User talk:Jacobolus/HalfTan

Suggestions
Nice article. In the hopes that it will be helpful, I'll give some criticisms. My intent is to help you make the article stronger, not to stop you from what you are doing or anything like that. I apologize in advance for any criticisms that aren't constructive in a positive sense. I'll organize them by section so that we can keep multiple parallel discussions going if we want to. — Q uantling (talk &#124; contribs) 20:55, 2 November 2022 (UTC)


 * Please don’t worry about saying what you think. It won’t hurt my feelings. (E.g. we may need to cut down or cut out some sections here for pushing the scope too broad, do some radical reorganization, rewrite sections, or the like.) There is still quite a lot to add, e.g. an introductory section at the top explaining the geometric context, a history section, one or more sections about applications in science/engineering, etc. –jacobolus (t) 23:26, 2 November 2022 (UTC)

Lede
In the lede, I'm new to the Cayley transform -- thanks for that pointer! When I go to that page, I see $(z − i) / (z + i)$ but on your page I see $−i (z − 1) / (z + 1)$. It looks like the latter can be written as $$\frac{1}{i}\frac{(iz)-i}{(iz)+i}$$. Would that be better in this article? (I'm a newbie when it comes to the Cayley transform, so maybe what you already have is standard fare to those with more familiarity?) — Q uantling (talk &#124; contribs) 20:55, 2 November 2022 (UTC)


 * My impression is that the name “Cayley transform” is not defined nearly so strictly as the Cayley transform article implies. That article should perhaps be researched more carefully and then expanded/rewritten. –jacobolus (t) 21:50, 2 November 2022 (UTC)
 * But I could also be wrong about that, I’m not an expert. We could say something like “a modified Cayley transform” instead, or just call it a Möbius transformation. I think some of the traditional conventions chosen in Complex analysis are suboptimal here. Mapping ∞ from the real line ↔ 1 on the complex unit circle is a mistake, because we want to map the identity under tangent-sum to the identity under complex multiplication, and preserve signs when mapping from angle measures ↔ half-tangents. It works okay when we want to think of the unit disk as the hyperbolic plane, but is not very helpful when we want to think of the complex unit circle as a model for planar rotations. (In a similar way, the conventional concept of the Riemann sphere mapping the origin in the extended complex plane to the "south pole" of the sphere is a mistake, because it reverses orientation.) –jacobolus (t) 23:46, 2 November 2022 (UTC)
 * Note that the version of the Cayley transform described at is analogous in our setting to the natural logarithm, and its inverse is our function $$E$$, analogous to the exponential function. See  § Circular functions › Inverse functions of this article. –jacobolus (t) 05:00, 5 November 2022 (UTC)
 * There is an alternative (perhaps more reasonable, but not very common) convention, which is to use pure imaginary values for half-tangents and angle measures, in which case we want a map from the complex plane that takes 1 ↦ 0, i ↦ i, -i ↦ -i, -1 ↦ ∞. Then the interior of the disk is mapped to the left half plane and the exterior of the disk to the right half plane. I would be fine using such a convention in personal work (say, in writing a new textbook or something), but I think sticking with real-valued angle measures and half-tangents is safer in the context of Wikipedia. –jacobolus (t) 23:53, 2 November 2022 (UTC)

"modified" Cayley transform is good enough for me as would be "rotated". I might tweak the formula to be
 * $$\frac1i \left(\frac{iz-i}{iz+i}\right)$$

but what you have also works. — Q uantling (talk &#124; contribs) 16:19, 5 November 2022 (UTC)


 * The way I think of it is that $$ih$$ is actually the [oriented] quantity of interest (if we were working with quaternions or geometric algebra, the $$i$$ could stand for any bivector, representing an angle or rotation oriented with some other plane), which we want to compute by $$z \mapsto ih.$$ Then $$h = -i(ih)$$ is a scalar ("real number") with the orientation stripped away. The quantity $$iz$$ is not so geometrically natural or interesting per se. –jacobolus (t) 18:20, 5 November 2022 (UTC)

Tangent addition
For Section 2, it looks like every formula applies not only to half-angle tangents (my personal preference name for the concept) but also to full-angle tangents, double-angle tangents, third-angle tangents, etc. Sure, only for half-angle tangents $h ∈ (−∞, ∞]$ is there a one-to-one and onto mapping to angles $θ ∈ (−\pi, π]$, but otherwise these formulas work quite a bit more generally, don't they? For example, I might put them in the article for tangent. But ... you put them here for a reason no doubt ... perhaps an introduction to Section 2 would explain that to people who would otherwise react to it as I have. — Q uantling (talk &#124; contribs) 20:55, 2 November 2022 (UTC)


