Talk:Hyperbolic angle

Image

The current image hyperboolhoek.svg is inappropriate for the article text which requires rays to (1,1) and (x,1/x) from the origin. I have placed a request for a similar diagram on hyperbolic sector. A good graphic will help show the unbounded nature of the hyperbolic angle in the first quadrant, an important distinction from the circular case. The current graphic is not so much wrong as it is inappropriate for the development of the concept of hyperbolic angle from fundamentals such as reference to the harmonic series.Rgdboer 23:34, 3 August 2007 (UTC)

Thanks to User:Rocchini the is now an appropriate image for this article and hyperbolic sector.Today I changed images to the more appropriate.Rgdboer 22:19, 13 September 2007 (UTC)

Kennelly

Today I included a 1912 reference to Kennelly's book on application of hyperbolic functions in electrical engineering. Chapter I of this book is "Angles in circular and hyperbolic trigonometry". The approach includes one based on area of a hyperbolic sector. Kennelly's text is flawed by a wrong equation for the hyperbola: yy - xx = 1 instead of xx - yy = 1, a flaw corrected on the page by a reader of the U Cal text digitized. On page 22 he comes to the point of saying that "Θ will be, in hyperbolic radians, double the sector area AOE in sq cm" on his definitive figure. Doubling the area is not necessary when one uses the hyperbola y = 1/x as in the article. On page 25 Kennelly defines hyperbolic sine by XE and hyperbolic cosine by 0X in his figure, similar to statements in hyperbolic triangle. An attentive student will see that XE and 0X give sinh and cosh for half the angle represented in Kennelly's figures. Thus we justly use in WP the hyperbola xy = 1 to establish hyperbolic angle and functions. The faults in Kennelly's text have been weighed against the significance of this reference in history and as a demonstration of the method of acquiring the ideas.

Rgdboer (talk) 23:34, 17 April 2009 (UTC)

Comparison with circular angle

These are the issues that call for attention in this section. Consider each paragraph: First, there is an incorrect understanding of hyperbolic orthogonality: it doesn’t mean opposite slopes as stated. Further, linking to pseudoeuclidean space or Minkowski space does not improve the explanation of hyperbolic angle.

The second paragraph has the obscure phrase "moving steadily in an orthogonal direction to a ray". The idea may be better presented in terms of a locus of equidistant points. The idea of hyperbolic angle is not advanced.

In the third paragraph of this section an irrelevant property of circular arc addition is advanced. It is an interesting property of a Lie trapezoid, the parallel sides being chords of the circle. This paragraph prepares for the analogy on the hyperbola in the next:

After stating the Lie trapezoid property for the hyperbola, one reads "it makes sense to define the hyperbolic angle …" The introduction of these Lie trapezoids detracts from the clear presentation of the hyperbolic angle. The mention of the logarithm here ignores the historic precedence of the hyperbolic angle in the quadrature of the hyperbola.

Paragraph five says "it can be shown with the above definition", instead of "as the definition says" in the discussion of area swept out by a radius. The final points of paragraph five are valid: in this configuration circular area is 2pi , this xy = 1 hyperbola is the one where we get the equality of area and angle, and the hyperbolic angle magnitude always exceeds the circular one for radial line.

In summary, paragraph 1 is erroneous, paragraph 2 is unhelpful, paragraphs 3 & 4 are distracting though there may be material for contribution elsewhere. A few phrases in paragraph 5 have merit.

Rgdboer (talk) 23:21, 10 August 2009 (UTC)

Dear Rgdboer,

Firstly, before I start, have you read the referenced link that the section is inspired by? (Bjørn Felsager, Through the Looking Glass - A glimpse of Euclid’s twin geometry, the Minkowski geometry, ICME-10 Copenhagen 2004, p.14; see also example sheets [1] [2] exploring Minkowskian parallels of some standard Euclidean results)

I accept I might perhaps have over-compressed the treatment here, but the original gives a really outstanding presentation of how pseudo-Euclidean geometry mirrors Euclidean geometry, and how hyperbolic angles fall out of that really naturally, as the natural counterparts to traditional circular angles.

So, on to the points you have raised.

