mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, an equidistant set (also called a midset, or a bisector) is a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

whose elements have the same distance (measured using some appropriate

distance function In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are a general setting fo ...

) from two or more sets. The equidistant set of two singleton sets in the Euclidean plane is the perpendicular bisector of the segment joining the two sets. The

conic section A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, tho ...

s can also be realized as equidistant sets. This property of conics has been used to generalize the notion of conic sections. 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 The United Nations Convention on the Law of the Sea (UNCLOS), also called the Law of the Sea Convention or the Law of the Sea Treaty, is an international treaty that establishes a legal framework for all marine and maritime activities. , 169 sov ...

(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. 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.

Definition

Let (''X'', ''d'') be a

metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

and ''A'' be a nonempty subset of ''X''. If ''x'' is a point of ''X'', the distance of ''x'' from ''A'' is defined as ''d''(''x'', ''A'') = inf. If ''A'' and ''B'' are both nonempty subsets of ''X'' then the equidistant set determined by ''A'' and ''B'' is defined to be the set . This equidistant set is denoted by . The study of equidistant sets is more interesting in the case when the background metric space is the Euclidean space.

Definition

Examples

Straight lines

Conics as equidistant sets

See also

References