Talk:Subtended angle

Subtend and insist

As far as I know, an angle is subtended by an arc, and not vice versa. Also, an object is subtended by a solid angle, and not vice versa. English is not my mother tongue. However, in my language there exists a second verb, which is used to express the opposite concept. I am not sure, but the most likely English translation of this verb is "to insist" (an angle insists on an arc = an angle defines the endpoints of an arc).

Thus, "subtended arc" is improper. The article should be called "subtended angle", and its text modified. Paolo.dL (talk) 17:20, 18 December 2007 (UTC)

Yes, this is very confusing. In the article listed as Solid Angle (http://en.wikipedia.org/wiki/Solid_angle), the following quote is made: "The solid angle, Ω, is the angle in three-dimensional space that an object subtends at a point." It appears there is confusion as to what subtend actually means. What should be the object of the verb? I must admit I see the statement that an object subtends an angle more frequently than visa versa. Should this article be corrected?Firth m (talk) 23:25, 24 February 2008 (UTC)

From a quick look at a few online dictionaries, it would seem to me that, in English at least, "subtend" appears to be a symmetric relation: if A subtends B, then B subtends A. The usage of a line/figure/object subtending a (solid) angle does seem to be somewhat more common, and is the only one given by MathWorld and thesaurus.maths.org, but the examples given in the general-purpose dictionaries strongly suggest that the opposite is linguistically valid as well. Indeed, some of the examples given, such as an arc subtending a chord, don't (directly) involve angles at all.

Incidentally, it is somewhat notable that most of the Google results for "subtend" are indeed dictionaries of some sort. It's just not a very commonly used word. —Ilmari Karonen (talk) 03:06, 3 November 2008 (UTC)

That's a lot of dictionaries. Let me add one more, the OED. According to the OED, "subtend" comes from the Latin subtendere, which comes from "sub-" meaning under and "tendere" meaning "stretch". As a transitive verb, it means "To stretch or extend under, or be opposite to: said esp. of a line or side of a figure opposite an angle; also, of a chord or angle opposite an arc." The sense of an angle subtending an arc goes back to the word's earliest use in English in Billingsley's 1570 translation of Euclid's Elements: "That angle is said to subtend a side of a triangle, which is placed directly opposite, and against that side." (I.IV.14) In the opposite direction, that of an arc subtending an angle, the earliest use is Leonard Digges, A geometrical practise named Pantometria (1571), which says, "This done conioyne their endes togither and the angle subtended of the longest staffe is a right." (I.xviii) So it seems that both "subtended angle" and "subtended arc", or equivalent constructions, have been used in English for over 450 years now. Ozob (talk) 15:28, 3 November 2008 (UTC)

It seems like Ozob's research is a bit more complete then mine. I just went to amazon and searched inside serveral geometry texts, but it does seem it is fine to use both constructions. Thenub314 (talk) 19:04, 3 November 2008 (UTC)

Another proposal: the arc subtends the angle; the angle intercepts the arc. Michael Hardy (talk) 19:13, 3 November 2008 (UTC)

If it helps any, people should remember that "sub" (as a prefix) does not always mean "under" in Classical Latin. It frequently means "coming up from below" (as in the verbs supporto, sustollo, and suffero). 216.99.198.254 (talk) 04:43, 21 June 2009 (UTC)

I think it is misleading to use dictionaries in this way. Dictionaries reflect usage, whether that usage is strictly-speaking correct. Also, this is part of a mathematics project, not English, so definitions and meaning could well be more specific.

Etymology is a much better method: But a word is coined to represent a specific concept – Just because another concept could be represented by that same word does not mean that was the original intention.

Original usage, therefore, is a good indicator of original intent, but that was 440 years ago: What should the modern meaning be – informed by years of use and interpretation and (mis)understanding?

Most disciplines have their own jargon, which is usually more (and extremely rarely, if ever, less) specific than common usage. And mathematics is surely the most disciplined of disciplines.

The most edifying observation is that MathWorld and thesaurus.maths.org only give one definition – a line subtends an angle.