• Firstly, what am I trying to do here? The key is in the first five words: "Hyperbolic angles can be motivated..." The lead paragraph gives a bald definition of a hyperbolic angle, but just by itself it can seem incredibly arbitrary. Why this definition? Why the hyperbola? Why is it so like an angle that that is what we call it? On its own this bald introduction fails to convey why circular angles and hyperbolic angles are such exact mirror image analogues of each other, and so gives no clue of (eg) the directness of the transition from seeing how complex numbers and quaternions characterise Euclidean rotations in terms of Euclidean angles, to seeing how entirely analogous it is for split-complex numbers and mixed Clifford algebras to characterise Lorentz boosts in terms of hyperbolic angles; a correspondence I know has been a particular area of interest of yours.

A proper understanding of "hyperbolic angle" should include an understanding of the closeness of the parallels with a Euclidean angle, and how the motivation for one from the other follows directly from flipping the switch as to what should constitute orthogonality.

Now, on to 'hyperbolic orthogonality', and you write: "it doesn’t mean opposite slopes as stated". Well, this depends what axes one has chosen. If one takes a space-like axis ε₁ and an orthogonal time-like axis ε₂, then two directions are hyperbolic-orthogonal if their slopes are mirror images in one of the 45° lines. That is the (only) convention currently presented in the hyperbolic orthogonality article. But those are not the axes that have been used in the picture at the top of this article. Instead, in this article axes have been chosen parallel to the light-cone directions, ie parallel to ε₁ ± ε₂. With this choice of axes it is true that hyperbolic orthogonality means opposite slopes, as stated.

• Next, second paragraph, and you write: "the obscure phrase 'moving steadily in an orthogonal direction to a ray'. The idea may be better presented in terms of a locus of equidistant points". I prefer what I wrote for two reasons. Firstly, because it shows the hyperbola growing because of a local property at each point along it. Secondly, because this local property is directly the {Euclidean/hyperbolic} orthogonality which has just been being discussed, which I am presenting as the essential switch which alternately leads to the idea of Euclidean or hyperbolic angle.

"The idea of hyperbolic angle is not advanced". It most certainly is: because it is this section which has introduced the idea of where specifically the hyperbola has come from: why it is the natural counterpart to the circle when one moves from Euclidean to pseudo-Euclidean notions of orthogonality.

• So, on to the Lie trapezoids. I didn't know that is what they were called, so I'm grateful to you for recognising and naming the beasts. I also see that Wikipedia doesn't have an article on them, either in the original Euclidean setting, or the hyperbolic analogue. This is an article that I think could usefully be created. And if it were, I think that would provide just exactly the necessary clarification for what I sensed when I wrote it, was perhaps the one slightly underdeveloped point in the exposition.

I'm very much following Felsager in introducing it, but I think the Lie trapezoids are useful because they arguably give a more primitive (in the sense of relying on less other material) characterisation of what it means to add two angles together and obtain a third. The Lie trapezoids rely only on being able to draw parallel straight lines; they don't require any notion of area, or any machinery (e.g. calculus) to calculate areas. Therefore in my view they represent a more fundamental characterisation of the properties of a Euclidean angle, which when translated over to the hyperbola gives a useful and intuitive rationale for the addition properties of hyperbolic angles.

Giving both this more geometrical construction, in addition to the area-based definition, IMO significantly helps show that the area-based definition has not been posited, deus-ex-machina, out of nowhere; but rather is the result of a deep geometrical parallelism between the Euclidean and the pseudo-Euclidean concepts.

• And thus, in paragraph 5, we can state that this definition built up in an intuitive way actually matches the much more rabbit-from-a-hat definition in the lead; and the reader's understanding of how everything connects together is improved.

So in summary, paragraph 1 is not erroneous; paragraph 2 is fundamental; paragraphs 3 and 4 give a more low-level characterisation and understanding of what it is to add angles, which underlines in a very "geometric" way the directly comparable nature of Euclidean and pseudo-Euclidean angles, and paragraph 5 brings everything together, and re-introduces the more technically difficult idea of area, setting up the way for the more historically-focused discussion which follows.

