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:{{merge done}} —[[User_talk:Quondum|Quondum]] 01:41, 23 February 2023 (UTC)
::Thanks for the merger done sometime ago. I've recently sourced a definition for [[angular displacement]] from an authoritative source ([[ISO 80000-3]]) which states explicitly it may be a signed quantity. Therefore, I've retargeted the redirect [[Angle of rotation]]. [[User:Fgnievinski|fgnievinski]] ([[User talk:Fgnievinski|talk]]) 08:52, 22 July 2023 (UTC)
== Math ==
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:::I'm not saying the concept of unsigned and unoriented "angle" doesn't have a place, but I looked and there aren't any obvious references that argue against the signed definition. So the relationship I infer from the sources is signed as canonical and unsigned as a simplification, rather than the unsigned being canonical. I guess we could call the small angles "reference angles" (0-90 degrees) or "proper angles" (0-180).
:::I'm not sure what your point is with spherical trigonometry, the angles are still rotations and the sets of rotations around a point forms a circle group. The fact that the rotations form SO(3) and can be labelled with dihedral angles or 3D bivectors is interesting but not really relevant to defining angles.
:::As far as writing for a non-technical audience, [[WP:TECHNICAL]] is of course the guide. But notably there is [[WP:OVERSIMPLIFY]]: "Encyclopedia articles should not 'tell lies to children'". So dumbing down the definition of angle is not the right approach. I would say the lead paragraph should give a good definition in terms of the circle group and define general concepts, and then a second paragraph can then go on to describe common examples. Lie groups are relevant to defining angles in higher dimensions, but the basic notion of angle is defined in a plane, so the Lie groups can go in a section. [[User:Mathnerd314159|Mathnerd314159]] ([[User talk:Mathnerd314159|talk]]) 18:33, 5 May 2023 (UTC)
::::Element of the circle group is too reductive (as you say, we should not [[WP:OVERSIMPLIFY]]). First, [[circle group]] explicitly describes such elements as complex numbers (not just as elements of an abstract group defined structurally with multiple possible representations), when the most popular representation of angles is as angle measures in a periodic interval. Second, this only covers angles in a particular plane, without allowing the possibility for angles to include orientation. Third, this does not allow for contexts where angle measure per se does not make sense, for example in referring to an angle in an affine space, or an angle in a space where the coordinates are elements of a finite field. Fourth, this does not allow for angles that can spin around many times, as seen in e.g. [[angular velocity]]. While we're at it it would be good (not necessarily in the lead but somewhere in the article) to describe the difference between absolute orientations vs. relative "angles" (quotients of orientations or if you like rotors transforming one rotation into another), cf. Baez (2009) [https://math.ucr.edu/home/baez/torsors.html "Torsors Made Easy"]. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 18:43, 5 May 2023 (UTC)
:::: Where I will agree with you though is that I think we should refocus the lead definition here away from talking about a reference to particular parts of a geometric figure (though this sense of "angle" should be described early in the article) and toward a general concept of relative orientation. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 18:59, 5 May 2023 (UTC)
:::::Yeah, the lead is the place to start. Maybe something like this:
::::::An angle is formed when an object is rotated around a point.<ref>{{cite book |last1=Young |first1=Cynthia Y. |title=Trigonometry |date=6 September 2017 |publisher=John Wiley & Sons |isbn=978-1-119-44520-3 |url=https://www.google.com/books/edition/Trigonometry/476ZDwAAQBAJ?hl=en&gbpv=1&dq=angle%20rotation&pg=PA4&printsec=frontcover |language=en}}</ref> In Euclidean geometry, a ray is rotated around its initial point, called the vertex of the angle, and the starting and ending rays are called the sides of the angle. The measure of an angle denotes its magnitude and is a real number with some angular unit such as degrees or radians. The measure is in general any real number, but it may be bounded to [0,180°) or [0,360°) depending on context. Angles are often drawn as circular arcs, and the measure of a counterclockwise angle in radians is the ratio of the length of the circular arc to its radius.
