Talk:Angle

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[edit] Angle Conventions

Should there not be a section on angle conventions ie. that clockwise rotations are generally positive within the cartesian plane, and that angles are usually measured from the x-axis? —The preceding unsigned comment was added by 195.83.126.10 (talk) 09:32, August 20, 2007 (UTC)


[edit] ERROR in the section Formal definition/using rotation

It is NOT true that there exists only one linear application sending vector u to vector v. I am not sure about the true statement needed here, something like "there exists only one positive isometry...". Can someone who knows better update that section? Thanks. Jeanot2432 18:36, 31 May 2006 (UTC)

[edit] Alternate definition

How about defining angle as the ?linear? codependance/independance of two curves, 0 being identical and 1 being orthogonal? I think the whole difficulty is defining angle in such a way as to make complex angles a logical extension of real angles, while keeping the definition extremely simple and clear. Kevin 2003.03.14

[edit] Angles in Complex Hilbert spaces

Here are some thoughts about angles in complex Hilbert spaces. I moved them from the main page because they don't qualify as encyclopedic knowledge. One could also use the absolute value of the dot product I suppose. --AxelBoldt

For complex Hilbert spaces, the formula (*) can be recycled to obtain a complex angle, but it is not entirely clear that this corresponds to a real-world notion of angle. An alternative is to use

(**) R(u·v)=cosθ ||u|| ||v||

where R denotes the real part. Definition (**) also special cases to (*) for real Hilbert spaces, so that may be a reasonable choice.

[edit] Definition?

The definition of angle on the main page seems rather vague. Perhaps a better definition would be: the fraction of the arc of a circle with a center at the origin of the angle.

That way degrees can be clearly defined as:

(s/c)×360

and radians as:

(s/c)×2π = s/r

where s = arc length, c = circumphrence, and r = radius.

[edit] Grad?

Scientific calculators often have a 3rd measure of angle for trig functions, abbreviated to "grad" , with 400 per circle -- what are they?

See gon. —Herbee 00:31, 2004 Mar 8 (UTC)
Often used in civil engineering in France, a gradian (or grad) is a unit of angle such that a right angle is 100 grad. While this may look smart, it stops people from dividing a right-angles into three equal parts... I think it was invented during the French revolution, when everybody thought it might be a good idead to count in tens rather than in twelves... A bit silly really, as 12 is dividible by 2,3,4 and 6 whereas 10 is only dividible by 2 and 5... The only reason we count in tens is because we have ten fingers, which proves the Babylonians were much more advanced than the French of 1789... Regards, -- Deimos 28 20:31, 18 March 2006 (UTC)

[edit] Dimensionless?

[quote] Note that angles are dimensionless, since they are defined as the ratio of lengths. [/quote] Let us not confuse too many things here with this statement. Pi is defined by the ratio of two lengths. Pi can then be used to define the dimension of direction on a unit circle. Certainly, units are associated with things that have dimension. Angles have units, and they require the correct unit for a specific equation (Perhaps it is helpful to think of Radians as dimension-less and all other forms of angular units as dimensions.). Also, angles are a dimension, in polar and spherical coordinates, they have the dimension of direction. Angles depict two different directions.

(William M. Connolley 07:39, 2 Sep 2004 (UTC)) Angles are dimensionless but they may have units (degrees). Use of an angular coordinate as a dimension in one application isn't the same question. Direction is not a dimension.
User: Nobody_EDN 2004.10.22

Then why do the first three dimensions, length, width, and breadth, have three different directions??? 'Different directions' is the basic property of dimension. ---- Please tell me what definition of dimension you are using where an angle is ruled out. (They have units and they can define a point in space. Just as width, and breadth can, using length as the third dimension for either.)

(William M. Connolley 20:09, 23 Oct 2004 (UTC)) You can sign your comments by using 4 tildes ~~~~ like that. Now, on: I'm using "dimension" is the sese of dimensional analysis. Since angles are length/length (or at least thats one good way to define them) they have dimensions of L/L = no dimensions.

Pi is a ratio of two lengths.

(William M. Connolley 20:09, 23 Oct 2004 (UTC)) Yes, exactly. Pi is a pure number, and an angle. Hence, angles are dimensionless. Another way of seeing it.

Angles are defined by the vertex of part of a disk, cut once or more through the center. Angles are measured in fractions of Pi, or fractions of the full circle. Angle measurements are chosen by taking a full circle and cutting it up in pieces by a set constant. That constant is 2Pi for radians, 360 for degrees, 400 for gradients (grads), 800 percent for percent grade (Such that 45 degrees is a 100% grade.), and 32 for compass points.

