Talk:Distance

(old)

Could somebody who knows how please write out the distance formula in symbols, too, just for clarity? I read the last paragraph a couple of times before I realized that it was talking about the formula I know :) Dreamyshade

I split proxemics and propinquity into separate articles. When people are looking for 'proxemics' they will likely be surprised to see a page about geometrical distance. Ditto for 'propinquity' -- I study both mathematics and psychology, and propinquity is a major topic in social psychology, but I have never heard it used in geometry.

--Johnkarp 11:11, 9 Oct 2004 (UTC)

removed not well-explained and unhelpful paragraph

I removed the following text from distance

Distance is one sort of interval between things or events, Time being the other.

I found it unclear. Oleg Alexandrov 22:20, 31 May 2005 (UTC)

But see Time. Banno 21:39, May 31, 2005 (UTC)

OK, you are writing at the very beginning of the time article:

Time is one sort of interval between things or events, distance being the other..

I am not sure this is a proper characterisation of time. I am not sure time and distance can be compared in that way. I am not sure you what you mean by interval in this sentence. I cannot even say that what you wrote is incorrect, I think it is ambiguous and confusing. Would you clarify me please? Oleg Alexandrov 22:14, 31 May 2005 (UTC)

An excellent point - I see the circularity in interval. I had just retained the word from a previous edit. Try:

Time is one sort of separation between things or events, distance being the other..

Failing that, I'm open to your suggestions. Banno 22:48, May 31, 2005 (UTC)

This sentence is true. But I do not think it is so important or so releveant as to be the very first sentence in either the distance article or in the time article. Oleg Alexandrov 23:37, 31 May 2005 (UTC)

If you wish to demote it, be my guest. But then, what is your suggestion for an introduction to time? The advantage of this sentence is that it draws attention to the link between time and distance; it is also succinct. Banno 23:48, May 31, 2005 (UTC)

"This sentence is true." - AFAIK in a classical Newtonian universe - yes, in an Einstein-Minkowski universe - no. Physicists nowadays know that time and space are not independent, consequently causing things like e.g. the Twin paradox. Furthermore I understand that extreme accurate measuring and comparing of horizontal and vertical distances in a gravitational field is not trivial and therefore might warrant its own page (but my current knowledge about physics isn't enough for that). Stevemiller (talk) 17:47, 22 January 2008 (UTC)

In the most general sense: Distance is a description or gauge of the expected strength of interaction of between two reference entities A and B. Strength of interaction also being the basis for ease of observability. Distance is a transitive property in that proposed interaction between A and C may be described in terms of AB and BC interactions. In general physics and the common realm of human perception, distance is normally referenced to gravity and EM fields, the macroscopic long forces.

One trivial result of this definition is that the universe will remain finite in size despite the "infinite" numerical space upon which it could be mapped. Given a finite number of elementary particles, all systems in which a given particle is separated beyond measurable interaction of forces with all other particles are essentially identical. Thus meaningful and recognizable definitions of the universe require that all elementary particles remain within limits of interaction with at least one other particle in the universe. Given a rough estimate of the number of particles in the universe an theoretical upper limit can be set on the size of a meaningful universe.

Alternatively, we could expect that the number of particles and mass of the universe can randomly change through "disappearance" and "appearance" of particles from beyond meaningful distance at the edge of the universe. The old philosophical issue of observability: "just because you can't see it or measure it doesn't mean it isn't there -- only that you are at the limits of your ability to measure via interactions". Of course at the quantum level where distance is a measure of interaction via dominant short forces, the edge of the universe might be at your very feet, if very rare. What happens when a given particle finds a position beyond the limits of short force interaction with any other particle? Essentially it meets the criteria for being unobservable and can fall into a crack and disappear "perpendicular" to our universe with its reappearance only a statistical probability. 69.23.124.142 (talk) 20:11, 9 January 2008 (UTC)

Reworked Article

I’ve made some rather significant changes to this article. In the course of doing so, I deleted the following material. Wherever possible, I’ve attempted to reincorporate this information elsewhere in the text, but some of it didn’t seem particularly pertinent to the discussion. As this is one of my first attempts, I would appreciate feedback from more experienced users. It would be nice if someone could help with the page layout or say something more about physics and informal treatments. I wasn't sure what to do with the "Euclidean spaces" section either, but it seems a like it might be a bit excessive for this article.

The distance between two points is the length of a straight line segment between them.