So what do we want? – I want a word to describe one concept and have one meaning. What is acceptable should, I feel, carry little weight. We need a word that describes the construction of an angle from a line or similar, at a position, and 'subtend' fulfils that role admirably.

The angle subtended by a chord at the major arc of a circle, is constant, and is π minus the angle subtended at the minor arc.

We also need a word that describes the construction of an arc or line segment from an angle, although this would be less common in the practical sciences. Insist (which I like, but I'd have to look-up) and intercept, have been proposed here, but I don't know if there is an established convention. If I were writing about a line, I would probably use delineated, or segmented (as in a line-segment; but might be confused with the segment of a circle). A line-segment could be defined by an angle (and a line). An arc could be constrained by an angle.

We don't need to resort to overloading subtend. I think we should actively avoid this, as the two links above have done. Ζετα ζ (talk) 00:44, 3 July 2013 (UTC)

Subtend doesn't originate with Classical Latin though, but with Ancient Greek ὑποτείνω (hupoteínō, as now found in the English word hypotenuse, which came from the Greek for "subtending side [of the right angle]"). It's not entirely clear to me what the original range of meanings of ὑποτείνω was in Greek. –jacobolus (t) 20:36, 6 January 2025 (UTC)

Jargon

Okay, I think it means...

An angle opposite a curve, used to create a triangle of sorts, with the curve in the place where a straight edge would be in an actual triangle...

It's a question. Is this what that means? ~ R.T.G 11:11, 18 September 2019 (UTC)

The angle opposite the curved edge, in a triangle with two straight edges and one curved edge...? (EDIT:also asked to reference desk just now) ~ R.T.G 11:12, 18 September 2019 (UTC)

No encyclopedic coverage

This article seems to run afoul of the policy that Wikipedia is not a dictionary, in that it simply defines what a subtended angle is and provides no encyclopedic coverage. This page seems to do little more than a dictionary definition could. However, it doesn't seem that much can be done about this, given that there isn't exactly a rich history of subtended angles or some other information that could give the article encyclopedic coverage. I originally considered nominating the article for deletion but wanted to check on the talk page for what other people seem to think. Rappatic (talk) 00:16, 29 July 2021 (UTC)

Possible examples:

• a parsec is defined as the distance the at which the mean radius of the earth's orbit subtends an angle of 1 arcsecond.
• the apparent size of an object is determined by the angle it subtends at the eye.
• total and annular solar eclipses occur as they do because the moon and sun coincidentally subtend almost exactly the same angle at the earth's surface.
• Subtended Angle Theorem: at any point on the circumference, a circle's diameter subtends an angle of 90°.
• An arc of a circle subtends double the angle at the circle's centre that it does anywhere on the remaining part of the circle.

—I think there's scope for a genuinely encyclopaedic entry. Musiconeologist (talk) 19:27, 9 September 2021 (UTC)

Text in image

An angle is subtended at a point, not subtended from, so the image is wrongly worded. Musiconeologist (talk) 17:54, 9 September 2021 (UTC)

Subtended angle intro

Moved from User talk:Jacobolus

Hi,

Obviously it's disappointing to have clarified something for the worse, but I'm happy to go with your reverting of it.

What I'm wondering, though, is whether there's a way to introduce readers in a really clear way to the central idea first, rather than (as it seems to me) hit them straight away with several definitions in succession, all applied to different situations, and a large number of wikilinks? That's the problem I was trying to solve, really, and it's clear from the talk page that at least one visitor had trouble understanding the article when they visited in 2019, though I've not checked what it looked like back then. Musiconeologist (talk) 21:13, 5 January 2025 (UTC)

Hi @Musiconeologist, sorry, I meant to start a talk page discussion after reverting your change but had to go cook breakfast. I agree that my version isn't amazing, and we can probably make an improvement. But I found "the angle which an object subtends at a point $P$ is the angle between two straight lines joining its endpoints to $P$." to be quite confusing compared to just saying that the a side of a triangle subtends the opposite angle, and I think we should try to avoid bullet lists if we can communicate the same thing with ordinary paragraphs. I'm not sure what the ideal content should be for this article: in my opinion it's most useful purpose is to be a link target for subtend to give something akin to a dictionary definition, but with more elaboration and hopefully more accessible than just dumping a reader to wiktionary. Maybe it would be helpful to just move the article to subtend. There are some difficulties/inconsistencies with a lot of the geometry/trigonometry terminology in English due to changing mathematical methods and style over time, so that words which originally had a clear meaning and place in Euclid-style geometry now often fit awkwardly or have been (subtly or unsubtly) redefined to fit different concepts and context.