Let me finally come back to one more comment you made, "The mention of the logarithm here ignores the historic precedence of the hyperbolic angle in the quadrature of the hyperbola". This seems to be a complaint along the lines of "That isn't how it was originally done". Maybe not; but it is my view that we should be trying to provide the most intuitive and foundational introduction to a subject that we can as of 2009, not 1909 or 1709. Sometimes a historical development is the best way to ease a reader to that point; but often it is not. It is interesting that the great 1911 Britannica also took this as a policy view in its science and mathematics articles: present the best current understanding of the material in the main part of the article first, and then at the end give a sketch of the historical development. Jheald (talk) 11:32, 11 August 2009 (UTC)

Thank you Jheald for your quick and detailed reply. Especially important is the point from paragraph one: the hyperbola xy = 1 is not the one presently presumed for the definition of hyperbolic-orthogonality. You may have noticed that I have put a note on Talk:Hyperbolic-orthogonality indicating that editing there is necessary. Recently I added some material relating to the reflection in an asymptote that produces the conjugate hyperbola, conjugate diameters, and for us, hyperbolic-orthogonal lines. This approach serves well to address the different interpretation with respect to slopes of lines. I had come to the conclusion that, indeed, your mention of hyperbolic-orthogonality is valid given the algebraic hyperbola as reference.

I have re-read Felsager and thank you and him for an excellent exposition; now that I see his context with xy = 1 it reads more clearly. With that block removed your contribution seems much better. Motivating and applying a novel concept is an important function of the teacher and encyclopedia contributor. The use of parallel chords was striking at first; to prove it true in the circle I resorted to the usual parametrization and wrote down the slopes of the lines. With some trigonometric identities I got convinced. Those parameters made me write Lie trapezoid for short. Do you have any other reference besides Felsager on this property? It has an attractive simplicity. I retract my objections to the paragraphs on comparison that you have contributed and defended.Rgdboer (talk) 21:58, 11 August 2009 (UTC)

Geometry name

Since Hermann Minkowski wrote Geometrie der Zahlen he is credited with the Geometry of numbers. It is inappropriate to redirect Minkowski geometry to Minkowski space because this relativity article does not pertain to the usual understanding of Minkowski's work in geometry. The change today to Non-Euclidean geometry provides an accurate alternative.Rgdboer (talk) 01:24, 14 November 2009 (UTC)

Diagrams

Here are two at a request (this article lacks the actual depiction of hyperbolic angle).

The hyperbolic angle u in a hyperbolic sector, within a hyperbolic triangle.

The hyperbolic angle u in a hyperbolic sector.

Which would be best for the article? Maschen (talk) 07:44, 5 December 2012 (UTC)

Thank you for the two further diagrams; the vertical orientation conforms to diagrams of the future in special relativity. For the moment, the diagram from Hyperbolic triangle has been used to replace to old sector. That diagram has the "standard position" property for setting the Hyperbolic triangle on an angle in standard position. The issue of "standard position" is more sensitive in the hyperbolic case than the circular because the symmetry is different. When considering the absolutely naive reader, landing on this page might be a mountain, so mercy should be our watchword. The business of the root 2 is one issue. Perhaps a diagram showing a circle tangent to the hyperbola x y = 1, center at the origin. Then a ray of slope less than one from the origin marks off a circular angle on the circle and a hyperbolic angle on the hyperbola. Projections from the intersections on to x = y then determine right triangles with legs √2 cos, √2 sin, √2cosh, and √2 sinh. Such a diagram would link the familiar sine and cosine projection lengths (modulo √2) to the hyperbolic functions. After all, the introduction to hyperbolic angle is frequently from the values it gives with hyperbolic functions. The other alternative approaches involve Saint Vincent's theorem or the projective approach with parallel secants of a conic.Rgdboer (talk) 20:53, 5 December 2012 (UTC)

In the process... Maschen (talk) 22:55, 5 December 2012 (UTC)

Done ↓ - although it's cluttered and unclear... Is this what you meant? I may have misinterpreted again so apologies... Maschen (talk) 23:59, 5 December 2012 (UTC)

Hyperbolic and circular angle construction from the rectangular hyperbola xy = 1 and unit circle x² + y² = 1.