Sometimes a distance thus indicated is ambiguous because the means of transport is neither mentioned nor obvious.

In the study of complicated geometries, we call the most common type of distance Euclidean distance, as we define it from the Pythagorean theorem.

===The distance formula===

The (Euclidean) distance, d, between two points expressed in Cartesian coordinates equals the square root of the sum of the squares of the changes of each coordinate.

Thus, in a two-dimensional space

$d={\sqrt {(\Delta x)^{2}+(\Delta y)^{2}}},$

and in a three-dimensional space:

$d={\sqrt {(\Delta x)^{2}+(\Delta y)^{2}+(\Delta z)^{2}}}.$

Here, "Δ" (delta) refers to the change in a variable. Thus, Δx is the change in x, pronounced as such, or as "delta-x". In mathematical terms, $\Delta x=\left|x_{1}-x_{0}\right|$, and so $(\Delta x)^{2}=(x_{1}-x_{0})^{2}$.

This distance formula can be seen as a specialized form of the Pythagorean theorem; it can also be expanded into the arc-length formula.

== Formal definition ==

A distance between two points P and Q in a metric space is d(P,Q), where d is the distance function that defines the given metric space.

We can also define the distance between two sets A and B in a metric space as being the minimum (or infimum) of distances between any two points P in A and Q in B.

Alternatively, the distance between sets may indicate "how different they are", by taking the supremum over one set of the distance from a point in that set to the other set, and conversely, and taking the larger of the two values (Hausdorff distance).

===Generalized distance in arbitrary dimensions: Norms===

== Distances in other spaces ==

==Distance covered==

--Fell Collar 06:39, 10 March 2006 (UTC)

Headers should have sentence-style capitalization (i.e. only one capital).

Also you made some odd deletions, e.g. in the section on the distance between sets.

Please do not start a line with a blank space, even on the talk page, it makes lines wider than the screen.--Patrick 14:38, 10 March 2006 (UTC)

I've put back the material about the Hausdorff metric in the section on distance between sets and fixed the headers. Are there any other deletions that struck you as inappropriate? I had read that it is considered polite to move deleted material to the talk page, but I wasn't sure how to format it. What is the convention? --Fell Collar 16:51, 10 March 2006 (UTC)

Aw, I see it right above: indent and italicize. I've restored a great deal more of the material I'd deleted. In retrospect, I'm not sure why I got rid of a lot of it. The remainder is either

• paraphrased elsewhere in the article
• a now-defunct header, or
• superfluous (e.g. the bit about "delta x", which should be fairly obvious from the way I've formatted the equation)

Fell Collar

Thanks!--Patrick 00:32, 11 March 2006 (UTC)

Negative Distance?

Someone (81.207.214.244) indicated that one could have negative distance if it was "predefined". I wonder if they (or someone else) could elaborate. If distance was negative, it certainly wouldn't be a metric, so it seems odd to me that we would call it "distance" in a formal setting. Can someone give an example? --Fell Collar 18:03, 13 March 2006 (UTC)

I'm moving the sentence here for now. It seems very suspect, and at any rate distance certainly won't be negative in Euclidean space (the section it was placed in). If someone could elaborate and put this where it belongs, that would be nice.

There is no such thing as a negative distance (unless it is predefined).

--Fell Collar 03:41, 16 March 2006 (UTC)

Ive changed the heading of what was the physics section, because, it appeared to have nothing much to do with physics! I made a new physics description, and put it at the top, but I think it generally covers what most people know of as distance. I deleted the "distinctions" of distance that had been written, that said things like "the distance formula is not true for curved surfaces", which cant be true, because the distance formula is part of the law of physics and is always true! they were true in a general sense, but i didnt see how they had any useful meaning.

if someone understands the special theory of reletivity better than i do, can they please give some information on how distance applies generally in physics.

ive been thinking that it might be overkill to seperate this topic into physics/geometry/maths, because distance is all of them at once, and it doesnt need to be explained over and over in slightly different senses.

(Comment)Isn't it impossible to have a negative distance, as it would just be a positive distance in another direction. Somebody, correct me if I'm wrong.24.22.22.113 (talk) 00:39, 16 September 2009 (UTC)

@24.22.22.113: There's no such thing as negative distance, distance uses absolute values therefore it is positive no matter what direction. PlanetStar 01:21, 28 April 2017 (UTC)

Content syncronizing

Please see Talk:Metric (mathematics)#Content syncronizing. `'Míkka 17:52, 8 October 2007 (UTC)

Distance between functions

I am of the opinion that a section on distance between functions be added:

For e.g. In the fourier series representation of a simple periodic function f(t) with time 't' as the independent variable, the fourier coefficients are so adjusted that the distance between the finite fourier sum and the function f(t) is minimized.

