2023-10-09 12:32:14
Introduction
In mathematics, a distance is an application which formalizes the intuitive idea of distance, that is to say the length which separates two points. It is through the analysis of the main properties of the usual distance that Fréchet introduces the notion of metric space (In mathematics, a metric space is a set within which a notion of distance enters…), developed (In geometry , the evolute of a plane curve is the locus of its centers of…) then by Hausdorff. It introduces a geometric language into many questions of analysis and number theory.
From the definition (A definition is a discourse which says what a thing is or what a name means. Hence the…) of a distance, view (View is the sense which allows one to observe and analyze the environment by reception and…) as an application satisfying certain axioms, other notions of distance can be defined, such as for example the distance between two parts, or the distance of a point (Graphics) to a part, without these latter responding to the primary definition of a distance.
Distance on a set
Definition
In mathematics (Mathematics constitutes a domain of abstract knowledge constructed using…), we call distance on a set E an application verifying the following properties:
Name Property symmetry separation triangular inequality
A set with a distance is called a metric space.
Noticed
In the definition of a distance, we generally ask that the arrival set be; we can simply assume that it is and invoke the sequence of inequalities valid for any pair (x,y) of real numbers:
using separation, triangular inequality and then symmetry respectively.
Property: Ultrametry
The distance is said to be ultrametric if in addition:
Name Property Ultrametry
An example of such a distance occurs crucially in the theory of p-adic valuations. The geometric interpretation of the triangular inequality in an ultrametric space leads us to say that all triangles are isosceles.
Distance between two sets
Let E1 and E2 be two parts of a metric space with a distance d, we define the distance between these two sets as:
NB This “distance” is not a distance over all the parts of E in the sense of the axioms defined above. In particular if the distance between two sets is zero, we cannot deduce that these sets are equal.
Nevertheless, it is possible to define a true distance between the compact parts of a metric space. For this, see: Hausdorff distance (In mathematics, and more precisely in geometry, the distance of…).
Algebraic distance
Consider two points A and B of an affine space (Historically, the notion of affine space comes from the shock due to the…) through which an oriented line passes (a line with a meaning, that is- that is to say generated by a vector (In mathematics, a vector is an element of a vector space, which allows…) non-zero). We call the algebraic distance from A to B the real such that:
its absolute value (In mathematics, the absolute value (sometimes called module) of a real number is its…) or the distance (defined above) between A and B if the value is non-zero: the real is positive in the case where the vector has the same direction as , that is to say equal to , with k > 0, negative otherwise.
We can demonstrate that the algebraic distance from A to B (denoted da(A,B)) is worth:
Be careful, the algebraic distance is not a distance, since it is non-symmetric:
da(A,B) = − da(B,A)
Distance on vector spaces
Distance from Manhattan (Manhattan is one of the five boroughs of New York City (the other four…) (red path (The color red has different definitions, depending on the color system we use.. .), yellow (There are (at least) five definitions of yellow which designate approximately the same…) and blue) against Euclidean distance in green (Green is a complementary color corresponding to light which has a length d ‘wave…)
In a standardized vector space, we can always canonically define a distance d from the norm (A norm, from the Latin norma (“square, ruler” ) designates a…). In fact, it is enough to ask:
In particular, in , we can define the distances between 2 points in several ways, although it is generally given by the Euclidean distance (or 2-distance). Given two points of E, (x1, x2, …,xn) and (y1, y2, …,yn), we express the different distances as follows:
Name Parameter (A parameter is in the broad sense an element of information to take into account…) Function Manhattan distance 1-distance Euclidean distance 2-distance Minkowski distance p-distance Chebyshev distance ∞-distance
The 2-distance makes it possible to generalize the application of the Pythagorean theorem (The Pythagorean theorem is a theorem of Euclidean geometry which…) to a dimensional space (In the common sense, the notion of dimension refers to size; the dimensions of a…) n. This is the most intuitive distance.
The p-distance is rarely used outside of the cases p = 1, 2 or ∞. The 1-distance has the amusing particularity of allowing the rigorous definition of square spheres (see oxymoron).
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