Very strong illustration. The only change necessary regards the circle which is xx + yy = 2 (not 1). The image has been posted in the Comparison with circular angle section. Everything is in place except the circle is labelled with xx + yy = 1. If the circular sector was colored in contrast, then the point about u being area would show. Thank you very much for your efforts to bring this material into common knowledge.Rgdboer (talk) 23:18, 6 December 2012 (UTC)

You're very welcome! It must be very tiresome for you to keep reminding me of that obvious √2 radius (instead of 1, which I am accustomed to...), apologies for that. ^_^ Maschen (talk) 00:03, 7 December 2012 (UTC)

Not tiresome at all. Very good that you now show the circular sector too by contrast.Rgdboer (talk) 01:13, 7 December 2012 (UTC)

question about the diagram of circular and hyperbolic angle

There might be a mistake in the diagram of circular and hyperbolic angle: File:Circular and hyperbolic angle.svg.

there might be a mistake in the diagram

There are two angles in the diagram. One is the hyperbolic angle related to the hyperbola xy=1, and let's call this hyperbolic angle u_hyp. The other is the circular angle related to the circle xx+yy=2, and let's call this circular angle u_cir. Right now in this diagram, these two angles are shown to be indentical, i.e. u_hyp=u_cir=u. I think such indentity should not exit in general, i.e. u_hyp is usually not equal to u_cir. By equations, u_hyp=ArcTanh[Tan[u_cir]]>=u_cir, and u_cir=ArcTan[Tanh[u_hyp]]<=u_hyp, where the equals sign holds only when u_hyp=u_cir=0. In other words, u_hyp is equal to the area of the yellow and red regions, while u_cir is equal to the area of yellow region only. Armeria wiki (talk) 03:19, 9 June 2013 (UTC)

I'll check over this, but the sin, cos trace out the circle and sinh and cosh trace out a hyperbola. All the diagram shows is a different orientation of the curves. Do you have a source to support what your'e saying? M∧Ŝc²ħεИ_τlk 06:53, 9 June 2013 (UTC)

No, I don't have a source. This is my own opinion.Armeria wiki (talk) 05:25, 10 June 2013 (UTC)

Perhaps you want File:HyperbolicAnimation.gif restored or made into a SVG static file? M∧Ŝc²ħεИ_τlk 07:09, 9 June 2013 (UTC)

I'd like to distinguish these two angles in the diagram. They are different.Armeria wiki (talk) 05:25, 10 June 2013 (UTC)

Which can be done by redrawing File:HyperbolicAnimation.gif as SVG, I'll do it soon. M∧Ŝc²ħεИ_τlk 05:52, 11 June 2013 (UTC)

another question about the diagram of hyperbolic angle

There might be a another mistake in the diagram of File:Hyperbolic angle2.svg.

there might be a mistake in the diagram

I think the hyperbolic angle should be defined between a ray from the origin point (or the symmetric point) and the ray of the symmetric axis of the hyperbola. In this diagram, neither of the two rays is the symmetric axis of the hyperbola, and the hyperbolic angle u defined in such diagram should have the wrong meaning.Armeria wiki (talk) 03:34, 9 June 2013 (UTC)

I follow what your saying, but then you could say the diagram shows the sum of two hyperbolic angles: ones either side of the symmetric axis you describe. So this diagram is correct. M∧Ŝc²ħεИ_τlk 06:53, 9 June 2013 (UTC)

I'd like to to call it as a difference of a positive hyperbolic angle and a negative one.Armeria wiki (talk) 12:06, 10 June 2013 (UTC)

I suspected you would mention this, and will show +/- senses a redrawn diagram, but you could consider the total circular angle between two angles measured relative to a radial line from a circle centre, either side of the line, as well. I guess it's just preferences in how one conceives angles as +/-ve or absolute. M∧Ŝc²ħεИ_τlk 05:52, 11 June 2013 (UTC)

Sorry to Maschen. After the discussion in Wikipedia:Reference desk/Mathematics about the hyperbolic angle, I think what you draw before with two arbitary rays is correct. The hyperbolic angle itself does not necessarily need a sepecial ray for origin. Two rays are enough to define an angle. A special ray for the origin is useful to provide a system of angle, but not necessary for the angle itself. And thank you for your work.Armeria wiki (talk) 01:47, 12 June 2013 (UTC)

