81:
71:
53:
742:, 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
22:
785:
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.
556:, 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 Knowledge, and had to look for other sources for a basic introduction. –
598:
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. --
612:
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
597:
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
336:
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
532:(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).
767:
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
706:
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
750:
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
784:
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
768:
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
702:
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
699:, 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.
635:
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.
763:
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
Knowledge articles is really
670:
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
650:
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.
235:. 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. −
155:
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.
799:
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
1317:
569:
133:
738:
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,
803:
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. –
312:
568:
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
352:
While there's still risk of misinterpretation in providing two interpretations and not clearly distinguishing them, I propose that the conformal interpretation be removed. --
334:
444:
552:
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
2021:
625:
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.
488:
464:
406:
127:
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517:, and pretty much only covers Benz's work. This article is about Laguerre transformations treated algebraically as fractional linear transformations.
828:
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
1352:
1564:
103:
366:
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. --
2016:
1974:
490:
is simply an indirect Euclidean isometry. The lesson is don't trust everything on Knowledge, and this article would benefit from proofs. --
175:
If you want to produce the sort of visualisations that you see in the article, and have some basic Python knowledge, then see this:
94:
58:
710:
These are problems that many math articles on Knowledge share (not to mention many other mathematical sources), including e.g. the
841:
593:), but not under those names. Algebraic representations are given for both. The axial inversions are not covered. Actually, they
1424:
664:
864:
Here’s a list in chronological order of references related to Laguerre geometry / transformations. Feel free to add more. –
573:
518:
33:
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1985:
Pacheco, Rui; Santos, Susana D. (2020). "Envelopes of circles and spacelike curves in the Lorentz–Minkowski 3-space".
1220:
524:
It seems like there should also be some space on Knowledge for describing the basic concepts developed by Laguerre of
1268:
739:
590:
711:
680:
1456:
1590:
Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques
1085:"Sur les anticaustiques par réfraction de la parabole, les rayons incidents étant perpendiculaires à l'axe"
628:
Regarding basic definitions, those can indeed be provided upfront, instead of buried somewhere in the text.
1343:
1056:
925:
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1782:"New Euclidean theorems by the use of Laguerre transformations – Some geometry of Minkowski (2+1)-space"
688:
274:
39:
1362:
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652:
528:(semi-straights, i.e. directed lines, "spears" or "rays" in more recent German/English literature) and
80:
194:
1327:
656:
514:
21:
1841:
Pottmann, Helmut; Wallner, Johannes (2001). "6.3.2 The Cyclographic Mapping and its Applications".
1809:
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848:
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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1942:
1867:. The summer school on Combinatorial Geometry and Optimisation, 4–10 June 2004, Brescia, Italy.
1509:
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1167:
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911:
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833:
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1963:"Inversion, Laguerre T.S.D.R., Euler polar tangential equation and d'Ocagne axial coordinates"
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The apex of the triangle is (√2, √2 m), so both base and altitude have factor √2. The area is
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the conformal interpretation less interesting, and the article doesn't focus much on it. --
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968:
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613:
of the relation to high school geometry people are more familiar with, no diagrams, etc. –
537:
1962:
1615:
1134:"Transformation de figures analogue a la transformation par rayons vecteurs réciproques"
470:
followed by an orientation reversal. It turns out there is no orientation reversal, and
1606:
1400:
Vorlesungen über Differentialgeometrie III: Differentialgeometrie der Kreise und Kugeln
865:
845:
804:
773:
752:
719:
676:
632:
614:
577:
557:
510:
1554:
473:
449:
391:
2010:
1872:
Knight, Robert D. (2005). "The Apollonius contact problem and Lie contact geometry".
1244:
1199:
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572:, which surfaced some other old German/French sources and some relevant material at
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1765:
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Dubikajtis, L. (1967). "Un modèle hyperbolique de la géométrie plane de Laguerre".
1542:
1522:
1106:
Recherches sur la géométrie de direction: Methodes de transformation anticaustiques
543:
Recherches sur la géométrie de direction: Methodes de transformation anticaustiques
236:
1744:
1721:
1532:
1973:
Bobenko, Alexander I.; Lutz, Carl O. R.; Pottmann, Helmut; Techter, Jan (2019).
