Distance

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Manhattan plan. The euclidian distance (green segment), does not correspond to the “shortest possible path” ente two points of that city, in addition to not only a path of lesser length.

In mathematics, the distance between two points in Euclidean space is equal to the length of the line segment that joins them, expressed numerically. In more complex spaces, such as those defined in non-Euclidean geometry, the "shortest path" between two points is a straight segment with curvature called a geodesic.

In physics, distance is a scalar quantity, expressed in units of length.

Distance in geometry with coordinates

Distance on the line

There is a bijection (an element to element correspondence) between the points of a line and the set $mathbb{R}$ of the real numbers, so that each real number corresponds to a single point, and at each point, exactly a real number. In order to do this it is precise of a point O and fixed of the straight and another point U, such that by definition 1 is the absciss of U. It denotes U(1). The points of positive absciss are on the right, on the left the points of negative absciss, and the O origin has abscised 0. Such a line provided with abscises for its points is called Real.

Yeah. ${displaystyle A(x_{1})}$ and ${displaystyle B(x_{2})}$ are two points of the actual straight, then the distance between points A and B is ${displaystyle d(A,B)=|x_{2}-x_{1}|}$

Distance of two points in the plane

Yeah. ${displaystyle A(x_{1},y_{1})}$ and ${displaystyle B(x_{2},y_{2})}$ are two points of a Cartesian plane, then the distance between these points is calculated as follows: Create a third point, call yourself ${displaystyle P(x_{2},y_{1})}$ from which a rectangle triangle is formed. Continue to use the Pythagorean Theorem with the AB segment as hypotenuse. ${displaystyle H^{2}=(cat_{1})^{2}+(cat_{2})^{2}}$ . Continue to replace the formula with the elements of each segment and perform the procedure:

{displaystyle d(AB)^{2}=AP^{2}+BP^{2}}

{displaystyle d(AB)^{2}=(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}

{displaystyle {sqrt {d(AB)^{2}}}={sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}}

{displaystyle d(AB)={sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}}

Distance in metric space

From a formal point of view, for a set of elements $X$ defined distance or Metric like any mathematical function or application $d(a,b)$ of $X times X$ in $mathbb{R}$ to verify the following conditions:

Not negativity:

{displaystyle forall a,bin X;:quad d(a,b)geq 0}

{displaystyle a,bin X,quad d(a,b)=0quad Longleftrightarrow quad a=b}

- I mean, the distance is zero if only induced on the same point

Symmetry:

{displaystyle forall a,bin X;:quad d(a,b)=d(b,a)}

Triangular inequality:

{displaystyle forall a,b,cin X;:quad d(a,b)leq d(a,c)+d(c,b)}

If we stop demanding that this last condition be fulfilled, the resulting concept is called pseudodistance or pseudometric.

The distance is the fundamental concept of the Topology of Metric Spaces. A metric space is nothing but a pair $(X,d)$ Where $X$ is a set in which we define a distance $d$ .

In case we had a couple $(X,d)$ and $d$ was a pseudo-distance $X$ Then we'd say we have a pseudometric space.

Yeah. $(X,d)$ is a metric and $E subset X$ , we can restrict $d$ a $E$ as follows: $d': E times E longrightarrow mathbb{R}$ So if $x,y in E$ then. $d'(x,y)=d(x,y)$ (i.e., $d'=d|_{E times E}$ ). Implementation $d'$ is also a distance over $d$ and as he shares on $E times E$ the same values as $d$ , it also denotes in the same way, that is, we will say $(E,d)$ is metric subspace $(X,d)$ .

Distance from a point to a set

Yeah. $(X,d)$ It's a metric space, $E subset X$ , $E ne varnothing$ and $xin X$ , we can define the distance of the point $x$ to the whole $E$ as follows:

d(x,E):= inf {d(x,y): y in E}

The following three properties are noteworthy:

First, in the given conditions, there will always be that distance, for $d$ it has by domain $X times X$ So for any $y in E$ There will be a single positive real value $d(x,y)$ . By completeness $mathbb{R}$ and since the image of d is lowered by 0, the existence of the infim of that set is guaranteed, that is, the distance from point to set.

