Talk:Laguerre transformations

Learning

I'm writing this article while still learning about Laguerre transformations. Early versions of the article did have a few mistakes and misconceptions, which I've tried to edit out. I apologise if there are still any mistakes. --Svennik (talk) 13:02, 20 June 2020 (UTC)

Software

If you want to produce the sort of visualisations that you see in the article, and have some basic Python knowledge, then see this: https://github.com/wlad-svennik/laguerre_transformations --Svennik (talk) 09:38, 23 June 2020 (UTC)

There's now something similar for hyperbolic Laguerre transformations. https://github.com/wlad-svennik/hyperbolic_laguerre_transformations --Svennik (talk) 10:17, 26 June 2022 (UTC)

Conformal section: Area of triangle

[removed] --Svennik (talk) 23:44, 17 December 2020 (UTC)

The apex of the triangle is (√2, √2 m), so both base and altitude have factor √2. The area is m. Note that a circle with radius √2 has area 2pi, so sector area of this circle corresponds to angle size. The right triangle, or differences of two of them, form "dual sectors", in analogy to circular and hyperbolic sectors. − Rgdboer (talk) 03:14, 18 December 2020 (UTC)

Oh yes, you're right. --Svennik (talk) 06:46, 18 December 2020 (UTC)

The two interpretations of the Laguerre transformations are not the same

There's the interpretation under which $PGL(2,\mathbb {D} )$ (where $\mathbb {D}$ is the dual numbers) consists of conformal transformations (where angles are slopes) and there's another interpretation under which it consists of transformations acting on oriented lines and circles. The two interpretations are not the same. I'm concerned that this might cause confusion. I also find the conformal interpretation less interesting, and the article doesn't focus much on it. --Svennik (talk) 13:40, 18 December 2020 (UTC)

While there's still risk of misinterpretation in providing two interpretations and not clearly distinguishing them, I propose that the conformal interpretation be removed. --Svennik (talk) 20:28, 21 December 2020 (UTC)

A less radical idea is to move the conformal stuff to the end of the article, so as to maintain the flow. I think that would make me happy. --Svennik (talk) 20:36, 21 December 2020 (UTC)

Embarrassing mistake

Earlier, it was claimed by me that if $V$ is a unitary dual-number matrix where $\det(V)=-1$ then $V$ represents an indirect Euclidean isometry followed by an orientation reversal. It turns out there is no orientation reversal, and $V$ is simply an indirect Euclidean isometry. The lesson is don't trust everything on Wikipedia, and this article would benefit from proofs. --Svennik (talk) 09:53, 12 September 2021 (UTC)

perhaps room for another article or two about Laguerre geometry

The article at Laguerre plane is about the incidence structure, was mostly copied from Hartmann "Planar Circle Geometries", and pretty much only covers Benz's work. This article is about Laguerre transformations treated algebraically as fractional linear transformations. Spherical wave transformation#Transformation by reciprocal directions is focused on an application to electromagnetic waves in special relativity.

It seems like there should also be some space on Wikipedia for describing the basic concepts developed by Laguerre of semi-droites (semi-straights, i.e. directed lines, "spears" or "rays" in more recent German/English literature) and cycles (directed circles), his concept of the power of a spear with respect to a cycle (dual concept to the power of a point with respect to a circle), and "transformations par semi-droites réciproques" (transformations by reciprocal spears, dual concept to inversion of the plane with respect to a circle).

Laguerre, Edmond (1885). Recherches sur la géométrie de direction: Methodes de transformation anticaustiques [Research on the geometry of direction: Methods of anticaustic transformation]. Gauthier-Villars. (A book collecting Laguerre's papers.)

Neither of the existing two articles make much attempt to establish these concepts, explain the basic geometry, show how they relate to Euclidean plane geometry or inversive geometry, or explain how they relate to the alternative models of the "Laguerre plane" that were developed afterward. As someone who didn’t know anything about Laguerre’s work before the past few days, I found I couldn’t really make heads or tails of it from the material currently on Wikipedia, and had to look for other sources for a basic introduction. –jacobolus (t) 18:37, 27 January 2023 (UTC)

For context I noticed a discussion of the "power of a great circle with respect to a small circle" in Todhunter & Leathem's spherical trigonometry book, which prompted me to ask this question about the 'power of a line' at the math reference desk, which surfaced some other old German/French sources and some relevant material at Spherical wave transformation#Transformation by reciprocal directions. –jacobolus (t) 19:20, 27 January 2023 (UTC)