Here the distance between f(t) and g(t) is given by : {(1/2pi integral((f(t)-g(t))sqrd.dt}sq. root.

It would be infromative if this is discussed in the article. Sridhar10chitta (talk) 08:48, 27 November 2007 (UTC)

distance of a function

It would probably be appropriate if a section describing the distance of a time dependent function maybe f(t) = at^sq + bt +c is added as well before discussing the distance between functions. Sridhar10chitta (talk) 16:56, 28 November 2007 (UTC)

minkowski unit circles

Here's an image I made that might be helpful for explaining Minkowski distances:

Purpose of this article

This article mainly talks about distance in mathematics. Most of that material is in Metric (mathematics), or could be moved there. There is little mention of distance in physics, and no mention of distance in everyday use. Why do we have this article? --Taeshadow (talk) 15:49, 11 March 2009 (UTC)

Removed section

The following seciton is removed since it is absolutely incomprehensible. While it may be OK for an expert in the field, who can read between lines, it is absolutely inadmissible in encuclopedia due to its lack of context, lack of explanation of notation and unclear opinions. Whoever can fix it, they are welcome. - Altenmann >t 15:33, 24 September 2009 (UTC)

Generalization of Euclidean distance to higher-dimensional manifolds

The Euclidean distance between two objects may also be generalized to the case where the objects are no longer points but are higher-dimensional manifolds, such as space curves. Additional concepts such as non-extensibility, curvature constraints, and non-local interactions that enforce noncrossing then become central to the notion of distance. The distance between the two manifolds is the scalar quantity that results from minimizing the generalized distance functional, which represents a transformation between the two manifolds:

${\mathcal {D}}=\int _{0}^{L}\int _{0}^{T}\,ds\,dt\left\{{\sqrt {\left({\partial {\vec {r}}(s,t) \over \partial t}\right)^{2}}}+\lambda \left[{\sqrt {\left({\partial {\vec {r}}(s,t) \over \partial s}\right)^{2}}}-1\right]\right\}$

If two discrete polymers are inextensible, then the minimal-distance transformation between them no longer involves purely straight-line motion, even on a Euclidean metric. There is a potential application of such generalized distance to the problem of protein folding This generalized distance is analogous to the Nambu–Goto action in string theory, however there is no exact correspondence because the Euclidean distance in 3-space is inequivalent to the space-time distance minimized for the classical relativistic string.

OK. I added a section on variational formulation of distance. This will make the above text much more accessible to an non-expert user. Nocompletechaos (talk) 07:42, 26 September 2009 (UTC)

References

Definition

Currently the article begins:

Distance is a numerical description of how far apart objects are.

Sometimes distance is not expressed numerically, but rather quantitatively as in "A is further away than B". Also, sometimes it is not about how far apart objectS are, but about how extensive a single object is - either numerically as "A is 3 metres long", or non-numerically (but still quantitatively) as "A is taller than B". It can aso be the distance travelled in a period of time, like "The Earth travels further in one orbit than Mercury does" JimWae (talk) 09:28, 12 July 2010 (UTC)

Prudent distance

@IM3847: What's the definition of prudent distance? I saw it in an article about Cassini's final days and said 'Cassini is orbiting Saturn at the same inclination it will have during the Grand Finale, but at a more prudent distance.' I learned the definition of prudence but how that relates to distance? PlanetStar 00:32, 28 April 2017 (UTC)

This is not a technical term, which is probably why no one has responded to your query. Some editor was just waxing eloquently when what was meant was a "safer" distance, that is, not so close as to spiral into Saturn as it will during the Grand Finale.--Bill Cherowitzo (talk) 04:26, 13 May 2017 (UTC)

Splitting Distance (mathematics)

I noticed you recently made some edits to distance and split out distance (mathematics). Do you have a vision for what should be on these pages? I've tried to think coherently about the "distance" page but I don't have a clear idea of its intended audience or purpose, either now or ideally in the future. What sort of person visits the Wikipedia page on "distance" and what do they hope to find there? (My angle here is that I'm a mathematician and recently thoroughly rewrote metric space, which has some overlap with both these pages but probably a different intended audience.) --platypeanArchcow (talk) 00:46, 14 September 2022 (UTC)

