Talk:Line (geometry)/Archive 1

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Let's mention that:

• a line is straight
• it's the shortest distance between two points, i.e., a line segment
• a line is infinite in 2 opposite directions, while a ray starts at one point and then goes "to infinity, and beyond!" (okay, leave out the joke...)

--Uncle Ed 21:01 Mar 11, 2003 (UTC)

As for: a line segment [... is] the shortest distance between two points

No (again!).

"Distance" is the measure of a geometric relation between two points/elements, while

"Line" (and "Line segment") is a certain set of infinitely many points/elements.

As for: a line is straight

Indeed. And importantly: how to measure whether a curve is straight (and therefore a line), or not.

Regards, Frank W ~@) R 22:02 Mar 16, 2003 (UTC)

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As far as I know (and I didn't get much past calculus), even the usual non-Euclidean geometry accepts the definition of a line as "the shortest distance between two points." It is also true that there are an infinite number of points in a line segment, but that's a consequence of treating distance as measurable by real numbers, rather than quantized. (Which is a closer map to the physical universe is not yet know, and may not be relevant.) Vicki Rosenzweig

No: the length of a line segment is the shortest distance between two points. But a distance is a number and neither a line nor a line segment is a number. Michael Hardy 22:07, 21 November 2005 (UTC)

As for: even the usual non-Euclidean geometry accepts the definition of a line as "the shortest distance between two points."

Then let that be represented as well, as the POV of (some of) those who are unable to distinguish between "infinitely many", and "(relation between) two".

As for: an infinite number of points in a line segment, but that's a consequence of treating distance as measurable by real numbers, rather than quantized.

1. Distance ratios may be real or rational numbers (e.g. "part_segment_distance" / "whole_segment_distance"); but not any one distance value by itself; therefore they're usually represented as a number and a "unit" or "unit distance".

2. As far as it makes a difference: treating distance ratios as measureable by rational numbers can have the "consequence" of an infinite number of points in a line segment, too.

3. It's plainly a matter of definition; if for any two distinct elements und corresponding distance values there were for whatever reason not yet another element and distance value "between", then -- that's not a "line" in this sense. "It", i.e. such a set of elements, might be called "line" in some other sense (and Wikipedia should appropriately disambiguate these notions); or "it" might be given some different name altogether (which Wikipedia could accomodate right away).

Regards, Frank W ~@) R 04:16 Mar 17, 2003 (UTC)

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In general, the straightness of any curve (and therefore, whether or not it constitutes a line) can be defined and evaluated from the distances measured (pairwise) between its elements, by Heron's formula.

This only applies to Euclidean spaces, not in general. Heron's formula on, for example, a sphere, does not yield empty triangles for the distances between three points on a line (i.e., a great circle). To define a straight line based on Heron's formula is essentially circular: straight lines, after all, make up the sides of a triangle. Chas zzz brown 05:49 Mar 17, 2003 (UTC)

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As for: Heron's formula on, for example, a sphere, does not yield empty triangles for the distances between three points on a [...] great circle

Right -- as indeed it should not, if the distances under consideration are to be taken for the Chords "through the bulk" of the sphere.

That's why, yes, from this "perspective" (i.e. considering a space in which the sphere is embedded), any three such points are called belonging to a circle (in distinction to "straight line").

Also correctly noted: the relevant defining evaluation is whether Heron's formula evaluates to zero.

We don't neccessarily need to consider the sphere as "embedded" in another space - although that is a convenient way of visualising it. Instead, we can simply define distance between points as the Euclidean distance as if the sphere were embedded in the next higher space. Then the "distance between two points" is not equal to "the length of the shortest curve between those two points". The relevant defining definition is not whether Heron's formula evaluates to zero - it is the more exact definition given at geodesic. Chas zzz brown 19:14 Mar 20, 2003 (UTC)

As for: for example, a sphere, does not yield empty triangles for the distances between three points on a line (i.e., a great circle).

No -- if the distances under consideration are to be taken for the (great circle) Arcs "on the surface" of the sphere (at least: within one-and-the-same hemisphere).

I.e., if distance is defined as the length of a straight line...

For these distances of three such points to each other, Heron's formula does "yield empty triangles".