 * Yes, this “tangent sum” operation also applies to tangents. It could be made into its own article, but there isn’t currently one on Wikipedia and I didn’t want to spin up several draft articles in parallel. Some of the same material could also be duplicated in an article about the tangent function but Wikipedia doesn’t have one of those either per se (tangent (trigonometry) redirects to Trigonometric functions). I included a substantial amount of material about the tangent sum here because it is the basic composition operator for angles represented as half-tangents, used throughout the rest of the article.–jacobolus (t) 22:07, 2 November 2022 (UTC)

I'd change the existing sentence "When rotations are represented as half-tangents, composition is a tangent addition operation..." to "When rotations are represented as tangents of some angle, composition is the tangent addition operation ...".

Where you have
 * $$\begin{aligned}

\infty \oplus h &= \frac{h + \infty}{1 - h\infty} = -\frac{1}{h} = \infty \boxminus h, \\[10mu] \infty \boxplus h &= \frac{h + \infty}{1 + h\infty} = \!\!\phantom{-}\frac{1}{h}\,\, = \infty \ominus h, \end{aligned}$$ I might instead drop the last expression for each line and give each its own new line. While it is the case that the last expression does equal the second-to-last expression (for each of these lines), it is not because of the third-to-last expression... at that is a bit confusing.

Bigger picture, I am still thinking that this material is not appropriate for an article on half-tangents. It could go in an existing article for the tangent function or a new one, but I find it out of place here. — Q uantling (talk &#124; contribs) 16:36, 5 November 2022 (UTC)


 * The last equality can probably be dropped there; it’s not really necessary to make the point. [Update: I dropped it.]
 * There isn’t an existing article for either the tangent addition operation or the tangent function (per se), but there probably should be both: the article trigonometric functions is too broad in scope and doesn’t leave much room for expansion about particular details. You are right that this material could be repeated there. Working in terms of tangents (rather than half-tangents) is a way to model the "elliptic line", which in 1 dimension is equivalent to the 1-sphere (circle) [whereas the elliptic plane and 2-sphere are topologically distinct, and likewise for higher dimensions]. Arguably, the half-tangent is as or more fundamental than the tangent; it’s historical happenstance that one is defined in terms of the other.
 * The reason for including what seems like a lot of material about the tangent addition operation here is that composition of angles is a fundamental operation for half-tangents, and interpreting/manipulating/deriving pretty much any other formula on this page depends on applying the basic algebraic relations of that operation. (The name tangent addition is one that has been used occasionally in previous work; it could just as well be called half-tangent composition, stereographic addition, or some similar kind of name, or something else; it’s a bit tricky to figure out the clearest nomenclature because it is a new operation somewhere in between ordinary addition and multiplication, and can be related to both, but is much less common in mathematics than addition or multiplication themselves.)
 * One basic tension is that most work (either in trigonometry directly or applying trigonometry) is written down in terms of angle measures; most of the time, half-tangents are treated as a special trick that is applied ad-hoc, with (a) as much manipulation as possible still done in terms of angle measures (or sometimes unit complex numbers), and (b) what manipulation is done to half-tangents handled using ordinary addition and multiplication, because authors are not familiar with the tangent addition operation.
 * There have been a few efforts to treat half-tangents as their own subject per se (i.e. the projectively extended real numbers with a tangent addition operation) and describe them systematically, but there aren’t any extended survey papers or textbooks or the like. This greatly improves conceptual clarity, at the expense of requiring fluency (or at least familiarity) with the tangent addition operation. –jacobolus (t) 17:04, 5 November 2022 (UTC)

Commutative group structure
Isn't it the case that the group of $(-1,1)$ under the $⊞$ operation is isomorphic to $R$ under ordinary addition? (That is, to add velocities with the relativistic addition formula, you can instead add the underlying rapidities.) — Q uantling (talk &#124; contribs) 16:53, 5 November 2022 (UTC)


 * Yes, you can use hyperbolic angle measure as an alternative model for hyperbolic angles, with ordinary addition. The advantage of using hyperbolic half-tangents in (–1, 1) is that you can avoid transcendental functions if you want to convert back and forth to unit-magnitude split-complex numbers.
 * This is analogous to the way you can use circular angle measure (in a periodic interval) as a model for circular angles, again with ordinary [wrapping] addition as an operation. –jacobolus (t) 17:09, 5 November 2022 (UTC)