Perhaps defining Pi, Circle, Circumference and Radius first would help in defining the Angle. Use a demonstration of the wrapping function.

A Circle, drawn with Radius of one unit, on a number line, centered at minus one, has zero located on the outer rim of the Circle. If the number line is then wrapped around the circle, the numbers one, two, three, four, five, and six will all be wrapped around the edge before we get back to the beginning of the circle where number zero is.

The number Pi is the ratio of Circumference over the Diameter, Or C/D. The Circumference over the Radius C/R is 2Pi. So the wrapping function has at 180 degrees, or the opposite side of the circle from zero, the number Pi. 2Pi coincides back again with the number zero. (A picture, or several, at this point would help.)

An angle is the position on the wrapped number line in Radians. Radians are in fractions and multiples of Pi. Pi is C/D and therefore length over length and is though of as dimension-less. Although they still can contain a dimension of direction at times, most mathematical and engineering equations eliminate any dimension from calculations. 2Pi times Radial length equals length of circumference..... 2Pi times Radius time rpm become velocity.....

Degrees are a different way of unitizing Radians. Degrees break up a circle into 360 degrees. Therefore, 2pi = 360 degrees. Once converted to Degrees the angle then carries the dimension of Degrees and all mathematical equations they are used in must deal with that dimension/unit, when doing dimensional/unit checks.

Other ways are points, grads, and percent. They carry the same dimension/unit requirement with them.

Supplied for creative incentive to improve the definition and description of an angle.

User: Nobody_EDN 2004.09.01

Asserting an angle is dimensionless because it is the ratio of two lengths is quite invalid, it merely establishes that it does not have the dimension of length, a claim not generally made. An angle is a measure of rotation in the same way that length is a measure of displacement and as such is a dimension. Rotation is susceptible to the same differential and integral operations as displacement, it is not some mathematicl mystery hidden away with obscure dimensionless properties. The sooner this article recognises these facts the better.--Damorbel (talk) 07:17, 5 June 2008 (UTC)

[edit] Link suggestions

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[edit] Reflex angle

Now, I'm certainly no math whiz... But do these exist? I googled for a diagram of such, but there was literally 0 search results, and the number one result for text-based material was this article. --69.132.195.212 18:07, 14 July 2005 (UTC)

Ack, sorry. This is User:Thorns among our leaves... just not logged in. --69.132.195.212 18:08, 14 July 2005 (UTC)
Uhm… this is my first hit: http://mathworld.wolfram.com/ReflexAngle.html –Gustavb 20:36, 3 March 2006 (UTC)

[edit] Supplementary angles

Quote:

   * The difference between an acute angle and a right angle is termed the complement of the angle
   * The difference between an angle and two right angles is termed the supplement of the angle.

Okay, so it is defined that you can only use an acute angle to compute the complementary. Does this mean though that for supplementary angles, you can only use either an acute angle or an obtuse angle? (effectively an angle in the range 0..180)

Is it illegal to ask, for example, for the supplementary angle of -10? The same question but for 190? --195.85.158.73 16:03, 10 March 2006 (UTC)

[edit] sin(theta) = sin(theta+pi)?

Under the section "A formal definition" under the subsection "Using trigonometric functions" there is a statement that sin(theta)= ... = y/x = -y/-x = sin(theta+pi). That is not true. sin(theta) does not equal sin(theta+pi). As an example consider when theta = pi/2. sin(theta) = 1, while sin(theta + pi) = -1.

No, the statement is that sin(θ)/cos(θ)=y/x=(−y)/(−x) = sin(θ+π)/cos(θ+π). This is true because sin(θ+π)=−sin(θ) and cos(θ+π)=−cos(θ). Bo Jacoby 14:17, 10 July 2006 (UTC)

[edit] Explementary Angles

Angles that sum to a full circle are explementary. (This continues the concept that angles that sum to a quarter circle are complemetary, and those that sum to a semicircle are supplementary.)

See http://mathforum.org/library/drmath/view/63015.html

[edit] Units

What's the name for the "clock face" measure of angle (e.g. where 12 o'clock is forwards, 9 o'clock is left, etc.)? Ojw 22:54, 23 August 2006 (UTC)

[edit] People misuse the word "equal" for angles

Saying that two angles of equal measure are equal is not quite as clear and accurate as saying that they are congruent. For example, if I construct an equilateral triangle ABC and then let D be a point between B and C, I cannot say that angles DBA and DCA are equal under Leibniz's law, because for the predicate of whether they contain point B, their truth values are different.

Angles of equal measure are instead referred to as congruent. This is sufficient to describe them, because two Euclidean angles are congruent if and only if they are of equal measure.