:::::[[User:Mathnerd314159|Mathnerd314159]] ([[User talk:Mathnerd314159|talk]]) 20:28, 5 May 2023 (UTC)
:::::: I would instead make the fundamental definition something like "An angle is the relationship between two orientations or directed lines." But maybe we can find some sources (historical surveys, mathematical dictionaries, ...) with a clear definition. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 22:04, 5 May 2023 (UTC)
:::::::Unless you specify it as a rotation, you don't get angles beyond 360 degrees. Hence "relationship" is just too vague. [[User:Mathnerd314159|Mathnerd314159]] ([[User talk:Mathnerd314159|talk]]) 14:43, 7 May 2023 (UTC)
{{reflist talk}}
== "[[:Angles]]" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]] ==
[[File:Information.svg|30px]]
The redirect <span class="plainlinks">[//en.wikipedia.org/w/index.php?title=Angles&redirect=no Angles]</span> has been listed at [[Wikipedia:Redirects for discussion|redirects for discussion]] to determine whether its use and function meets the [[Wikipedia:Redirect|redirect guidelines]]. Readers of this page are welcome to comment on this redirect at '''{{slink|Wikipedia:Redirects for discussion/Log/2023 May 23#Angles}}''' until a consensus is reached. <!-- from Template:RFDNote --> —[[User:Lights and freedom|Lights and freedom]] ([[User talk:Lights and freedom|''talk'']] ~ [[Special:Contributions/Lights and freedom|'''contribs''']]) 00:50, 23 May 2023 (UTC)
== Euclidean Geometry only? ==
The lead sentence specifies euclidean geometry, but angles are also a thing in spherical geometry and hyperbolic geometry as well. Is there a common definition to all three that we can use in the lead sentence, and then briefly define the specifics for each geometry? It's a disservice to specify euclidean and then not discuss the other geometries as well, as doing so implies that only euclidean geometry has angles-- which is not true. [[User:Fieari|Fieari]] ([[User talk:Fieari|talk]]) 07:05, 23 May 2023 (UTC)
:I went ahead and was [[WP:BOLD]] and did it. I welcome anyone to review my additions... I believe everything I said is accurate and lead-worthy, but I'd love a 2nd eye on it. [[User:Fieari|Fieari]] ([[User talk:Fieari|talk]]) 07:21, 25 May 2023 (UTC)
::Your addition would make sense in a specific section named for example "In other geomeries". It is definitively too [[WP:TECHNICAL]] for the lead: most people interested in triangles do not know anything on non-Euclidean geometries. So, I'll revert your edit. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 10:14, 25 May 2023 (UTC)
== what about grad angle system? And forgot about circular angle? ==
I think we should add the following topics [[User:Yuthfghds|Yuthfghds]] ([[User talk:Yuthfghds|talk]]) 12:05, 4 June 2023 (UTC)
:What do you mean by "circular angle"? Grads are discussed under {{slink|Angle#Units}}. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 20:35, 4 June 2023 (UTC)
::[[User:Jacobolus|@Jacobolus]] Sorry my fault I didn't noticed the grad angle system.I am talking about degree measure in which 1°= 60' and 1'= 60''. Here we are only talking about linear angles not about circular degree measure.Thanks for pointing out my mistake, I am a new editor please avoid my faults.I still need feedback from your kind of editors.
::Thank you
::Yuthfghds [[User:Yuthfghds|Yuthfghds]] ([[User talk:Yuthfghds|talk]]) 05:49, 5 June 2023 (UTC)
:::I still do not understand what you are trying to say. What is the difference between "linear angles" vs. "circular degree measure"? This article does discuss degrees, minutes, and seconds. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 06:03, 5 June 2023 (UTC)
::::[[User:Jacobolus|@Jacobolus]]
::::Linear Angles:
::::Linear angles are measured in degrees or radians and represent the amount of rotation or deviation between two lines or line segments.
::::They are used to measure angles in two-dimensional space, such as in geometry or trigonometry.
::::Linear angles have a fixed starting point (usually a reference line) and are measured in a straight line.