1850:
1654:
672:
99:
1932:
1915:
1893:
Knight, Robert D. (2008). "A Euclidean area theorem via isotropic projection".
1906:
1885:
1475:(1936). "Une application de la transformation par semi-droites réciproques,".
1407:
1072:
76:
1830:
894:
521:
is focused on an application to electromagnetic waves in special relativity.
1998:
920:
707:
rendering computer pictures) but should not be taken as a basic definition.
1645:
1628:
1269:"Untersuchungen über die Geometrie der Speere in der Euklidischen Ebene"
570:
ask this question about the 'power of a line' at the math reference desk
267:
The two interpretations of the Laguerre transformations are not the same
1800:
1772:
1695:
1580:
1444:
1312:
1284:
1236:
193:
There's now something similar for hyperbolic Laguerre transformations.
1328:"Über die Laguerresche Geometrie orientierter Geraden in der Ebene. I"
988:
591:
Laguerre transformations § Points, oriented lines and oriented circles
943:(in German). Coppenrathsche Buch- und und Kunsthandlung. pp. 227–229.
574:
Spherical wave transformation#Transformation by reciprocal directions
519:
Spherical wave transformation#Transformation by reciprocal directions
1529:. Translated by Shenitzer, Abe. Mathematical Association of America.
1249:"Sur le problème d'Apollonius et sur quelques propriétés des cycles"
1461:
Comptes rendus hebdomadaires des séances de l'Académie des sciences
195:
https://github.com/wlad-svennik/hyperbolic_laguerre_transformations
734:: I looked at Yaglom's book. In my opinion the entire section 2.9
1491:(1939). "Sur les transformations par semi-droites réciproques".
714:
article, which should really lead with some basic discussion of
1379:
505:
perhaps room for another article or two about Laguerre geometry
1709:"On the circular transformations of Möbius, Laguerre, and Lie"
15:
1565:"Laguerre- und Blaschke-Modell der ebenen Laguerre-Geometrie"
1429:"Eine Axiomatik der Kreisgeometrie und der Laguerregeometrie"
1735:. In Davis, Chandler; Grünbaum, Branko; Sherk, F.A. (eds.).
1732:"The geometry of cycles, and generalized Laguerre inversion"
1712:. In Davis, Chandler; Grünbaum, Branko; Sherk, F.A. (eds.).
1417:
Vorlesungen über Darstellende Geometrie II: Die Zyklographie
1168:"3. App. VIII. Transformation par semi-droites réciproques"
881:"10. Beweise einiger auf der Kugel Statt findenden Sätze"
513:
is about the incidence structure, was mostly copied from
177:
https://github.com/wlad-svennik/laguerre_transformations
1617:Принцип относительности Галилея и неевклидова геометрия
1477:
Det Kongelige Norske Videnskabers Selskab Forhandlinger
1221:"Über duale Zahlen und ihre Anwendung in der Geometrie"
1975:"Laguerre geometry in space forms and incircular nets"
1832:
Using Laguerre geometry to discover Euclidean theorems
1608:
A Simple Non-Euclidean Geometry and Its Physical Basis
1527:
Geometric Transformations IV: Circular Transformations
1457:"Sur la solution de Laguerre du problème d'Apollonius"
1174:(in French) (6th ed.). Gauthier-Villars. pp. 298–308.
1041:"Sur les courbes de direction de la troisième classe"
476:
452:
414:
394:
320:
277:
1117:"Sur l'hypercycle et la théorie des cycles polaires"
98:, a collaborative effort to improve the coverage of
1551:. Translated by Primrose, Eric J.F. Academic Press.
1829:
1415:Müller, E.A. (1929). Krames, Josef Leopold (ed.).
482:
458:
438:
400:
328:
306:
132:This article has not yet received a rating on the
1061:Proceedings of the Edinburgh Mathematical Society
1861:Divisible designs, Laguerre geometry, and beyond
1752:Rigby, J. F. (1991). "Cycles and Tangent Rays".
