center of the ellipse. The distance from this center M to the two vertices S_1 and S_2 is a, respectively, and to the vertices S_3 and S_4 is b, respectively, with b^2+e^2=a^2 (according to the Pythagorean theorem; see also the exhibits “Pythagoras” and “Proof without words: Pythagoras to lay”), i.e., b=\sqrt{a^2-e^2}.

The connecting line between a focal point F_1 or F_2 (focus) and a point of the ellipse is called leading ray or focal ray. The names focal point and focal ray result from the property that the angle between the two focal rays at a point of the ellipse is bisected by the normal (straight line, perpendicular to the tangent) at that point. Thus, the angle of incidence formed by one focal ray with the tangent is equal to the angle of reflection formed by the tangent with the other focal ray. Consequently, a light beam originating from one focal point, e.g. F_1, is reflected at the elliptical tangent in such a way that it hits the other focal point. Thus, for an elliptical mirror, all light rays emanating from one focal point meet at the other focal point.

If the eccentricity e=0, F_1=F_2 is valid. The ellipse becomes a circle with radius r=a=b.

A simple way to draw an ellipse exactly is the so-called gardener’s construction. It directly uses the ellipse definition:

To create an ellipse-shaped flowerbed, you drive two stakes into the focal points and attach to them the ends of a string with length 2a. Now stretch the string and run a marking tool along it. Since this method requires additional tools besides a compass and a ruler — namely a string — it is not a construction of classical geometry. In MATHEMATICS ADVENTURE LAND, this construction can be understood by means of a simple experiment.

Figure 2: Gardener’s construction of an ellipse in MATHEMATICS ADVENTURE LAND

And now … the mathematics of it:

In the following the ellipse equation is derived from the “gardener construction” described above:

For a point (x,y) of the ellipse — corresponding to figure 1 above — \lvert\overline{F_1P}\rvert+\lvert\overline{F_2P}\rvert=2a, i. e. i.e., if we set F_1=(-e,0) and F_2=(e,0), we get the equation

    \[\sqrt{y^2+(x+e)^2}+\sqrt{y^2+(x-e)^2}=2a.\]

Squaring this equation gives

    \[y^2+(x+e)^2+y^2+ (x-e)^2-4a^2=-2\sqrt{(y^2+(x+e)^2)(y^2+(x-e)^2)}\]

und somit

    \[2y^2+2x^2+2e^2-4a^2=-2\sqrt{(y^2+(x+e)^2)(y^2+(x-e)^2)}.\]

Squaring again yields

    \[4(y^2+x^2+e^2-2a^2)^2= 4(y^2+(x+e)^2)(y^2+(x-e)^2)\]

and by simplification — i.e., suitable “truncation” — we get:

    \[a^2x^2-e^2x^2+a^2y^2+a^2e^2-(a^2)^2=0,\]

d.h.

    \[(a^2-e^2)x^2+a^2y^2=a^2(a^2-e^2).\]

Because of b^2=a^2-e^2 (see above), the normal form (also “midpoint form”) of an elliptic equation is then

    \[\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2=1.\]


Comments:

The so-called first Kepler’s law (“ellipse theorem”, “planet theorem”) states that the orbit of a satellite is an ellipse. One of its foci lies in the center of gravity of the system. This law follows from Newton’s law of gravitation, provided that the mass of the central body is much greater than that of the satellites and the interaction of the satellite with the central body can be neglected. Consequently, according to Kepler’s first law, all planets move in an elliptical orbit around the sun, with the sun at one of the two foci. The same applies to the orbits of recurring (periodic) comets, planetary moons or double stars.


Literature

Schupp, H.: Kegelschnitte, Mannheim, 1988.