No need to apologize! I think it's better to show negative as well as positive senses, thanks to you raising the point. ^_^ M∧Ŝc²ħεИ_τlk 05:40, 12 June 2013 (UTC)

The article, right from the start, has File:Hyperbolic angle.svg. So you think this should replace the other one completely? M∧Ŝc²ħεИ_τlk 07:09, 9 June 2013 (UTC)

The diagrams in the article are correct (using symmetric axis). The diagrams not using the 45° line found in this discussion are not used in the article so they merely guide discussion here. Concerns of Armeria refer to these other images. Maschen's work is good.Rgdboer (talk) 21:03, 9 June 2013 (UTC)

I modified File:Hyperbolic angle2.svg, however my File:Circular and hyperbolic angle.svg is definitely wrong as

• the line segment from the origin to the hyperbola extends along the alongside the hyperbolic asymptote,
• while the line from the origin to the circle simply rotates around in the circular path

so the triangle constructed with respect to the hyperbola would not follow the triangle constructed with respect to the circle. Therefore as Armeria wiki says the angles cannot be identical. I originally interpreted the request to show show the sin, cos and sinh, cosh functions relate to the same parameter u in a geometric way. I'll replace my drawing with the previous animation by Sam Derbyshire which is correct and shows far clearer and nicer the difference between the angles. M∧Ŝc²ħεИ_τlk 08:05, 11 June 2013 (UTC)

Ok. The use of letter u to mean two different measures when used as the argument of two different types of trigonometric functions causes confusion. Thank you for your attention to this important learning site.Rgdboer (talk) 22:20, 11 June 2013 (UTC)

Sense of angle

The current diagrams indicate hyperbolic angle evolves counter-clockwise like circular angle. Can this assertion be confirmed in references? Probably not, since the concept of hyperbolic angle is tied up with logarithm as made clear in the article hyperbolic sector. There the angle evolves in the opposite direction. Another reason for using the opposite orientation to circular angle lies in the application for rapidity where diagrams generally have frames moving to the right. I suggest that the diagrams be altered accordingly.Rgdboer (talk) 19:25, 12 March 2014 (UTC)

Sinh from sin( i x )

An editor from Alpharetta, Georgia, changed the formula for sinh in from sin(ix). For clarity, note the series:

$\textstyle \sinh x=\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)!}}$

Now note that sin has the series converted to an alternating series. Note further

$i^{2n+1}=i(i^{2n})=i(-1)^{n}$ so that the alternating factor is canceled out.Rgdboer (talk) 01:08, 9 November 2014 (UTC)

Contradiction between lead image and text?

The hyperbolic angle u is a real number that is the argument of the hyperbolic functions sinh and cosh. It determines a hyperbolic sector (red) that has area u. The legs of the hyperbolic triangle (yellow) are proportional to sinh(u) and cosh(u).

Sorry if I'm just confused here, but it seems to me that the text in the lead contradicts the accompanying image. Currently the text of the lead says

A hyperbolic angle in standard position is the angle at (0, 0) between the ray to (1, 1) and the ray to (x, 1/x) where x > 1....The hyperbolic angle in standard position is considered to be negative when 0 < x < 1.

But in the accompanying image, reproduced here, u is defined over 0 < x < 1, with the caption saying that u is the area and not its negative. Contradiction? Loraof (talk) 14:51, 14 April 2015 (UTC)

I'll try to resolve this by adding an explanatory sentence to the caption. Revisions welcome! Loraof (talk) 16:00, 14 April 2015 (UTC)

Thank you for your interest in consistency for the sense of a hyperbolic angle. Since the work of G. de St. Vincent the sense has usually been positive for x > 1. However, some people have noted that this direction is opposite the counter-clockwise sense for circular angles. The issue arises in images. Furthermore, transformation of the axes determining these senses may be simplified to a rotation and dilation if the senses are made to agree. Thus the image used produces a contradiction that you have noted. The issue was raised before in this Talk, but you are the first to remark on it. An image consistent with historic development and hyperbolic sector is preferred.Rgdboer (talk) 19:01, 14 April 2015 (UTC)

Sense

The hyperbolic angle u is a real number that is the argument of the hyperbolic functions sinh and cosh. It determines a hyperbolic sector (red) that has area u (defined to be negative for the case shown here because x < 1 at the point on the hyperbola). The legs of the hyperbolic triangle (yellow) are proportional to sinh(u) and cosh(u).