Yeah. $x in E$ then. $d(x,E)=0$ .

Maybe. $d(x,E)=0$ but $x notin E$ for example if $x$ is a point of adherence $E$ . In fact, the closure $E$ is precisely the set of points $X$ distance 0 to $E$ .

The cases of distance from a point to a line or of distance from a point to a plane are nothing more than particular cases of the distance from a point to a set, when considering the Euclidean distance.

The following method can be used: Given a point (n,m) that does not lie on the line f(x), 1) Find the equation of the line perpendicular to f(x) that passes through (n,m). This involves two steps: finding the slope (perpendicular slope) and finding the ordinate to the origin (by plugging in the point (n,m) and solving). 2) Find the intersection between these two lines. This involves two steps: finding the x-intercept by matching, finding the y-intercept by substituting the x into either of the two equations. With this we obtain the point (o,p) 3) Find the distance between (n,m) and (o,p).

Distance between two sets

Yeah. $(X,d)$ It's a metric space, $Asubset X$ and $B subset X$ , $A ne varnothing$ , $B ne varnothing$ , we can define the distance between the sets $A$ and $B$ as follows:

d(A,B):= inf {d(x,y): x in A, y in B}

For the same reason as before, it is always defined. Plus $d(A,A)=0$ , but it may happen $d(A,B)=0$ and yet $A ne B$ . Moreover, we can have two closed sets whose distance is 0 and yet disjoined, and even have disjoined closures.

For example, the whole $A:= {(x,0): x in mathbb{R}}$ and the whole $B:= {(x,e^x): x in mathbb{R}}$ . On the one hand, $A=operatorname{cl}(A)$ , $B=operatorname{cl}(B)$ and $A cap B = varnothing$ and for another $d(A,B)=0$ .

The distance between two lines, the distance between two planes, etc. they are no more than particular cases of the distance between two sets when the Euclidean distance is considered.

References and notes

↑ Howard E. Taylor; Thomas L. Wade: Two-dimensional analytical geometry Subsets of the plane. Editorial Limusa S.A. de C.V, México D.F. (1986) ISBN 968-18-0038-9
↑ D. Kleténik: Problems of analytical geometry. Editorial Mir, Moscow (1968); reviewed by N. Efimov, translation of Emilio Apparition Bernardo.
↑ Walter. Rudin: Principles of mathematical analysis. McGraw-Hill Books, printed in Mexico D-F. (1980). It is translated by Miguel Irano checks Luis Briseño.
↑ V.A. Trenoguin; B.M. Pisarievki; T.S. Baseball: Functional analysis problems and exercises. Editorial Mir, Moscow (1984); translated from Russian, Andriánova M.A; printed in the USSR. https://www.academia.edu/44703968/Problemas_y_Ejercicios_de_An%C3%A1lisis_Funcional_V_A_Trenoguin_MIR
↑ Trenoguin and others: Op. cit.

Additional bibliography

Weisstein, Eric W. Distance. Wolfram Mathworld (in English). Consultation on 5 October 2022.
«Distance Between 2 Points». Math is fun (in English). Consultation on 5 October 2022.
Chan, T.; Zhu, W. (from 2005). «Level Set Based Shape Prior Segmentation». 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05) (in English) 2: 1164-1170. doi:10.1109/CVPR.2005.212.
"The Directed Distance". Information and Telecommunication Technology Center (in English). University of Kansas. Archived from the original on November 10, 2016. Consultation on September 18, 2018.
Malladi, R.; Sethian, J.A.; Vemuri, B.C. (from NaN). «Shape modeling with front propagation: a level set approach». IEEE Transactions on Pattern Analysis and Machine Intelligence (in English) 17 (2): 158-175. doi:10.1109/34.368173.
Elena Deza, Michel Deza (2006). Dictionary of Distances (in English). Elsevier. ISBN 0-444-52087-2.

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