Equidistant set

The conic sections can also be realized as equidistant sets.

[1] The concept of equidistant set is used to define frontiers in territorial domain controversies.

For instance, the United Nations Convention on the Law of the Sea (Article 15) establishes that, in absence of any previous agreement, the delimitation of the territorial sea between countries occurs exactly on the median line every point of which is equidistant of the nearest points to each country.

[1] Though the usage of the terminology is quite old, the study of the properties of equidistant sets as mathematical objects was initiated only in 1970's.

The study of equidistant sets is more interesting in the case when the background metric space is the Euclidean space.

Animation showing the equidistant set of two singleton sets in a Euclidean plane.
Image showing equidistant set of two straight lines in a Euclidean plane.
Animation showing the generation of parabola as an equidistant set of a singleton point and a straight line.
Animation showing the generation of an ellipse as the equidistant set of two circles.
Animation showing the generation of one branch of a hyperbola as the equidistant set of two circles.