# Circle and Ellipse

An *ellipse* is the set of all points of the -plane for which the sum of the distances to two given points and is equal (). Ellipses belong to the class of *conic sections* (see the exhibit “Conic Sections“)

An *ellipse* is the set of all points of the -plane for which the sum of the distances to two given points and is equal (). Ellipses belong to the class of *conic sections* (see the exhibit “Conic Sections“)

The points and are called *focal points*. The centre of their connecting line (of length , — *eccentricity*) is called the *centre* of the ellipse. The distance from this centre to the two vertices and is in each case and to the vertices and is in each case with (according to the *Pythagorean theorem*; see also the exhibit “Proof without words: Pythagoras for laying“), i.e. .

The connecting line between a focal point or (*focus*) and a point of the ellipse is called the *guide ray* or *focal ray*. The names focal point and focal ray result from the property that the angle between the two focal rays at one point of the ellipse is bisected by the *normal* (straight line, perpendicular to the *tangent*) at that point. Thus, the *angle of incidence* that one focal ray forms with the tangent is equal to the *angle of divergence* that the tangent forms with the other focal ray. Consequently, a light ray emanating from one focal point, e.g. , is reflected at the elliptical tangent in such a way that it hits the other focal point. With an elliptical mirror, all light rays emanating from one focal point therefore meet at the other focal point. If the eccentricity , then applies. Die Ellipse wird zu einem *Kreis* mit dem *Radius* .

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

To create an elliptical flower bed, drive two stakes into the focal points and attach the ends of a *string* of length to them. Now stretch the string and run a marking device along it. Since this method requires additional tools besides a *circle* and a *ruler* – a string – it is not a construction of *classical geometry*. In ADVENTURE LAND MATHEMATICS, this construction can be understood by means of a simple experiment.

In the following, the ellipse equation is derived from the “*gardener’s construction*” described above:

For a point of the ellipse — according to figure 1 above — , i.e. if we set and , we get the equation

Squaring this equation gives

and thus

Squaring again gives:

and by simplification — i.e. suitable “*shortening*” — we get:

i.e.

Because of (see above), the *normal form* (also “*midpoint form*“) of an elliptic equation is

The so-called *first Kepler’s law* (“*ellipse theorem*“, “*planet theorem*“) states that the orbit of a *satellite* is an ellipse. One of their focal points is in the *centre of gravity* of the system. This law results from *Newton’s law of gravitation*, provided that the mass of the *central body* is considerably 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.

Schupp, H.: *Kegelschnitte*, Mannheim, 1988.