Terminology
not only to half-angle tangents (my personal preference name for the concept) [copied from above]


 * The name “half-angle tangent” gets to be a real mouthful when you need to repeat it over and over again, use it as an adjective phrase, etc. Personally I think that Norman Wildberger’s name “half-slope” is a better name (because it doesn’t get confused with the tangent meaning a line just touching a curve), but that one is only used by one or two sources and is not really supportable as a primary name. I picked the name “half-tangent” because it was historically the first name used (was a standard term for centuries) and has been used by a wide variety of authors, but is still reasonably concise. In the past century or so there’s really not an accepted common name for this concept; the most common is to not name it at all, or when it is named there are a wide variety of names chosen, none with a clear majority usage. –jacobolus (t) 22:07, 2 November 2022 (UTC)
 * Indeed it sounds like a problem with no easy solution. I suggest mentioning some of the also rans as in:
 * In mathematics, the half-tangent (also known as the half-angle tangent and half-slope) is ...
 * — Q uantling (talk &#124; contribs) 13:27, 3 November 2022 (UTC)
 * That's why I added a 'terminology' section after the lead here. If we want to list names that are sometimes used, there are also "semi-tangent", "tangent half-angle", "(one-dimensional) stereographic projection", "half-tan", "half-angle tan", "tan-half-angle", "tan½", or more descriptively "tangent of half the angle", "tangent of the half angle", etc., or as descriptions of the half-tangent function/substitution, "half angle substitution", "universal trigonometric substitution", "universal substitution", "rationalizing substitution", "t substitution", "Weierstrass substitution", etc. (Not to mention names in Latin, French, German, ...) It gets overwhelming and I expect doesn't help readers all too much. I don’t think "half-slope" can be fairly added to the lead section over all of those other names, though it deserves some credit for being used in a source that tries to develop a theory of this subject per se instead of just applying it in passing. –jacobolus (t) 16:13, 3 November 2022 (UTC)
 * It’s a somewhat similar situation with split-complex numbers, which have been given like a dozen different names in different sources. (The name "split-complex" is frankly horrible.) –jacobolus (t) 16:45, 3 November 2022 (UTC)

Even if in this Terminology section, I would put in bold those terms that you could imagine setting up page redirects for. — Q uantling (talk &#124; contribs) 16:21, 5 November 2022 (UTC)


 * I did think about it quite a bit. In my opinion it’s not really that helpful, since any potential redirect would go to the lead section, and I wouldn’t expect readers to have trouble figuring out from there whether they were in the right place or not. In the terminology section bolding every possible synonym would be overwhelming/distracting, and there aren’t really any I would pick out as needing to attract special visual attention away from the flow of ordinary reading. The difference between e.g. half-tangent, half-angle tangent, tangent half-angle, half-angle-tan, or tangent of half the angle is not so great that I expect a typical reader to be confused if one name is substituted for the other, assuming there is a clear enough definition in the lead. Terms like half-slope or semi-tangent are used rarely enough that I don’t really expect any appreciable number of readers to come through there or be confused hunting for them. Many Wikipedia pages are a bit overzealous in using bold text; it can be helpful when used for occasional redirect targets (e.g. I bolded half-tangent function in § Half-tangent function in case we ever try to redirect Tangent half-angle formula to there), but in general bold fonts should be used sparingly. –jacobolus (t) 16:39, 5 November 2022 (UTC)

Chordal distance
There's probably a nice formula in terms of $$h^{\oplus 1/2}$$. — Q uantling (talk &#124; contribs) 17:04, 5 November 2022 (UTC)


 * Yes, this is what we call $$\bigl|S\bigl((h_1 \ominus h_2)^{\oplus 1/2}\bigr)\bigr| = \bigl|S_{1/2}(h_1 \ominus h_2)\bigr| = \bigl|S_{1/2}\bigr(h_1 \oplus (-h_2)\bigl)\bigr|. $$ We have an identity for $$S_{1/2}\bigr(h_1 \oplus h_2 \bigl)$$ in § Half-angle identities. Note that $$S_{1/2}(h) := \bigl(h^{\oplus1/2}\bigr)\vphantom{)}^{\boxplus2}$$ so we could write $$\bigl|\bigl((h_1 \ominus h_2)^{\oplus1/2}\bigr)\vphantom{)}^{\boxplus2}\bigr|$$  –jacobolus (t) 18:43, 5 November 2022 (UTC)
 * But with that said, I don’t know if it’s helpful to add material about in the section § Chordal distance; it might just be a distraction. –jacobolus (t) 19:33, 5 November 2022 (UTC)