::::Linear angles range from 0° to 180° (or 0 to π radians) in a half-circle and extend up to 360° (or 2π radians) in a full circle.
::::Circular Angles:
::::Circular angles are measured in terms of arc length or fraction of a circle and are used to describe rotational positions or displacements.
::::They are commonly used in applications that involve circular or periodic motion, such as physics, engineering, or navigation.
::::Circular angles are based on the concept of a full circle, where one complete revolution is equal to 360 degrees or 2π radians.
::::Circular angles can exceed 360° or 2π radians, representing multiple revolutions or fractional parts of a revolution. [[User:Yuthfghds|Yuthfghds]] ([[User talk:Yuthfghds|talk]]) 06:17, 5 June 2023 (UTC)
:::::"Linear angles" is not a term used in technical literature or curriculum that I have seen. Most of the results in a search for "linear angle" seem to be using it as a synonym for "straight angle" (i.e. a half turn, 180°). Where did you get these names? Do you have a source? I would call this instead something like "angular distance" or "angular separation" (or just "angle"). I agree that this article should do a better job about talking about different concepts of angle, and I have started gathering sources for hopefully an eventual rewrite/rework. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 06:45, 5 June 2023 (UTC)
::::::[[User:Jacobolus|@Jacobolus]] Sir, here are some references for linear and circular angles:- help you.
::::::Linear Angle:
::::::Elementary Geometry for College Students by Daniel C. Alexander and Geralyn M. Koeberlein.
::::::Chapter 1: Basic Geometry Concepts.
::::::Chapter 2: Reasoning and Proof.
::::::Geometry: A Comprehensive Course by Dan Pedoe.
::::::Chapter 2: Angles.
::::::Circular Angle:
::::::Trigonometry by James Stewart, Lothar Redlin, and Saleem Watson.
::::::Chapter 1: Trigonometric Functions.
::::::Trigonometry by I.M. Gelfand and Mark Saul.
::::::Chapter 1: Angles.
::::::Elementary Trigonometry by H. S. Hall and S. R. Knight.
::::::Chapter 1: Trigonometrical Ratios of Acute Angles.
::::::sir if you still have some queries please question me.Thanks for feedback.
::::::Yuthfghds [[User:Yuthfghds|Yuthfghds]] ([[User talk:Yuthfghds|talk]]) 07:25, 5 June 2023 (UTC)
{{dedent|6}} Unless I am doing something wrong with my text searches at the internet archive, the term "linear angle" does not appear in either of the books you listed by [https://archive.org/details/elementarygeomet0000alex_o5k5_4ed/page/6/mode/2up?q=%22linear+angle%22 Alexander] or [https://archive.org/details/geometrycomprehe0000pedo/page/n5/mode/2up?view=theater&q=%22linear+angle%22 Pedoe]. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 11:13, 5 June 2023 (UTC)
:[[User:Jacobolus|@Jacobolus]] Please recheck,if you still don't find it I can't do anything.Because I don't have those books that I mentioned you.Because I red those books few years ago.And I think linear angle is informal term mentioned by the author.And I argued here so it's my fault but still we can add these topics in this article.If I tell you the definition of linear angle and circular angle it would be like this:-
:linear angle- Angle measured between two rays in degrees.
:circular angle- Angle measured between two rays also in degrees but including minutes and hours. Technically angles that are placed inside a unit circle with 3 divisions namely degrees, minutes and hours.