If you're interested in a broader discussion we can also move this to Wikipedia talk:WikiProject Mathematics. --platypeanArchcow (talk) 00:47, 14 September 2022 (UTC)

@PlatypeanArchcow: I wanted to give more space for Distance (mathematics) to grow and flourish. Then Distance would remain as a WP:broad-concept article. A successful example is Surface and Surface (mathematics). fgnievinski (talk) 00:53, 14 September 2022 (UTC)

Thanks for the links, that's helpful. In the case of surfaces, there really is a well-defined mathematical idea of a surface which can be thought of as the mathematical formalization of the broad concept. This is not so clear in the case of distance. Thinking aloud:

• The mathematical formalization of the colloquial notion of distance (whether literal or metaphorical) is the metric in a metric space. Note that distance function already redirects to metric space. Moreover, the ideas described in the sections "In physics and geography", "In computer science", as well as some of those in "In statistics", can be formalized as metrics. (In particular, they are all mathematical concepts, so separating distance (mathematics) from them seems perverse.)
• Not everyone who comes to Wikipedia to understand the mathematics of distance is well-served by the article "metric space". If they are interested in distances in Euclidean geometry or something like distance between two skew lines, then it will be needlessly intimidating for them. However, these elementary topics are fairly well-summarized in Euclidean distance.
• Not every concept described in distance (mathematics) is a metric. Specifically, Distance (mathematics)#Between sets or a point and a set describes something that's not a metric. Perhaps distance between sets deserves to be a separate article.
• The current distance (mathematics) is a random collection of topics that doesn't serve anyone particularly well. Most of these topics are covered in depth in the articles "Euclidean distance", "metric space", and "geodesic".

In summary, I'm not convinced that distance (mathematics) needs to exist, but there does need to be some sort of disambiguation/launching point for anyone who is looking for information on the various topics that could be there. IMO this could easily fit within the main broad-topic article (which itself needs a lot of improvement).

--platypeanArchcow (talk) 04:22, 14 September 2022 (UTC)

@PlatypeanArchcow: Thanks for your input. Your first and second points illustrate well the challenge: "distance function" may suffice for a mathematician, but it doesn't seve well for a layperson. But blanking Distance (mathematics) and redirecting it to Euclidean distance would not do justice to the richness of mathematical concepts related to "distance". Ideally, Distance (mathematics) would be a either summary style article (possibly with automatic excerpts from other articles) or an introduction to metric space (as in [1]). fgnievinski (talk) 03:03, 15 September 2022 (UTC)

@Fgnievinski I find unnecessary -- and, frankly, a bit weird -- to have two articles. I agree with @PlatypeanArchcow that it isn't clear there is a need for Distance (mathematics), and that the content of this article could easily fit in a subsection of Distance.

I think that the fact that there isn't a need for two separate articles is well illustrated by the fact that Distance (mathematics) currently starts with "In mathematics, a metric space is a set for which distances between all members of the set are defined." and that Metric space starts with "In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points."

I would also like add another point against keeping Distance (mathematics): I think it can hide Metric space for some users (or at least make it a bit less direct to find the right article). One of the main reasons why I use math articles such as Metric space is to quickly check a specific technical point that I have forgotten or -- as a non-native English speaker having studied math outside of the English-speaking world -- to remind myself of the correct way to name some objects in English (e.g, "is there a standard name for that axiom saying that d(x, y) = 0 iff x = y?" or "is that axiom called the 'triangular' inequality, or just the 'triangle' inequality"; etc). I think those are fairly common use cases.

With the new article, I land by default on Distance (mathematics), which is not the article I need. Since the name suggests to me that I'm on the right article, I don't even bother looking around for Metric space: I just assume that what I'm looking for is not on Wikipedia, and I leave.

So, for all these reasons: personally not a big fan of the split. Malparti (talk) 13:11, 16 September 2022 (UTC)

@Malparti, thanks for the comments. With your implied blessing I reorganized the article in a way that hopefully demonstrates that a separate "(mathematics)" sub-article is unnecesssary :) --platypeanArchcow (talk) 00:59, 18 September 2022 (UTC)

See also Length#Use_in_mathematics. fgnievinski (talk) 06:51, 21 September 2022 (UTC)

Definition source

Is there any source for the definition given?131.114.35.40 (talk) 14:11, 26 October 2022 (UTC)