From congruence (geometry):

In a Euclidean system, congruence is fundamental; it's the counterpart of an equals sign in numerical analysis.

In this article I have corrected a few uses of "equal" for angles, and in other articles I plan to do the same.

Benzi 17:00, 26 November 2006 (UTC)

In old books angles can be refer to as equal but in new books they can not. Zginder 20:05, 1 May 2007 (UTC)

Actually, "equals" is not a misuse. Both have a 90 degree angle. DBA and DCA above are triangles, if they were not triangles, then neither could contain a point in the first place. Consider, each makes a 90 degree angle, and obviously 90+90=180, as should be. However, DBA+DCA is not a number. It is a number, however, if you mean to say that DBA is not a set in the plane, but the angle formed; in which case, B is not in it as it is not a set. To be short, you are equivocating.Phoenix1177 07:11, 2 November 2007 (UTC)

a more meaningful explanation for the reason why 'equals' is a misuse would be by definition: an angle is defined either as a set of points of a plane limited by two rays with common endpoint including them, or just the rays, so an angle is a set regardless of the definition you choose, so the relation of equality between angles is the relation of equality between sets, and so it mustn't be confused with cogruence, as shown by benzi's example.--Ghazer (talk) 12:48, 11 February 2008 (UTC)

This is not essentially different from the common abuse of language when saying "two equal line segments" instead of "two line segments of equal measure". While somewhat sloppy, everyone understands the intended meaning.  --Lambiam 21:19, 11 February 2008 (UTC)
nonetheless, its still a mistake. wikipedia's aim is to be encyclopedic and thus accurate, and so ambiguities and plain mistakes should be avoided as much as possible - you could just as well say there's nothing wrong with replacing each 'they're' with 'their' & vice versa - most people would understand it, but would it be correct? no. common abuse is still an abuse.--Ghazer (talk) 14:27, 13 February 2008 (UTC)
There exist different definitions of what an angle is. Mathworld defines an angle as an "amount of rotation".[1] PlanetMath uses a notion of "free" angle for one in which (for example) all right angles are equal.[2] Mathematicians have no qualms in employing "common abuse of language" when it simplifies the discourse, as when they say "the function f(x)", as long as they are understood. This specific abuse is also rampant on Wikipedia, as are many other common abuses. Good luck in rooting it out. Personally I think it is more fruitful to explain that when angles are said to be equal, this usually and normally means they are equal in size.  --Lambiam 10:36, 14 February 2008 (UTC)

[edit] Old-school definition?

Just wondered if anyone has though of adding the old-school definition in? When I was in school all those years ago, we were taught to never refer to angles as things meeting at a point, but "when a line rotates about one of it's extremities in one plane, from one position to another.", in the trigonometric sense anyhoo :-) ♥♥ ΜÏΠЄSΓRΘΠ€ ♥♥ slurp me! 15:53, 28 May 2007 (UTC)

[edit] Unnecessarily complicated?

Does anyone else think that the introduction of this "k" throughout the "Units of measure for angles" section is unnecessarily complicated? In my view it's just likely to totally confuse people. Matt 20:23, 30 June 2007 (UTC).

Well, I have rejigged this section now. Matt 19:47, 18 July 2007 (UTC).

[edit] Semi Angle

Any chance of adding Semi Angle (e.g. Half the angle of the tip of a cone)- the only place I could find it was in my University Journal collection- not all web users have access to journals... —Preceding unsigned comment added by 124.181.109.145 (talk) 03:51, 16 September 2007 (UTC)

Not sure if this is very common? Probably not worth adding unless it is in standard textbooks. -- Steelpillow 06:57, 16 September 2007 (UTC)

OBLIQUE ANGLE

suggestion: ADD oblique angle definition to article.

I did not see a definition for "oblique angle" in the article. —Preceding unsigned comment added by 199.80.66.119 (talk) 19:13, 29 January 2008 (UTC)

[edit] Reflex Angles

When we say angle ABC, how do we know that it refers to the acute/obtuse angle between AB and BC? Surely, there is always a reflex angle there too? What if we wanted to refer to that instead? —Preceding unsigned comment added by 81.159.24.29 (talk) 23:18, 7 February 2008 (UTC)

[edit] Angle is not a figure

An angle is certainly not a figure! Using the term "ray" is associated with Hilbert's modification of Euclidian geometry, much can be found in this link http://www.libraryofmath.com/.

A useful definition of an angle for the non mathematitician is that an angle is "a measure of turning". Definitions based on set theory tend to have difficulty with infinities and zeros (empty sets) that tend to arise with day to day mathematics so I think it is advisable that set theory considerations should be dealt with by references for further study instead of in the article.