Top: Positive and negative hyperbolic angles. Bottom: The difference between the two positive angles is shown as Δu = u₂ − u₁.

An angle can have two senses according to motion along the bounding curve (circle or hyperbola). As mentioned above, historically the hyperbolic angle has positive sense opposite the direction of circular angle's positive sense. The diagram at the top of the article confounding the topic with two senses has been replaced with a modest diagram without sense. — Rgdboer (talk) 01:22, 12 March 2019 (UTC)

Standard position

As seen in the section Definition, standard position refers to an angle with one side on the line y=x. Using the phrase "standard position" in the lead to mean the hyperbola xy=1, is misleading. Therefore a change has been made to remove this phrase from the lead of the article. As for confusion with the unit hyperbola xx-yy=1, this article provides an introduction to the argument of hyperbolic functions, and leaves for later development the description of the unit hyperbola as a parametric curve depending on hyperbolic functions. Thank you for your interest in clarifying this topic. — Rgdboer (talk) 01:13, 14 September 2019 (UTC)

Real analysis

@Jacobolus: There is no need to put complex function theory in this article as the split in the exponential series holds for all variables. Infinite series facts relate circular and hyperbolic functions. Notes are included to clarify how the imaginary unit plays in the infinite series to distinguish the types of functions. Rgdboer (talk) 00:50, 24 September 2022 (UTC)

What do you mean “complex function theory”? The “imaginary angle” section inherently involves complex numbers, and the section already discussed power series. I didn’t add any new concepts, just made the discussion more explicit and hopefully clearer. –jacobolus (t) 02:49, 24 September 2022 (UTC)

Good that Bivector is gone; very distracting! Rgdboer (talk) 01:20, 24 September 2022 (UTC) Jacobolus Rgdboer (talk) 01:21, 24 September 2022 (UTC)

There should absolutely be a discussion of the relevance here to different geometries (the plane with pseudo-Euclidean metric of signature 1,1; the hyperbolic plane; vectors and bivectors in the Euclidean plane; etc.) but it should probably be a separate section. –jacobolus (t) 02:59, 24 September 2022 (UTC)

For reference we have WP:Make technical articles understandable which says, among other things,

Write "for the largest possible general audience" and write "one level down".

Note that imaginary unit < imaginary number < complex number. A reader consulting this article is likely a calculus student interested in the argument of a hyperbolic function. The section refers to infinite series only to clarify the trigonometric functions. Introduction of complex variable z = x + iy presumes too much. The exponential series splits into even and odd terms for all types of arguments taken from a Banach algebra, not just complex numbers as related by a statement in your text. Nevertheless, cos and sin are holomorphic functions (a link you removed), and as Howard Eves writes in his Functions of a Complex Variable (1966), page 154,

sin z = sin x cosh y + i cos x sinh y and

cos z = cos x cosh y – i sin x sinh y.

Thus in complex analysis cosh and sinh are absorbed into sin and cos. These equations are beyond the article, but are given here to show the deviance of your text which purports to present cosh and sinh in a fashion to your liking. The first edit summary you gave said some cleanup was needed but no faults were noted in this Talk. Another edit summary said "what does real analysis topic mean?"; no credit to your learning. You did not use Talk until the Ping. Obscurantism is detected here (for example, Bivector ?!). A return to the real analysis about hyperbolic angle as an imaginary number is recommended to minimize your exposure. Rgdboer (talk) 20:33, 24 September 2022 (UTC)

equations are beyond the article – first, an article can serve multiple audiences; and second, high school students learn about complex numbers in their introductory calculus class (if they haven’t already been introduced the year prior). no credit to your learning – what do you mean by this? –jacobolus (t) 00:16, 25 September 2022 (UTC)