The cycles and spears are covered here (Laguerre transformations § Points, oriented lines and oriented circles), but not under those names. Algebraic representations are given for both. The axial inversions are not covered. Actually, they are sort of covered in the classification theorem of Laguerre transformations: It's shown that a Laguerre transformation is either a Euclidean motion followed by an "axial dilation", or a Euclidean motion followed by an "axial inversion". (Note that I say "axial dilation" instead of "axial dilatation" as a matter of personal style, but I can see why some people might criticise that as a deviation from standard terminology). The resulting decomposition of a matrix resembles the Singular Value Decomposition in a suggestive way, and indeed it leads to the Singular Value Decomposition for all matrices over the dual numbers. --Svennik (talk) 19:25, 28 January 2023 (UTC)

Sure, and someone with a math degree who already spent a month thinking about this topic will be able to make some sense of that section. But it is not introduced or described in such a way that a non-expert reader is going to be able to follow it at all. There are no basic definitions, no discussion of the relation to high school geometry people are more familiar with, no diagrams, etc. –jacobolus (t) 19:48, 28 January 2023 (UTC)

You're the first guy to provide feedback. So thanks for that. I guess the article might need work, and might indeed benefit from being covered in a different way.

Regarding basic definitions, those can indeed be provided upfront, instead of buried somewhere in the text.

Also, some people might prefer the linear algebraic style of presentation in this article to the "high school geometry" approach you're pushing for. I've provided animations, but maybe those aren't clear to a newcomer. (How would I even know?) I've generated these visualisations from a program I wrote. Svennik (talk) 19:57, 28 January 2023 (UTC)

I'm not suggesting this article needs to be changed necessarily. I just don't think it sufficiently covers the basic concepts of Laguerre's work or its relations to other subjects. (Which might better fit into a different article, e.g. Laguerre geometry, power of a line, inversion in a line or transformation by reciprocal directions, or the like.)

If you lead off with "analogue of Möbius transformations over the dual numbers" that is both too technical and too vague as a basic definition for non-expert readers. You are assuming people already know about or are willing to go read about dual numbers, complex numbers, Möbius transformations, line coordinates, skew-Hermitian matrices, Lie sphere geometry, Minkowski space, and all of their many prerequisites (i.e. about 2 full-time years of post-secondary math coursework, probably best researched in textbooks rather than wiki articles) before coming to this article. That is going to scare away about 99% of potential readers.

Laguerre (and other geometers following) did not work with dual numbers, or the Minkowski 2+1 space, or even a Cartesian coordinate system, but made descriptions in terms of lines, circles, planar distance, and other basic synthetic/metrical geometry of the Euclidean plane.

This article also makes the choice to conflate "this particular formal model of a structure" with "the structure itself", if that makes sense. Laguerre transformations are not inherently coordinate-based, and imposing an arbitrary choice of coordinate system can be practically convenient (e.g. for rendering computer pictures) but should not be taken as a basic definition.

These are problems that many math articles on Wikipedia share (not to mention many other mathematical sources), including e.g. the Möbius transformation article, which should really lead with some basic discussion of inversive geometry, but instead doesn't even link to there. –jacobolus (t) 20:23, 28 January 2023 (UTC)

@Svennik: I looked at Yaglom's book. In my opinion the entire section 2.9 "Dual Numbers as Oriented Lines of a Plane" is a required prerequisite to understanding this article: without knowledge of that section it is very difficult for readers to make sense of anything here. The corresponding section here, Laguerre transformations § Line coordinates, is too compressed and decontextualized to bring new readers along. We should perhaps move that section to be first in the article and should for now direct people to read Yaglom's chapter as a prerequisite, but ideally could expand the section significantly, including an explanation of basic transformations. The terms "x-axis" and "y-intercept" should probably be avoided, because they are unrelated to the quantities x and y as used elsewhere for dual numbers. Instead it might be better to use terms such as "origin axis" or Yaglom's name "polar axis" (if you prefer an oriented line could be called a "ray", "spear", "semi-line", or whatever instead of an "axis"), and just talk about the "oriented distance" to another axis. –jacobolus (t) 17:18, 2 February 2023 (UTC)

And I hope I’m not leaving the wrong impression with my criticism. I am not trying to tear down the work done so far; you’ve got some great material here. Figuring out the appropriate audience(s) and then writing Wikipedia articles is really tricky.