1297:"Zur Geometrie der Speere im Euklidischen Raume"
1166:Rouché, Eugène; De Comberousse, Charles (1891).
415:
1836:(Thesis). University of California, San Diego.
1059:(1884). "Radical Axes in Spherical Geometry".
1001:"Transformations par semi-droites réciproques"
1916:"Laguerre Arc Length from Distance Functions"
1556:Комплексные числа и их применение в геометрии
1204:"Sur la géométrie de direction dans l'espace"
1121:Bulletin de la Société Mathématique de France
977:Bulletin de la Société Mathématique de France
8:
1808:Pottmann, Helmut; Peternell, Martin (1998).
1810:"Applications of Laguerre geometry in CAGD"
1780:Fillmore, Jay P.; Springer, Arthur (1995).
1737:The Geometric Vein: The Coxeter Festschrift
1714:The Geometric Vein: The Coxeter Festschrift
1675:(1975). "Laguerre's Axial Transformation".
1510:"2.38. Die Geradenabbildungen von Laguerre"
740:Laguerre transformations § Line coordinates
736:"Dual Numbers as Oriented Lines of a Plane"
718:, but instead doesn't even link to there. –
47:
1931:
1914:Barrett, David E.; Bolt, Michael (2010).
1644:
1612:. Translated by Shenitzer, Abe. Springer.
475:
451:
413:
393:
322:
321:
319:
297:
296:
276:
1629:"A Forgotten Geometrical Transformation"
589:The cycles and spears are covered here (
1665:Vorlesungen über Geometrie der Algebren
1353:A Treatise on the Circle and the Sphere
665:transformation by reciprocal directions
271:There's the interpretation under which
49:
19:
1559:(in Russian). Moscow: Fizmatgiz. 1963.
953:"Sur la règle des signes en Géometrie"
548:(A book collecting Laguerre's papers.)
408:is a unitary dual-number matrix where
388:Earlier, it was claimed by me that if
2022:Unknown-priority mathematics articles
1301:Monatshefte für Mathematik und Physik
1273:Monatshefte für Mathematik und Physik
1225:Monatshefte für Mathematik und Physik
1111:(A book collecting the above papers.)
7:
1943:"A Curious Geometric Transformation"
1021:"Sur quelques propriétés des cycles"
92:This article is within the scope of
631:Also, some people might prefer the
515:Hartmann "Planar Circle Geometries"
307:{\displaystyle PGL(2,\mathbb {D} )}
212:Conformal section: Area of triangle
38:It is of interest to the following
1620:(in Russian). Moscow: Nauka. 1969.
1538:(in Russian). Moscow: GITTL. 1956.
1363:"Note on Laguerre transformations"
1253:Nouvelles annales de mathématiques
1208:Nouvelles annales de mathématiques
1188:Nouvelles annales de mathématiques
1138:Nouvelles Annales de Mathématiques
1089:Nouvelles Annales de Mathématiques
1045:Nouvelles Annales de Mathématiques
1025:Nouvelles Annales de Mathématiques
1005:Nouvelles Annales de Mathématiques
957:Nouvelles Annales de Mathématiques
14:
836:in 1832, proved in his 1835 book
112:Knowledge:WikiProject Mathematics
1941:Beluhov, Nikolai Ivanov (2013).
1425:Van der Waerden, Bartel Leendert
1378:Witwiński, Romuald (1923–1924).
1332:Archiv der Mathematik und Physik
907:"Lehrsätze und Aufgaben. No. 14"
842:User:Jacobolus/Gudermann Spharik
115:Template:WikiProject Mathematics
79:
69:
51:
20:
1817:Computer Aided Geometric Design
1427:; Smid, Lucas Johannes (1935).
1184:"Sur la géométrie de direction"
973:"Sur la géométrie de direction"
1766:10.1080/0025570X.1991.11977599
1689:10.1080/0025570X.1975.11976432
500:09:53, 12 September 2021 (UTC)
424:
418:
301:
287:
1:
1950:Journal of Classical Geometry
1534:Геометрические преобразования
941:Lehrbuch der niederen Sphärik
879:Gerwien, Karl Ludwig (1826).
