Talk:Laguerre plane

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This seems to be a somewhat limited/limiting take on the subject

Laguerre's original research was about oriented circles and lines in the plane (later generalized to oriented hyperspheres/hyperplanes). Such a structure seems to be of independent interest, with e.g. applications in CAGD and analogies to special relativity. This particular article seem to instead be entirely from Benz's point of view (presumably including this model with the parabolas; I don’t read German and haven’t looked at Benz’s book), ignoring Laguerre and others following him.

It seems to me that the original structure and context should be put into some primary position in this page, with an alternate model sharing the same incidence structure described afterwards in a section. The relation between these incidence structures per se can be discussed at leisure in Benz plane.

But I am not an expert in this subject. Does anyone have a list of good sources? –jacobolus (t) 00:45, 19 January 2023 (UTC)[reply]

That's true. The original definition uses oriented lines ("Speere","Fährten") and oriented circles ("Zykel") (see BENZ: Geometrie der Algebren, p. 11). I can add a hint to this model. I introduced here the most common description, which is similar to that of a Möbius-plane (geometry of circles in the plane/ on a sphere) and a Minkowski-plane (geometry of hyperbolas in the plane/ circles on a hyperboloid). Ag2gaeh (talk) 09:13, 19 January 2023 (UTC)[reply]

There is some discussion of this topic in

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.

For more Helmut & Wallner recommend Coolidge:

Coolidge, Julian Lowell (1916). "X. The Oriented Circle". A Treatise on the Circle and the Sphere. Clarendon. pp. 351–407.

and also Müller & Krames (1929) Vorlesungen über Darstellende Geometrie II: Die Zyklographie, as well as several papers about specific applications e.g. to CAGD. –jacobolus (t) 00:20, 20 January 2023 (UTC)[reply]

Knight's (2000) UCSD PhD dissertation looks like a great exposition, but unfortunately only the first few pages are available online, and I don't want to pay $40 to ProQuest for a copy. –jacobolus (t) 08:48, 27 January 2023 (UTC)[reply]

A few more sources:

Pedoe, Daniel (1972). "A Forgotten Geometrical Transformation". L'Enseignement Mathématique. 18: 255–267. doi:10.5169/seals-45376.

Pedoe, Daniel (1975). "Laguerre's Axial Transformation". Mathematics Magazine. 48 (1): 23–30. doi:10.1080/0025570X.1975.11976432.

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.

–jacobolus (t) 17:29, 27 January 2023 (UTC)[reply]