The statement "Angles are considered dimensionless" has no basis in any mathematics, a brief consideration of spherical coordinates http://en.wikipedia.org/wiki/Spherical_coordinates should dispel any doubts (if anyone is in doubt spherical coordinates define points in three dimensional space by two angles and a length). Justification by a reference to dimensional analysis http://en.wikipedia.org/wiki/Dimensional_analysis is completely irrelevant. Dimensional analysis is a sort of checksum for consistency in formulae so that one avoids defining length by seconds or amps or other blush causing idiocies. As yet I haven't analysed the article sufficiently well to find the consequences of this whopper, they may well be profound, if someone else sees what is wrong and want to revise I will be very pleased, but I have never rewritten a Wiki article before.--Damorbel (talk) 20:54, 30 May 2008 (UTC)

The source you mention (Library of Math) defines an angle as a subset of the Euclidean plane, namely as the union of two rays, themselves sets of points.[3]. It has this to say about its measure:
Each angle ∠ A B C is associated with a unique real number between 0 and 180, called its measure and denoted angle measure m ∠ A B C. No angle can have measure 0 nor 180.[4]
This contradicts everything you write. Angles are figures, and the measure is a real number, which is dimensionless. You'll have to come up with other sources if you want to add alternative definitions. The definitions given at LOM (which, by the way, does not fit our definition of "reliable source") are perfectly fine for Euclid-style geometry.  --Lambiam 16:20, 31 May 2008 (UTC)

You maintain that an angle is a figure and that a source is required for alternative definitions. Do you have a source for your "figure" definition? After all the word "figure" has a number of meanings.

Saying that "This contradicts everything you write." is rather general, paricularly when you cite "No angle can have measure 0 nor 180". My understanding is that this kind of statement has origins in set theory, set theory tries to constrain mathematics by logic, a constraint which is OK if you confine your definition of mathematics to what a digital computer does, unfortunately mathematics got there long before computers, the upstart should not be putting restrictions on the senior subject. --Damorbel (talk) 11:56, 5 June 2008 (UTC)

I'm not maintaining any particular notion of what an angle "is". I'm happy to maintain, though, that the certainty expressed in the emphatic statement "An angle is certainly not a figure!" is baseless. There are various ways of defining the notion of angle, which is reflected in the article.  --Lambiam 07:17, 6 June 2008 (UTC)

One moment you say "The definitions given at LOM (which, by the way, does not fit our definition of "reliable source"), the next you cite "If you want a source for the definition of angle as a figure, well, much can be found in this link: http://www.libraryofmath.com/." which is of course LOM. Is this a variable source or a source dependent on who cites it?

You claim an angle is a figure, then you maintain an angle is dimensionless. But a figure, of whatever sort, requires the dimension length for its very existence. While I agree that an angle does not have length as a dimension, it seems to me to be wildly inconsistent to define it with a representation wholely dependent on length.

The current article even says "the angle is the "amount of rotation" that separates the two rays". "Separates" is imprecise, it can mean displaced as well as inclined. And the use of "two rays" is inconsistent when used to define a dimension that has no length. Curves can be rotated through an angle, it's happening to solids all the time! But for rotation there is no requirement for any representation of length. Similarly definitions based on a "vertex" are equally inappropriate.

To quote "I'm not maintaining any particular notion of what an angle "is" ". I do think an encyclopedia should give the clearest notion (let us say idea) of the subject of the article's title. In the case of an angle it is important that it is free of all connections with length is very important, witness polar coordinates http://en.wikipedia.org/wiki/Polar_coordinate_system and spherical coordinates http://en.wikipedia.org/wiki/Spherical_coordinate_system--79.79.30.151 (talk) 20:43, 6 June 2008 (UTC)

  1. I was responding to Damorbel – I don't know if you are the same editor – and I just gave the same source this editor mentioned in support of the statement that an angle is not a figure. For a reliable source, see the reference given in the article itself: Sidorov, L.A. (2001), “Angle”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 .
  2. I didn't write that an angle is dimensionless, but that the measure of an angle is dimensionless. However, as the article explains, the term "angle" is used interchangeably for both the geometric configuration itself and for its angular magnitude (which is simply a numerical quantity).
  3. If several non-equivalent definitions of some concept are current, then, in my opinion, we should present all, and not just the "clearest".
  4. I don't understand why you think that a figure, of whatever sort, requires the dimension length for its very existence. A single point is also a (very simple) figure, as is a line. Neither has a length. I also don't understand why you ascribe a particular significance to the use of angles in spherical coordinate systems, and why that should be incompatible with the present article.
 --Lambiam 02:48, 7 June 2008 (UTC)