The subject seems like a fascinating under-appreciated 19th century idea with big implications for metrical geometry considered broadly, worthy of further research and with potential practical applications. After hunting around I can’t really find any amazing sources describing Laguerre geometry in English (I don’t speak German and my French is not great). I still need to spend more time thinking about doodling around on paper to make full sense of the geometry for myself. But then I hope I can add more material, especially some basic diagrams, here and possibly at Laguerre geometry or other related pages. –jacobolus (t) 11:43, 3 February 2023 (UTC)

I'm wondering if there's a purely technological solution to the problem of translating material from German or French into English: Some combination of OCR (like the one in Mathpix -- albeit last time I checked, it chewed up German umlauts) and machine translation software, like DeepL or Google Translate. Beware that machine translation software might destroy syntactically correct Latex. There are also programs which are "forgiving" in their ability to import Latex, like GNU TexMacs. I've tried all this recently but got stuck by DeepL ruining the Latex / Texmacs source code, and Texmacs not being ability to display some of the resulting content. Svennik (talk) 13:51, 3 February 2023 (UTC)

That would be neat. I can mostly figure out what various past sources have said, either directly by looking at them carefully or by piecing together their content based on later references to them in languages I can read better.

What I was getting at is: I think there’s room for a longer wikipedia article (or perhaps several articles) to provide a valuable public service by covering this topic more completely. –jacobolus (t) 01:06, 4 February 2023 (UTC)

A mention of the 'power of a line' (discussed on a sphere but with the planar analog mentioned) apparently came several decades before Laguerre from Christoph Gudermann in a problem sent to Crelle's Journal in 1832, proved in his 1835 book Lehrbuch der niederen Sphärik, §§296–297. See User:Jacobolus/Gudermann Spharik. –jacobolus (t) 01:57, 9 February 2023 (UTC)

List of references

Here’s a list in chronological order of references related to Laguerre geometry / transformations. Feel free to add more. –jacobolus (t) 01:15, 6 March 2023 (UTC)

