Examples
Straight lines
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Conics as equidistant sets
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In mathematics, an equidistant set (also called a midset, or a bisector) is a set each of whose elements has the same distance (measured using some appropriate distance function) 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 sections can also be realized as equidistant sets. This property of conics has been used to generalize the notion of conic sections. [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. [1] [2]
Let (X, d) be a metric space 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{ d(x, a): a in A}. 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 {x in X: d(x, A) = d(x, B)}. This equidistant set is denoted by { A = B }.
The study of equidistant sets is more interesting in the case when the background metric space is the Euclidean space. [1]
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In mathematics, specifically general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples of compact spaces include a closed real interval, a union of a finite number of closed intervals, a rectangle, or a finite set of points. This notion is defined for more general topological spaces in various ways, which are usually equivalent in Euclidean space but may be inequivalent in other spaces.
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment . For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex.
Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension, including the three-dimensional space and the Euclidean plane. It was introduced by the Ancient Greek mathematician Euclid of Alexandria, and the qualifier Euclidean is used to distinguish it from other spaces that were later discovered in physics and modern mathematics.
In mathematics, a metric space is a set together with a metric on the set. The metric is a function that defines a concept of distance between any two members of the set, which are usually called points. The metric satisfies a few simple properties. Informally:
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.
In mathematics, open sets are a generalization of open intervals in the real line. In a metric space—that is, when a distance function is defined—open sets are the sets that, with every point P, contain all points that are sufficiently near to P.
Distance is a numerical measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria. The distance from a point A to a point B is sometimes denoted as . In most cases, "distance from A to B" is interchangeable with "distance from B to A". In mathematics, a distance function or metric is a generalization of the concept of physical distance; it is a way of describing what it means for elements of some space to be "close to", or "far away from" each other. In psychology and social sciences, distance is a non-numerical measurement; Psychological distance is defined as "the different ways in which an object might be removed from" the self along dimensions such as "time, space, social distance, and hypotheticality.
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset.
In real analysis the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states:
In mathematics, a ball is the volume space bounded by a sphere; it is also called a solid sphere. It may be a closed ball or an open ball.
In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane. For each seed there is a corresponding region, called a Voronoi cell, consisting of all points of the plane closer to that seed than to any other. The Voronoi diagram of a set of points is dual to its Delaunay triangulation.
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.
In mathematics, hyperbolic geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
In geometry, a locus is a set of all points, whose location satisfies or is determined by one or more specified conditions.
In combinatorics, a Helly family of order k is a family of sets in which every minimal subfamily with an empty intersection has k or fewer sets in it. Equivalently, every finite subfamily such that every -fold intersection is non-empty has non-empty total intersection. The k-Helly property is the property of being a Helly family of order k.
In mathematics, a metric or distance function is a function that gives a distance between each pair of point elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set, but not all topologies can be generated by a metric. A topological space whose topology can be described by a metric is called metrizable.
In mathematics, convex metric spaces are, intuitively, metric spaces with the property any "segment" joining two points in that space has other points in it besides the endpoints.
In mathematics, a conic section is a curve obtained as the intersection of the surface of a cone with 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, though historically it was sometimes called a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties.
A zone diagram is a certain geometric object which a variation on the notion of Voronoi diagram. It was introduced by Tetsuo Asano, Jiri Matousek, and Takeshi Tokuyama in 2007.
In mathematics, a generalized conic is a geometrical object defined by a property which is a generalization of sums defining property of the classical conic. For example, in elementary geometry, an ellipse can be defined as the locus of a point which moves in a plane such that the sum of its distances from two fixed points – the foci – in the plane is a constant. The curve obtained when the set of two fixed points is replaced by an arbitrary, but fixed, finite set of points in the plane is called an n–ellipse and can be thought of as a generalized ellipse. Since an ellipse is the equidistant set of two circles, the equidistant set of two arbitrary sets of points in a plane can be viewed as a generalized conic. In rectangular Cartesian coordinates, the equation y = x2 represents a parabola. The generalized equation y = xr, for r ≠ 0 and r ≠ 1, can be treated as defining a generalized parabola. The idea of generalized conic has found applications in approximation theory and optimization theory.