838:Lehrbuch der niederen Sphärik
376:20:36, 21 December 2020 (UTC)
362:20:28, 21 December 2020 (UTC)
347:13:40, 18 December 2020 (UTC)
259:06:46, 18 December 2020 (UTC)
245:03:14, 18 December 2020 (UTC)
226:23:44, 17 December 2020 (UTC)
106:and see a list of open tasks.
2017:C-Class mathematics articles
1961:Masurel, Christophe (2017).
1920:Asian Journal of Mathematics
1828:Knight, Robert Dean (2000).
1745:10.1007/978-1-4612-5648-9_26
1722:10.1007/978-1-4612-5648-9_25
1151:"Twee Cirkel-Transformaties"
852:01:57, 9 February 2023 (UTC)
811:01:06, 4 February 2023 (UTC)
795:13:51, 3 February 2023 (UTC)
780:11:43, 3 February 2023 (UTC)
759:17:18, 2 February 2023 (UTC)
726:20:23, 28 January 2023 (UTC)
646:19:57, 28 January 2023 (UTC)
621:19:48, 28 January 2023 (UTC)
608:19:25, 28 January 2023 (UTC)
584:19:20, 27 January 2023 (UTC)
564:18:37, 27 January 2023 (UTC)
329:{\displaystyle \mathbb {D} }
1851:10.1007/978-3-642-04018-4_6
1843:Computational Line Geometry
1633:L'Enseignement Mathématique
1548:Complex Numbers in Geometry
1493:Norsk matematisk tidsskrift
1384:Prace Matematyczno-Fizyczne
1380:"La géométrie de direction"
1367:Tôhoku Mathematical Journal
1155:Nieuw Archief voor Wiskunde
468:indirect Euclidean isometry
2038:
1933:10.4310/AJM.2010.v14.n2.a3
439:{\displaystyle \det(V)=-1}
1907:10.1007/s00022-008-1936-0
1886:10.1007/s00022-005-0009-x
1845:. Springer. pp. 366–383.
1739:. Springer. pp. 355–378.
1716:. Springer. pp. 345–353.
1704:Yaglom, Isaak Moiseevitch
1602:Yaglom, Isaak Moiseevitch
1543:Yaglom, Isaak Moiseevitch
1408:10.1007/978-3-642-50823-3
1361:Kubota, Tadahiko (1919).
1356:. Clarendon. pp. 351–407.
1073:10.1017/S0013091500037305
872:01:15, 6 March 2023 (UTC)
772:or other related pages. –
207:10:17, 26 June 2022 (UTC)
189:09:38, 23 June 2020 (UTC)
166:13:02, 20 June 2020 (UTC)
131:
64:
46:
1660:"§1.2 Laguerregeometrie"
1614:Originally published as
1553:Originally published as
1531:Originally published as
1523:Yaglom, Isaak Moiseevich
1517:. Birkhäuser. pp. 51–54.
1348:"X. The Oriented Circle"
1219:Grünwald, Josef (1906).
1115:Dautheville, S. (1886).
895:10.1515/crll.1834.11.130
249:Oh yes, you're right. --
134:project's priority scale
1999:10.1515/forum-2019-0092
1858:Havlicek, Hans (2004).
1563:Mäurer, Helmut (1966).
1402:(in German). Springer.
1344:Coolidge, Julian Lowell
1149:Coelingh, Derk (1889).
1132:Coelingh, Derk (1888).
921:10.1515/crll.1832.9.100
689:skew-Hermitian matrices
95:WikiProject Mathematics
1667:(in German). Springer.
1419:(in German). Deuticke.
1007:. Ser. 3 (in French).
959:. Ser. 2 (in French).
889:(in German): 130–135.
681:Möbius transformations
484:
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330:
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28:This article is rated
1987:In Forum Mathematicum
1729:Rigby, J. F. (1981).
