Eigelenk

Eigelenk is a term used in some German-language mathematical texts to refer to what is more commonly called an ellipse. In modern usage, ellipse is the standard name, while Eigelenk may appear in historical or regional writings. An Eigelenk is the planar curve consisting of all points for which the sum of distances to two fixed points, called foci, is constant. This locus is a closed, convex curve with mirror symmetry about its major and minor axes.

In Cartesian coordinates with the center at the origin and the major axis on the x-axis, the standard equation of an Eigelenk is x^2/a^2 + y^2/b^2 = 1, where a ≥ b > 0. The major axis length is 2a and the minor axis length is 2b. The focal distance satisfies c^2 = a^2 − b^2, and the foci lie at (±c, 0). The eccentricity e is defined as e = c/a, with 0 ≤ e < 1. A polar representation with one focus at the origin is r(θ) = (b^2 / a) / (1 + e cos θ). A circle is a special case of an Eigelenk when a = b.

Properties and applications include: a constant-sum distance definition, area A = πab, and a well-known optical reflection property: rays emanating from one focus reflect to the other. Ellipses model planetary orbits to a good approximation and appear in optics, architecture, and design. The perimeter has no simple closed form, but there are accurate approximations such as Ramanujan’s formula.

Historically, ellipses were studied by ancient Greek mathematicians and later named in broader European literature; today, Eigelenk is largely supplanted by Ellipse in technical usage.