• Gerwien, Karl Ludwig (1826). "10. Beweise einiger auf der Kugel Statt findenden Sätze" [Proof of some theorems taking place on the sphere]. Crelle's Journal (in German): 130–135. doi:10.1515/crll.1834.11.130.
• Gudermann, Christoph (1832). "Lehrsätze und Aufgaben. No. 14" [Theorems and Problems. No. 14]. Crelle's Journal (in German): 102. doi:10.1515/crll.1832.9.100. Figure 4.
• Gudermann, Christoph (1835). "§§ 296–297". Lehrbuch der niederen Sphärik [Textbook of Basic Spherics] (in German). Coppenrathsche Buch- und und Kunsthandlung. pp. 227–229.
• Laguerre, Edmond (1870). "Sur la règle des signes en Géometrie" [On the rule of signs in geometry]. Nouvelles Annales de Mathématiques. Ser. 2 (in French). 9: 175–180.
• Laguerre, Edmond (1880). "Sur la géométrie de direction" [On the geometry of direction]. Bulletin de la Société Mathématique de France (in French). 8: 196–208. doi:10.24033/bsmf.207.
• Laguerre, Edmond (1882). "Transformations par semi-droites réciproques" [Transformations by reciprocal semi-lines]. Nouvelles Annales de Mathématiques. Ser. 3 (in French). 1: 542–556.
• Laguerre, Edmond (1883). "Sur quelques propriétés des cycles" [On some properties of cycles]. Nouvelles Annales de Mathématiques. Ser. 3. 2: 65–74.
• Laguerre, Edmond (1883). "Sur les courbes de direction de la troisième classe" [On the curves of direction of the third class]. Nouvelles Annales de Mathématiques. Ser. 3. 2: 97–109.
• Allardice, Robert (1884). "Radical Axes in Spherical Geometry". Proceedings of the Edinburgh Mathematical Society. 3: 59–61. doi:10.1017/S0013091500037305.
• Laguerre, Edmond (1885). "Sur les anticaustiques par réfraction de la parabole, les rayons incidents étant perpendiculaires à l'axe" [On the anticaustics by refraction of the parabola, the incident rays being perpendicular to the axis]. Nouvelles Annales de Mathématiques. Ser. 3. 4: 5–16.
• Laguerre, Edmond (1885). Recherches sur la géométrie de direction: Methodes de transformation anticaustiques [Research on the geometry of direction: Methods of anticaustic transformation]. Gauthier-Villars. (A book collecting the above papers.)
• Dautheville, S. (1886). "Sur l'hypercycle et la théorie des cycles polaires" [On the hypercycle and the theory of polar cycles]. Bulletin de la Société Mathématique de France. 14: 45–67.
• Coelingh, Derk (1888). "Transformation de figures analogue a la transformation par rayons vecteurs réciproques" [Transformation of figures analogous to the transformation by reciprocal vector rays]. Nouvelles Annales de Mathématiques. Ser. 3. 7: 133–147.
• Coelingh, Derk (1889). "Twee Cirkel-Transformaties" [Two Circle Transformations]. Nieuw Archief voor Wiskunde (in Dutch). 16 (2): 116–159.
• Rouché, Eugène; De Comberousse, Charles (1891). "3. App. VIII. Transformation par semi-droites réciproques" [Transformation by reciprocal semi-lines]. Traité de Géometrie [Treatise of Geometry] (in French) (6th ed.). Gauthier-Villars. pp. 298–308.
• Bricard, Raoul (1906). "Sur la géométrie de direction" [On the geometry of direction]. Nouvelles annales de mathématiques. Ser. 4. 6: 159–179.
• Bricard, Raoul (1906). "Sur la géométrie de direction dans l'espace" [On the geometry of direction in space]. Nouvelles annales de mathématiques. Ser. 4. 6: 433–454.
• Grünwald, Josef (1906). "Über duale Zahlen und ihre Anwendung in der Geometrie" [On dual numbers and their application in geometry]. Monatshefte für Mathematik und Physik (in German). 17: 81–136. doi:10.1007/BF01697639.
• Bricard, Raoul (1907). "Sur le problème d'Apollonius et sur quelques propriétés des cycles" [On the problem of Apollonius and on some properties of cycles]. Nouvelles annales de mathématiques. Ser. 4. 7: 491–506.
• Blaschke, Wilhelm (1910). "Untersuchungen über die Geometrie der Speere in der Euklidischen Ebene" [Studies on the geometry of spears in the Euclidean plane]. Monatshefte für Mathematik und Physik (in German). 21: 3–60. doi:10.1007/BF01693218.
• Blaschke, Wilhelm (1910). "Zur Geometrie der Speere im Euklidischen Raume" [On the geometry of spears in Euclidean space]. Monatshefte für Mathematik und Physik (in German). 21: 201–307. doi:10.1007/BF01693226. PDF
• Blaschke, Wilhelm (1911). "Über die Laguerresche Geometrie orientierter Geraden in der Ebene. I" [On the Laguerre geometry of oriented lines in the plane]. Archiv der Mathematik und Physik (in German). 18: 132–140.
• Coolidge, Julian Lowell (1916). "X. The Oriented Circle". A Treatise on the Circle and the Sphere. Clarendon. pp. 351–407.
• Kubota, Tadahiko (1919). "Note on Laguerre transformations". Tôhoku Mathematical Journal. 15: 227–231.
• Witwiński, Romuald (1923–1924). "La géométrie de direction" [Geometry of direction]. Prace Matematyczno-Fizyczne. 33 (1): 99–114.
• Blaschke, Wilhelm (1929). Vorlesungen über Differentialgeometrie III: Differentialgeometrie der Kreise und Kugeln [Lectures on differential geometry III: Differential Geometry of Circles and Spheres] (in German). Springer. doi:10.1007/978-3-642-50823-3.