For example, as I understand it (which might be substantially wrong; I'm not an expert), in Greek geometry a polygon meant the surface content of the shape (while today it often refers to the boundary), and the Greek precursor of the word angle originally referred to something like the vertex of a polygon, but by implication meant the "filled" portion from the direction of that vertex. Originally line segments were conflated with their lengths and shapes were conflated with their areas, so that they could be compared by magnitude. There was originally no explicit concept of angle measure per se, but angles could be compared as larger or smaller, were called "equal" when they might be superimposed, and angles could be doubled or bisected, etc. Later, trigonometry developed, with a conceptual relation between circular arcs and chords, with circular arcs used as a way of measuring angles, but still not quite the modern concept of "angle measure". Chords, and later sines, cosines, and tangents were considered to be specific line segments rather than numbers, but it was well understood that these could be looked up in a table for a reference circle and then scaled as appropriate to fit a circle of arbitrary size. Later an alternative form of trigonometry took quantities such as sines or tangents to be ratios of line segments. Finally a third form of trigonometry took such quantities to be numbers, and more generally geometry tended to be redefined in terms of arithmetic in a coordinate system. Nowadays the word angle most often means a numerical angle measure (which can be signed or unsigned), but also sometimes means a configuration of rays or line segments, or a wedge-shaped portion of the plane extending infinitely from a point, ...

So the word subtend has picked up differences in meaning through those changes in context. When circular arcs were used as a proxy for angle measurement, it makes sense to say that a chord subtends an arc, and that is more or less the same idea as saying that the chord subtends the angle, since the arc can be used to measure the angle. I'm not sure if there's any great secondary source analyzing the meaning of subtend in different times or context, but we can somewhat figure out the meanings by looking up primary sources which use the word in various ways.

I think it's useful here to e.g. describe that the word can sometimes be used in the opposite sense – like "angle A subtends side BC" – because readers may encounter the word used in such a way and come looking for what that means. –jacobolus (t) 21:54, 5 January 2025 (UTC)

@Jacobolus I definitely agree that the triangle explanation is simpler than my attempt at a more general one. I didn't like the fact that I found myself needing to label $P$ instead of using plain English. I wonder if all that's actually needed is something like

[Triangle explanation]. Similarly, [chord explanation] . . .

and so on—i.e. small links and tweaks to make it obvious that all three are essentially versions of the same thing and that once someone has understood the triangle definition, the others are just extensions of it. I think that might also do the job I was trying to do with the bullet points: they were to help the reader more easily focus on each situation in turn, rather than feel they must understand everything at once.

The fact that originally arcs effectively "were" angles hadn't occurred to me as a connection here (despite knowing that in Euclid a triangle's angles add up to "half a circle"). I think it should be mentioned, since it does make the idea of "subtending an arc" far more intuitive. I think people understand and remember by seeing connections, and that piece of information gives them the connection. Musiconeologist (talk) 00:36, 6 January 2025 (UTC)

I tried to tighten and clarify. Is that better? –jacobolus (t) 20:56, 6 January 2025 (UTC)

Afterthought: maybe a better diagram would go some way. Show one angle, with a a line segment, an circular arc and an arbitrary curve all subtending it (and having the same endpoints?). We can also say the line segment subtends both the arcs if we want, and everything subtends the angle. Musiconeologist (talk) 21:44, 5 January 2025 (UTC)

I agree, a better diagram would be helpful. –jacobolus (t) 21:56, 5 January 2025 (UTC)