Formal definition
It looks like $$\tan \theta/2 = \frac{\sin \theta/2}{\cos \theta/2}$$. The power series for the denominator looks like it has some typos including that it should start with a constant of 1. — Q uantling (talk &#124; contribs) 21:18, 2 November 2022 (UTC)


 * Yes, fixed. Thanks! Is there another typo? –jacobolus (t) 21:51, 2 November 2022 (UTC)


 * I took the liberty of fixing the typos myself; I hope that is okay. — Q uantling (talk &#124; contribs) 13:20, 3 November 2022 (UTC)
 * Thanks! That’s what I get for copy/pasting bits of LaTeX markup without double-checking to change all of the parts. –jacobolus (t) 16:40, 3 November 2022 (UTC)

Relation to other circular functions
I'd be curious how this looks with hyperbolic functions too.

And if it isn't too far afield and proves to be illuminating, show the connection for the hyperbolic formulas to special relativity, where energy $E = mc cosh η$, momentum $p = mc sinh η$, and velocity $v = c tanh η$ for a rapidity of $η$. For example, is it all meaningful that?: $$\tanh \eta/2 = \frac{pc}{mc^2 + E}$$ — Q uantling (talk &#124; contribs) 13:48, 3 November 2022 (UTC)


 * The hyperbolic half-tangent is certainly also a thing worth discussing at length somewhere on Wikipedia, and at least a bit on this page, both as a model for hyperbolic rotations (Lorentz boosts) and lengths in hyperbolic space, and as a function of hyperbolic angle measure.
 * I already added a sub-section about the hyperbolic tangent addition operation (here called $$\boxplus$$; Ungar uses the symbol $$\oplus$$ and calls it "Einstein addition"), which was as far as I know first used by Poincaré in describing composition of (spatially) parallel relativistic velocities, which are naturally hyperbolic tangents. $$\boxplus$$ can also be employed when directions in Minkowski space are described using stereographic projection (half-tangent); I intend to still add a "hyperbolic trigonometry" section paralleling the spherical trigonometry section, which would describe some of the relevant relationships there. You’ll notice that the circular tangent addition function shows up in hyperbolic geometry and the hyperbolic tangent addition function shows up in spherical geometry.
 * Here the tangent addition operations $$\oplus$$ and $$\boxplus$$ are described as operations on scalar quantities ("real numbers"), but they can alternately be seen as part of a single $$\boxplus$$ operation, applied to bivector-valued arguments (or vector-valued, trivector-valued, ...), which square to either positive or negative scalars depending on the geometry. Here we mention $$i (h_1 \oplus h_2) = (ih_1) \boxplus (ih_2), $$ but for a unit bivector $$u$$ in a space of metric signature (1, 1) such that $$u^2 = 1$$, we also have the hyperbolic analog $$u (h_1 \boxplus h_2) = (uh_1) \boxplus (uh_2). $$
 * It is also worth somewhere mentioning the Gudermannian function (an article I recently cleaned up/expanded and made figures for), which relates a circular angle measure to a hyperbolic angle measure by identifying the associated circular half-tangent and hyperbolic half-tangent.
 * It’s a bit hard to figure out the right organization structure and scope for this article. I don’t really want to go beyond 2-dimensional geometry of the plane/2-sphere/hyperbolic plane, because it gets to be too much material, but I think those are helpful as core applications.
 * It might even be better to try to publish some of this material in an expository journal paper somewhere before or instead of Wikipedia: it’s not really novel or original research, but as far as I can tell it also hasn’t been systematically collated anywhere, and the half-tangent is nearly always employed as a one-off trick. –jacobolus (t) 16:37, 3 November 2022 (UTC)

Higher dimensions
> I don’t really want to go beyond 2-dimensional geometry of the plane/2-sphere/hyperbolic plane