1569:Mathematische Annalen
1515:Analytische Geometrie
1433:Mathematische Annalen
832:in a problem sent to
712:Möbius transformation
485:
461:
441:
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331:
309:
1754:Mathematics magazine
1677:Mathematics Magazine
933:Gudermann, Christoph
903:Gudermann, Christoph
474:
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412:
392:
384:Embarrassing mistake
318:
275:
118:mathematics articles
1895:Journal of Geometry
1874:Journal of Geometry
1789:Journal of Geometry
1646:10.5169/seals-45376
1172:Traité de Géometrie
1109:. Gauthier-Villars.
830:Christoph Gudermann
693:Lie sphere geometry
661:inversion in a line
546:. Gauthier-Villars.
1801:10.1007/BF01406828
1581:10.1007/BF01429050
1489:Arvesen, Ole Peder
1473:Arvesen, Ole Peder
1453:Arvesen, Ole Peder
1445:10.1007/BF01448057
1313:10.1007/BF01693226
1285:10.1007/BF01693218
1237:10.1007/BF01697639
915:(in German): 102.
860:List of references
716:inversive geometry
554:inversive geometry
480:
456:
436:
398:
326:
304:
87:Mathematics portal
34:content assessment
1505:Blaschke, Wilhelm
1396:Blaschke, Wilhelm
1324:Blaschke, Wilhelm
1293:Blaschke, Wilhelm
1265:Blaschke, Wilhelm
1057:Allardice, Robert
989:10.24033/bsmf.207
840:, §§296–297. See
770:Laguerre geometry
653:Laguerre geometry
483:{\displaystyle V}
459:{\displaystyle V}
401:{\displaystyle V}
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1967:
1957:
1947:
1937:
1935:
1910:
1901:(1–2): 141–155.
1889:
1868:
1866:
1854:
1837:
1835:
1824:
1814:
1804:
1786:
1776:
1748:
1734:
1725:
1711:
1699:
1668:
1662:
1650:
1648:
1613:
1611:
1597:
1584:
1552:
1530:
1518:
1512:
1500:
1484:
1468:
1448:
1420:
1411:
1391:
1374:
1357:
1339:
1316:
1288:
1260:
1240:
1215:
1195:
1175:
1162:
1145:
1128:
1110:
1101:Laguerre, Edmond
1096:
1081:Laguerre, Edmond
1076:
1052:
1037:Laguerre, Edmond
1032:
1017:Laguerre, Edmond
1012:
997:Laguerre, Edmond
992:
969:Laguerre, Edmond
964:
949:Laguerre, Edmond
944:
924:
912:Crelle's Journal
898:
886:Crelle's Journal
834:Crelle's Journal
685:line coordinates
633:linear algebraic
547:
538:Laguerre, Edmond
489:
487:
486:
481:
465:
463:
462:
457:
445:
443:
442:
437:
407:
405:
404:
399:
335:
333:
332:
327:
325:
313:
311:
310:
305:
300:
120:
119:
116:
113:
110:
89:
84:
83:
73:
66:
65:
55:
48:
31:
25:
24:
16:
2037:
2036:
2032:
2031:
2030:
2028:
2027:
2026:
2007:
2006:
2005:
1984:
1977:
1972:
1965:
1960:
1945:
1940:
1913:
1892:
1871:
1864:
1857:
1840:
1827:
1812:
1807:
1784:
1779:
1751:
1728:
1702:
1671:
1653:
1623:
1600:
1587:
1562:
1541:
1521:
1503:
1487:
1471:
1451:
1423:
1414:
1394:
1377:
1360:
1342:
1322:
1291:
1263:
1243:
1218:
1198:
1178:
1165:
1148:
1131:
1114:
1099:
1079:
1055:
1035:
1015:
995:
967:
947:
931:
901:
878:
862:
697:Minkowski space
677:complex numbers
667:, or the like.)
657:power of a line
536:
509:The article at
507:
472:
471:
448:
447:
410:
409:
390:
389:
386:
316:
315:
273:
272:
269:
214:
173:
153:
117:
114:
111:
108:
107:
85:
78:
32:on Knowledge's
29:
12:
11:
5:
2035:
2033:
2025:
2024:
2019:
2009:
2008:
2004:
2003:
1993:(3): 693–711.