Heron's formula is used to determine the area of a triangle in Euclidean space, given the length of the sides of the triangles. This formula does not yield the correct area of triangles unless we are in Euclidean space, except for the trivial case where the triangle has 0 area, and we use the same definitions for length and distance.

So, if you define "distance between points" as identical to "length of the shortest curve bewteen two points", then you will get "empty triangles" for three points on such a curve. But this is due to the (trivial) observation that, in that case, given points A, B, and C, we must have that AB + BC = AC, so that (AB + BC - AC) = 0. So your formula is not really derived from Heron, but follows from your definition of distance being identical to length - that's why it's essentially a circular definition. Chas zzz brown 19:14 Mar 20, 2003 (UTC)

From this "perspective" (strictly within the surface) such points do constitute straight lines, by Heron's formula. (The cases which extend beyond one hemisphere may be separately considered; involving an appeal to the "connectedness" of the curve under consideration.)

No, you've got it backwards. Defining distance as the length of the shortest curve (i.e., straight line) causes Heron's formula to become degenerate - not the other way round. That's why Heron's formula is the wrong way to go here. Chas zzz brown 19:14 Mar 20, 2003 (UTC)

As for: To define a straight line based on Heron's formula is essentially circular: straight lines, after all, make up the sides of a triangle.

Not at all: As a definition of straightness, Heron's formula does not refer to "sides" at all, but only to the distances of three particular elements, pairwise wrt each other.

Heron's formula is used to determine the area of triangle in Euclidean space. In Euclidean space, distance and length of the straight line segment are identical. Why use Heron in non-Euclidean spaces, where it is incorrect as a measure of triangle area, and where it only yields "empty triangles" if special conditions are met (i.e., distance = length)? Chas zzz brown 19:14 Mar 20, 2003 (UTC)

Now, the present definition of straightness in the article to be talked about refers to "the shortest curve [...]". -- What's that?, i.e. how might the author(s) who introduced this notion have supposed to evaluate and compare "shortness" of various curves?? (Are "distance" values to be determined? Can the Triangle inequality be employed? May straightness a la Heron be obtained at least as a special case? ...)

Certainly, I considered including a short definition of length as "the limit of the sum of the distances between points on the curve", but this starts to get quite technical. We can appeal to the fact that there is a homeomorphism between the real line and this curve to acheive this in a sensible fashion - but there's a lot of detail to fill in, and this is supposed to be an elementary article. Want more detail? See geodesic. Chas zzz brown 19:14 Mar 20, 2003 (UTC)

As far as that's understood, and in view of the example cases extending beyond one hemisphere, the present definition of straight line as a curve which contains the shortest curve between any two seems to make it known that a great many curves (namely: "sufficiently long" arcs of a great circle) are excluded from being identified as straight lines, despite being proper (and connected) subsets of a curve which does contain the shortest curve between any two (namely: an entire great circle).

A subset of the great circle (for example, a long arc) is a line segment - a connected subset of a line. Chas zzz brown 19:14 Mar 20, 2003 (UTC)

But of course, precisely this may be known as a "Line (in) Mathematics" ... If so, a more coherent definition may have to be disambiguated as "Line (in) Physics".

Why do you believe that physics uses a different definition than mathematics? Chas zzz brown 19:14 Mar 20, 2003 (UTC)

Regards, Frank W ~@) R 08:04 Mar 18, 2003 (UTC)

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These pages that treat several nearly unrelated concepts that happen to be called by the same name are silly. This needs to be made a disambiguation page, or else a page that treats only one topic, a separate page being called line (disambiguation) pointing to the onther "line" topics (e.e., line (electrical engineering). Michael Hardy 01:51, 23 May 2004 (UTC)

I fully agree with you. I'm adding interlanguage links between the Finnish and English Wikipedias. Here are some translations from English to Finnish: a straight line (geometry) -> suora, a line (graphics) -> viiva, a line of text -> rivi, a line of people (a queue) -> jono, a telephone line -> puhelinlinja. There is no way we are going to put all these concepts on one page in the Finnish Wikipedia. Other languages probably have this problem too. -- Lakefall 21:10, 4 Jun 2004 (UTC)