You likely have ideas for how to generalize to more dimensions. My wish list would include a short, one-or-two paragraphs section on the projective-space interpretation and higher dimensions. It is sort of discussed in the Cayley transform article in the Real Homology section. Specifically, instead of defining $h = y / (1 + x)$, one writes $h = (1 + x, y)$ as the projective-space value. This form easily generalizes, to projective spaces with more dimensions. One gets $h = (1 + w, x, y, z)$ for $(w, x, y, z)$, for example. (That's what it would be for a 3-sphere (surface of a 4-ball) projecting to $(−1, 0, 0, 0)$, but one could get crazy and also include quadrics other than the hyperspheres and the standardized hyper-hyperbolic surfaces; see the section Quadric — disclaimer: which I wrote — where the discussions about rational numbers aren't relevant to this article but $W$ plays the role of $h$ and $y$ is $(−1, 0, 0, 0)$ in the hypersphere example.) Okay, that parenthetical could be too far, but a start towards that direction would be nice. — Q uantling (talk &#124; contribs) 18:43, 3 November 2022 (UTC)
 * Such discussion can probably fit in stereographic projection or somewhere. The reason I started this article is that I want to rewrite / expand the stereographic projection article adding many more figures, more about history, etc., and wanted to start with the one-dimensional case since it is easiest to illustrate and important as a slice through higher-dimensional cases. But there’s too much to say about the one-dimensional stereographic projection (half-tangent) per se for the scope and narrative structure of that article. So instead the idea was to make a short summary (a few paragraphs) of the 1-dimensional case there, linking to this as a "main" article. –jacobolus (t) 18:46, 3 November 2022 (UTC)
 * FWIW, I segregated the discussion of rational and integer solutions from the discussion of stereographic-like projection at Quadric. In case that helps with your article's discussion of higher dimensions....  — Q uantling (talk &#124; contribs) 17:44, 11 November 2022 (UTC)

Continued fraction
$$h^{\oplus w}$$ has a neat continued fraction expansion:
 * $$h^{\oplus w}=\cfrac{wh}{1-\cfrac{(w^2-1^2)h^2}{3-\cfrac{(w^2-2^2)h^2}{5-\cfrac{(w^2-3^2)h^2}{7-\ddots}}}}$$ A1E6 (talk) 01:38, 5 November 2022 (UTC)
 * Fun! Did you derive this one or find it somewhere? –jacobolus (t) 03:59, 5 November 2022 (UTC)
 * The formula can be found in Analytic Theory of Continued Fractions (p. 346, eq. 90.13) by H. S. Wall. A1E6 (talk) 10:10, 5 November 2022 (UTC)
 * Thanks. Link to the page for anyone curious: https://archive.org/details/analytictheoryof0000wall/page/346/ –jacobolus (t) 18:28, 5 November 2022 (UTC)
 * It can also be found in Khovanskii (1963) https://archive.org/details/the-applications-of-continued-fractions-and-their-generalizations-to-problems-in/page/108/ –jacobolus (t) 18:31, 5 November 2022 (UTC)

My remaining tasks/goals before trying to move this article to the main namespace
As I said above, I don’t think anything here is really “original” or “novel” research per se. It’s more like bringing together material from a wide variety of sources and trying to adapt it to common conventions. For example, there are a few turn-of-the-20th-century sources which worked out spherical trigonometry in terms of half-tangents, but they differed in whether to look at interior or exterior angles, whether to look at half-tangents or half-cotangents, and wrote everything down in terms of somewhat arbitrary and illegible variable names (which may have made sense in context). I think superficial changes like that are within the realm of WP:CALC, but it means that I don’t want to push my luck too far.

–jacobolus (t) 04:37, 5 November 2022 (UTC)
 * I want to make sure I have have some inline citations for as many formulas as I can, ideally including the earliest reference(s) I can find for an idea (usually written in classical trigonometry language, either in roughly modern notation or as prose sentences about proportions), the best reference I can find about the history of a particular idea or formula, and references to forms closer to what I put in the article here whenever available. I would also like to locate and link to (and add if necessary) those formulas written in classical trigonometry style in the relevant other wikipedia articles (e.g. spherical trigonometry).
 * I want to add at least several more figures; several of the sections are not sufficiently illustrated. Some of this may take writing some code to draw figures.
 * I want to add more explanatory prose for context. In particular:
 * a section at the top explaining the basic geometric idea accessible to the widest possible audience
 * a history section
 * a section about applications to directional statistics, kinematics of linkages, mathematical origami, geodesy/cartography, etc. (If anyone knows good sources, please chime in)
 * I want to add a section about trigonometry in the hyperbolic plane (Lobachevsky plane), paralleling the planar and spherical trigonometry sections. (But I don’t want to get too much into the weeds about even more geometries, 3+ dimensional space, etc.)
 * I want to clean up the references (figure out which can be cited just in footnotes, which should be in a dedicated list at the bottom, and which can be removed)
 * I need to decide whether to use interior or exterior angles. Exterior angles are a bit less common in classical trigonometry literature, but make formulas clearer (and in the spherical case make the polar triangle more precisely dual). I made a fork at User:Jacobolus/HalfTan/Ext.