1982:
1970:
1958:
1938:
1926:(2): 213–234.
1911:
1890:
1869:
1855:
1838:
1825:
1805:
1777:
1760:(3): 155–167.
1749:
1726:
1700:
1669:
1651:
1621:
1598:
1585:
1560:
1539:
1519:
1501:
1485:
1469:
1449:
1439:(1): 753–776.
1421:
1412:
1392:
1375:
1358:
1340:
1320:
1289:
1261:
1245:Bricard, Raoul
1241:
1216:
1200:Bricard, Raoul
1196:
1180:Bricard, Raoul
1176:
1163:
1146:
1129:
1112:
1097:
1077:
1053:
1033:
1013:
993:
965:
945:
929:
899:
875:
861:
858:
856:
826:
825:
824:
823:
822:
821:
820:
819:
818:
817:
816:
815:
814:
813:
801:
765:
728:
708:
704:
700:
668:
629:
626:
550:
549:
511:Laguerre plane
506:
503:
479:
466:represents an
455:
435:
432:
429:
426:
423:
420:
417:
397:
385:
382:
381:
380:
379:
378:
324:
303:
299:
295:
292:
289:
286:
283:
280:
268:
265:
264:
263:
262:
261:
213:
210:
172:
169:
152:
149:
146:
145:
142:
141:
138:
137:
130:
124:
123:
121:
104:the discussion
91:
90:
74:
62:
61:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
2034:
2023:
2020:
2018:
2015:
2014:
2012:
2000:
1996:
1992:
1988:
1983:
1976:
1971:
1964:
1959:
1955:
1951:
1944:
1939:
1934:
1929:
1925:
1921:
1917:
1912:
1908:
1904:
1900:
1896:
1891:
1887:
1883:
1879:
1875:
1870:
1863:
1862:
1856:
1852:
1848:
1844:
1839:
1834:
1833:
1826:
1823:(2): 165–186.
1822:
1818:
1811:
1806:
1802:
1798:
1794:
1790:
1783:
1778:
1774:
1771:
1767:
1763:
1759:
1755:
1750:
1746:
1742:
1738:
1733:
1727:
1723:
1719:
1715:
1710:
1705:
1701:
1697:
1694:
1690:
1686:
1682:
1678:
1674:
1673:Pedoe, Daniel
1670:
1666:
1661:
1656:
1652:
1647:
1642:
1638:
1634:
1630:
1626:
1625:Pedoe, Daniel
1622:
1619:
1618:
1610:
1609:
1603:
1599:
1595:
1591:
1586:
1582:
1578:
1574:
1570:
1566:
1561:
1558:
1557:
1550:
1549:
1544:
1540:
1537:
1535:
1528:
1524:
1520:
1516:
1511:
1506:
1502:
1498:
1495:(in French).
1494:
1490:
1486:
1482:
1478:
1474:
1470:
1466:
1462:
1458:
1454:
1450:
1446:
1442:
1438:
1435:(in German).
1434:
1430:
1426:
1422:
1418:
1413:
1409:
1405:
1401:
1397:
1393:
1389:
1385:
1381:
1376:
1372:
1368:
1364:
1359:
1355:
1354:
1349:
1345:
1341:
1337:
1334:(in German).
1333:
1329:
1325:
1321:
1319:
1314:
1310:
1306:
1303:(in German).
1302:
1298:
1294:
1290:
1286:
1282:
1278:
1275:(in German).
1274:
1270:
1266:
1262:
1258:
1254:
1250:
1246:
1242:
1238:
1234:
1230:
1227:(in German).
1226:
1222:
1217:
1213:
1209:
1205:
1201:
1197:
1193:
1189:
1185:
1181:
1177:
1173:
1169:
1164:
1161:(2): 116–159.
1160:
1156:
1152:
1147:
1143:
1139:
1135:
1130:
1126:
1122:
1118:
1113:
1108:
1107:
1102:
1098:
1094:
1090:
1086:
1082:
1078:
1074:
1070:
1066:
1062:
1058:
1054:
1050:
1046:
1042:
1038:
1034:
1030:
1026:
1022:
1018:
1014:
1010:
1006:
1002:
998:
994:
990:
986:
982:
979:(in French).