• Müller, E.A. (1929). Krames, Josef Leopold (ed.). Vorlesungen über Darstellende Geometrie II: Die Zyklographie [Lectures on Descriptive Geometry II: Cyclography] (in German). Deuticke.
• Van der Waerden, Bartel Leendert; Smid, Lucas Johannes (1935). "Eine Axiomatik der Kreisgeometrie und der Laguerregeometrie" [An Axiomatization of Circle Geometry and Laguerre Geometry]. Mathematische Annalen (in German). 110 (1): 753–776. doi:10.1007/BF01448057.
• Arvesen, Ole Peder (1936). "Sur la solution de Laguerre du problème d'Apollonius". Comptes rendus hebdomadaires des séances de l'Académie des sciences. 203: 704–706.
• Arvesen, Ole Peder (1936). "Une application de la transformation par semi-droites réciproques,". Det Kongelige Norske Videnskabers Selskab Forhandlinger. 9: 13–15.
• Arvesen, Ole Peder (1939). "Sur les transformations par semi-droites réciproques". Norsk matematisk tidsskrift (in French). 21: 9–12.
• Blaschke, Wilhelm (1954). "2.38. Die Geradenabbildungen von Laguerre" [Laguerre's straight lines]. Analytische Geometrie [Analytic Geometry]. Birkhäuser. pp. 51–54.
• Yaglom, Isaak Moiseevich (2009) [1956]. Geometric Transformations IV: Circular Transformations. Translated by Shenitzer, Abe. Mathematical Association of America. Originally published as Геометрические преобразования, Vol. 2 (in Russian). Moscow: GITTL. 1956.
• Yaglom, Isaak Moiseevitch (1968) [1963]. Complex Numbers in Geometry. Translated by Primrose, Eric J.F. Academic Press. Originally published as Комплексные числа и их применение в геометрии (in Russian). Moscow: Fizmatgiz. 1963.
• Mäurer, Helmut (1966). "Laguerre- und Blaschke-Modell der ebenen Laguerre-Geometrie". Mathematische Annalen. 164: 124–132. doi:10.1007/BF01429050.
• Dubikajtis, L. (1967). "Un modèle hyperbolique de la géométrie plane de Laguerre". Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques. 15: 619–626.
• Yaglom, Isaak Moiseevitch (1979) [1969]. A Simple Non-Euclidean Geometry and Its Physical Basis. Translated by Shenitzer, Abe. Springer. Originally published as Принцип относительности Галилея и неевклидова геометрия (in Russian). Moscow: Nauka. 1969.
• Pedoe, Daniel (1972). "A Forgotten Geometrical Transformation". L'Enseignement Mathématique. 18: 255–267. doi:10.5169/seals-45376.
• Benz, Walter (1973). "§1.2 Laguerregeometrie". Vorlesungen über Geometrie der Algebren [Lectures on the Geometry of Algebras] (in German). Springer.
• Pedoe, Daniel (1975). "Laguerre's Axial Transformation". Mathematics Magazine. 48 (1): 23–30. doi:10.1080/0025570X.1975.11976432. JSTOR 2689289.
• Yaglom, Isaak Moiseevitch (1981). "On the circular transformations of Möbius, Laguerre, and Lie". In Davis, Chandler; Grünbaum, Branko; Sherk, F.A. (eds.). The Geometric Vein: The Coxeter Festschrift. Springer. pp. 345–353. doi:10.1007/978-1-4612-5648-9_25.
• Rigby, J. F. (1981). "The geometry of cycles, and generalized Laguerre inversion". In Davis, Chandler; Grünbaum, Branko; Sherk, F.A. (eds.). The Geometric Vein: The Coxeter Festschrift. Springer. pp. 355–378. doi:10.1007/978-1-4612-5648-9_26.
• Rigby, J. F. (1991). "Cycles and Tangent Rays". Mathematics magazine. 64 (3): 155–167. doi:10.1080/0025570X.1991.11977599. JSTOR 2691294.
• Fillmore, Jay P.; Springer, Arthur (1995). "New Euclidean theorems by the use of Laguerre transformations – Some geometry of Minkowski (2+1)-space" (PDF). Journal of Geometry. 52: 74–90. doi:10.1007/BF01406828.
• Pottmann, Helmut; Peternell, Martin (1998). "Applications of Laguerre geometry in CAGD" (PDF). Computer Aided Geometric Design. 15 (2): 165–186.
• Knight, Robert Dean (2000). Using Laguerre geometry to discover Euclidean theorems (Thesis). University of California, San Diego.
• Pottmann, Helmut; Wallner, Johannes (2001). "6.3.2 The Cyclographic Mapping and its Applications". Computational Line Geometry. Springer. pp. 366–383. doi:10.1007/978-3-642-04018-4_6.
• Havlicek, Hans (2004). Divisible designs, Laguerre geometry, and beyond (PDF). The summer school on Combinatorial Geometry and Optimisation, 4–10 June 2004, Brescia, Italy.
• Knight, Robert D. (2005). "The Apollonius contact problem and Lie contact geometry". Journal of Geometry. 83. doi:10.1007/s00022-005-0009-x.
• Knight, Robert D. (2008). "A Euclidean area theorem via isotropic projection". Journal of Geometry. 90 (1–2): 141–155. doi:10.1007/s00022-008-1936-0.
• Barrett, David E.; Bolt, Michael (2010). "Laguerre Arc Length from Distance Functions". Asian Journal of Mathematics. 14 (2): 213–234. doi:10.4310/AJM.2010.v14.n2.a3.
• Beluhov, Nikolai Ivanov (2013). "A Curious Geometric Transformation" (PDF). Journal of Classical Geometry. 2: 11–25.
• Masurel, Christophe (2017). "Inversion, Laguerre T.S.D.R., Euler polar tangential equation and d'Ocagne axial coordinates" (PDF).
• Bobenko, Alexander I.; Lutz, Carl O. R.; Pottmann, Helmut; Techter, Jan (2019). "Laguerre geometry in space forms and incircular nets" (PDF).
• Pacheco, Rui; Santos, Susana D. (2020). "Envelopes of circles and spacelike curves in the Lorentz–Minkowski 3-space". In Forum Mathematicum. 32 (3): 693–711. doi:10.1515/forum-2019-0092.