:I am glad that you understand my perspective and debate over this topic. [[User:Yuthfghds|Yuthfghds]] ([[User talk:Yuthfghds|talk]]) 13:03, 5 June 2023 (UTC)
::The angle between two rays does not depend on whether or not a circle is drawn around their vertex! — When you say "including minutes and hours", are you referring to [[right ascension]] (part of the coordinates of a star in the sky) which is at least sometimes given in hours? — The subdivisions of a degree are called ''minutes'' and ''seconds''. As far as I know, hours and degrees are never used together, partly because of the confusion between two kinds of "minutes". —[[User:Tamfang|Tamfang]] ([[User talk:Tamfang|talk]]) 21:01, 7 July 2023 (UTC)
:::@[[User:Tamfang|Tamfang]] You are absolutely right.These are just "informal" terms to distinguish between trigonometric angles and geometric angles. [[User:Yuthfghds|Yuthfghds]] ([[User talk:Yuthfghds|talk]]) 05:25, 9 July 2023 (UTC)
::::Why do they need distinguishing? Is there ever a circumstance in which mistaking a trigonometric angle for a geometric angle (or vice versa) or a linear angle for a circular angle (or vice versa) (are these the same distinction?) would lead to wrong results? —[[User:Tamfang|Tamfang]] ([[User talk:Tamfang|talk]]) 20:08, 5 August 2023 (UTC)
:As I mentioned above linear angle is an informal term referred to angles that are measured in only degrees like in protector when we measure angle we say 12.5° or 45.2° just because we can't expand it but in circular angles we expand this system.linear angles are helpful in geometry but circular angles are helpful in trigo [[User:Yuthfghds|Yuthfghds]] ([[User talk:Yuthfghds|talk]]) 13:12, 5 June 2023 (UTC)
:well the references that I mentioned you are generated by Chatgpt.But I red the book by padeo don't know linear angle is in that book or not but I red somewhere about linear angle in a random book but when Ai generated that book I thought it cover that topic sorry for the inconvenience that you suffered but still together we can make this article near excellence. [[User:Yuthfghds|Yuthfghds]] ([[User talk:Yuthfghds|talk]]) 14:14, 5 June 2023 (UTC)
::Please don't ever ask ChatGPT for help writing Wikipedia, and be careful using it for any other purpose. It will confidently make up total nonsense. ChatGPT is a statistical language model, not a reference librarian. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 15:36, 5 June 2023 (UTC)
:::[[User:Jacobolus|@Jacobolus]] You are absolutely right ,my bad. [[User:Yuthfghds|Yuthfghds]] ([[User talk:Yuthfghds|talk]]) 15:47, 5 June 2023 (UTC)
== Negative angle definition ==
The following was removed: "Although the definition of the measurement of an angle does not support the concept of a negative angle," as unhelpful. Of course a definition can be contrived. Essentially one refers to area of a circular sector, using [[signed area]] according to the area being above or below the horizontal line of symmetry. This definition is for angles fixed in a standard position. Reference to [[area]] for definition is the historic basis of hyperbolic angle, so angle reference to area is one of the [[unifying theories in mathematics]]. [[User:Rgdboer|Rgdboer]] ([[User talk:Rgdboer|talk]]) 23:03, 9 November 2023 (UTC)
:This article needs to be clarified and expanded to more clearly describe and discuss the several different concepts of "angle" which are in common use. This is something I plan to do eventually; I've been occasionally gathering sources at [[User:Jacobolus/Angle]] but am not yet ready to write a solid survey. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 00:45, 10 November 2023 (UTC)
Thank you for those sources on directed angle. The issue of angle definition came to a head in 1893 in Chicago when Felix Klein shot down Alexander Macfarlane’s paper on the topic which was presented to the mathematical congress held in connection with the [[Columbia Exposition]]. The paper was included in ''Papers in Space Analysis'' (1894) as the ''Proceedings'' of the congress only noted the title. The notion of an area-based definition referring to the sector of a circle was included in ''The Elements of Plane Trigonometry'' (1892) by R. Levitt & C. Davison (page 158). [[Robert Baldwin Hayward]] noted the complete analogy between the circular and hyperbolic cases when area of sectors are used to define the circular and hyperbolic angles. — [[User:Rgdboer|Rgdboer]] ([[User talk:Rgdboer|talk]]) 01:19, 12 November 2023 (UTC)
:Of course, both circular and hyperbolic angle measure can be defined either via area or via arc length, with the latter using a Lorentzian (pseudo-Euclidean) concept of distance in the plane. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 02:25, 12 November 2023 (UTC)
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