978:
974:
970:
966:
962:
958:
954:
950:
946:
942:
938:
934:
930:
927:
922:
918:
914:
913:
908:
904:
900:
896:
892:
888:
887:
882:
877:
876:
874:
873:
870:
867:
859:
857:
854:
853:
850:
847:
843:
839:
835:
831:
812:
809:
806:
802:
798:
797:
796:
792:
788:
783:
782:
781:
778:
775:
771:
766:
762:
761:
760:
757:
754:
749:
745:
741:
737:
733:
729:
727:
724:
721:
717:
713:
709:
705:
701:
698:
694:
690:
686:
682:
678:
674:
669:
666:
662:
658:
654:
649:
648:
647:
643:
639:
634:
630:
627:
624:
623:
622:
619:
616:
611:
610:
609:
605:
601:
596:
592:
588:
587:
586:
585:
582:
579:
575:
571:
566:
565:
562:
559:
555:
545:
544:
539:
535:
534:
533:
531:
527:
522:
520:
516:
512:
504:
502:
501:
497:
493:
477:
469:
453:
433:
430:
427:
421:
395:
383:
377:
373:
369:
365:
364:
363:
359:
355:
351:
350:
349:
348:
344:
340:
293:
290:
284:
281:
278:
266:
260:
256:
252:
248:
247:
246:
242:
238:
234:
230:
229:
228:
227:
223:
219:
211:
209:
208:
204:
200:
196:
191:
190:
186:
182:
178:
170:
168:
167:
163:
159:
150:
135:
129:
126:
125:
122:
105:
101:
97:
96:
88:
82:
77:
75:
72:
68:
67:
63:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
1990:
1986:
1953:
1949:
1923:
1919:
1898:
1894:
1877:
1873:
1860:
1842:
1831:
1820:
1816:
1792:
1788:
1757:
1753:
1736:
1713:
1683:(1): 23–30.
1680:
1676:
1664:
1655:Benz, Walter
1636:
1632:
1616:
1607:
1593:
1589:
1572:
1568:
1555:
1547:
1533:
1526:
1514:
1496:
1492:
1480:
1476:
1464:
1460:
1436:
1432:
1416:
1399:
1390:(1): 99–114.
1387:
1383:
1370:
1366:
1351:
1335:
1331:
1304:
1300:
1276:
1272:
1256:
1252:
1228:
1224:
1211:
1207:
1191:
1187:
1171:
1158:
1157:(in Dutch).
1154:
1141:
1137:
1124:
1120:
1105:
1092:
1088:
1064:
1060:
1048:
1044:
1028:
1024:
1008:
1004:
980:
976:
960:
956:
940:
937:"§§ 296–297"
910:
884:
863:
855:
837:
827:
747:
743:
673:dual numbers
594:
567:
551:
542:
529:
526:semi-droites
525:
523:
508:
467:
387:
270:
232:
215:
192:
174:
154:
93:
40:WikiProjects
1639:: 255–267.
1575:: 124–132.
1307:: 201–307.
983:: 196–208.
109:Mathematics
100:mathematics
59:Mathematics
2011:Categories
1596:: 619–626.
1467:: 704–706.
1373:: 227–231.
1338:: 132–140.
1259:: 491–506.
1255:. Ser. 4.
1231:: 81–136.
1214:: 433–454.
1210:. Ser. 4.
1194:: 159–179.
1190:. Ser. 4.
1144:: 133–147.
1140:. Ser. 3.
1091:. Ser. 3.
1047:. Ser. 3.
1027:. Ser. 3.
1011:: 542–556.
963:: 175–180.
1795:: 74–90.
1604:(1979) .
1545:(1968) .
1525:(2009) .
1067:: 59–61.
1051:: 97–109.
866:jacobolus
846:jacobolus
805:jacobolus
774:jacobolus
753:jacobolus
720:jacobolus
615:jacobolus
578:jacobolus
558:jacobolus
1956:: 11–25.
1706:(1981).
1657:(1973).
1627:(1972).
1536:, Vol. 2
1507:(1954).
1483:: 13–15.
1455:(1936).
1398:(1929).
1346:(1916).
1326:(1911).
1295:(1910).
1279:: 3–60.
1267:(1910).
1247:(1907).
1202:(1906).
1182:(1906).
1127:: 45–67.
1103:(1885).
1083:(1885).
1039:(1883).
1031:: 65–74.
1019:(1883).
999:(1882).
971:(1880).
951:(1870).
935:(1835).
926:Figure 4
905:(1832).
540:(1885).
171:Software
151:Learning
1773:2691294
1696:2689289
1499:: 9–12.
1095:: 5–16.
800:better.
787:Svennik
764:tricky.
751:axis. –
732:Svennik
638:Svennik
600:Svennik
492:Svennik
368:Svennik
354:Svennik
339:Svennik
314:(where
251:Svennik
237:Rgdboer
218:Svennik
199:Svennik
181:Svennik
158:Svennik
30:C-class
703:plane.
530:cycles
36:scale.
1978:(PDF)
1966:(PDF)
1946:(PDF)
1865:(PDF)
1813:(PDF)
1785:(PDF)
1770:JSTOR
1693:JSTOR
446:then
791:talk
746:and
642:talk
604:talk
496:talk
372:talk
358:talk
343:talk
255:talk
241:talk
222:talk
203:talk
185:talk
162:talk
1995:doi
1928:doi
1903:doi
1882:doi
1847:doi
1797:doi
1762:doi
1741:doi
1718:doi
1685:doi
1641:doi
1577:doi
1573:164
1465:203
1441:doi
1437:110
1404:doi
1318:PDF
1309:doi
1281:doi
1233:doi
1069:doi
985:doi
917:doi
891:doi
869:(t)
849:(t)
844:. –
808:(t)
777:(t)
756:(t)
723:(t)
663:or
618:(t)
595:are
581:(t)
576:. –
561:(t)
416:det
128:???
2013::
1991:32
1989:.
1952:.
1948:.
1924:14
1922:.
1918:.
1899:90
1897:.
1880:.
1878:83
1876:.
1821:15
1819:.
1815:.
1793:52
1791:.
1787:.
1768:.
1758:64
1756:.
1691:.
1681:48
1679:.
1663:.
1637:18
1635:.
1631:.
1594:15
1592:.
1571:.
1567:.
1513:.
1497:21
1479:.
1463:.
1459:.
1431:.
1388:33
1386:.
1382:.
1371:15
1369:.
1365:.
1350:.
1336:18
1330:.
1305:21
1299:.
1277:21
1271:.
1251:.
1229:17
1223:.
1206:.
1186:.
1170:.
1159:16
1153:.
1136:.
1125:14
1123:.
1119:.
1087:.
1063:.
1043:.
1023:.
1003:.
975:.
955:.
939:.
909:.
883:.
793:)
695:,
691:,
687:,
683:,
679:,
675:,
659:,
655:,
644:)
606:)
498:)
431:−
374:)
360:)
345:)
257:)
243:)
224:)
216:--
205:)
197:--
187:)
179:--
164:)
156:--
2001:.
1997::
1980:.
1968:.
1954:2
1936:.
1930::
1909:.
1905::
1888:.
1884::
1853:.
1849::
1803:.
1799::
1775:.
1764::
1747:.
1743::
1724:.
1720::
1698:.
1687::
1649:.
1643::
1583:.
1579::
1481:9
1447:.
1443::
1410:.
1406::
1315:.
1311::
1287:.
1283::
1257:7
1239:.
1235::
1212:6
1192:6
1142:7
1093:4
1075:.
1071::
1065:3
1049:2
1029:2
1009:1
991:.
987::
981:8
961:9
928:.
923:.
919::
897:.
893::
789:(
748:y
744:x
730:@
640:(
602:(
494:(
478:V
454:V
434:1
428:=
425:)
422:V
419:(
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294:,
291:2
288:(
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282:G
279:P
253:(
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220:(
201:(
183:(
160:(
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42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
