|WikiProject Mathematics||(Rated C-class, High-importance)|
"affine set" redirects here. An affine set is not explicitly defined anywhere in this article. A separate article for "affine set> should be created.
The semi-direct product formula as written in has a couple of problems: (a) format (doesn't display on my browser) (b) for those fussy about invariance, it chooses a basis.
Charles Matthews 08:24, 18 Aug 2003 (UTC)
Call for axioms
I am disappointed to see that this entry states no axioms. Affine geometry can be derived by adding a few axioms, and a primitive notion or two, to projective geometry.126.96.36.199 01:30, 15 March 2006 (UTC)
I agree, i found this definition: Affine geometry is the study of parallel lines which meet at infinity, where there is no perpendicular lines —Preceding unsigned comment added by IMBlackMath (talk • contribs) 04:59, 12 March 2009 (UTC)
I am an intelligent member of the public who wanted to find out about affine geometry. This article failed to inform me of what affine geometry is, as it is explained in terms of affine geometry and other geometries in a roundabout of unilluminating self-references.
This is an encyclopedia, not a text-book for the elite. An encyclopedia brings explanation to difficult ideas. This article failed to achieve that, and looks more like an exercise in 'what I know, and you don't know'. Please edit it to say what affine geometry is.--IvorJ 18:28, 29 September 2007 (UTC)
- I think the introduction is now much more illuminating: "In mathematics affine geometry is the study of geometric properties which remain unchanged by affine transformations, i.e. non-singular linear transformations and translations." Δεκλαν Δαφισ (talk) 22:11, 21 February 2009 (UTC)
An editor's efforts
My Revision of 15:53, 9 December 2007 is a significant improvement to the article on affine geometry. I don't understand why User:AlexFusco5 reverted my edits as vandalism. To be sure, I did significantly restructure the page. A few of my edits may be controversial and I would encourage the community to think about them and either revert or further improve them to accommodate all prior editors' contributions as well as extend them for content and clarity. But please revert the one or two items with which you disagree, this wholesale reversion by User:Alexfusco5 is frustrating as I spent a lot of time trying to make my contribution as incisive as possible. Looking at his significant contributions, I assume he noted the major changes and hastily jumped to the wrong conclusion. Since this is the first article that I have changed significantly, it may be that I missed some elements of Wikipedia protocol (e.g., I forgot to provide an edit summary and didn't I post in the discussion thread ahead of the edits). When I re-institute my edits, I will document them better. my home page] has my e-mail address if someone might care to apprise me of protocols which I haven't adduced as yet.
I think the neutral point of view is violated in the original article by statements such as this: "That may be desirable from a geometric point of view, rather than finding a heavy-handed proof using analytic geometry. But it's then a question of axiomatic study (so-called synthetic point of view)." My version took the ideas of the author of those sentiments and accommodated them, but changed the point of view to a more neutral stance. Certainly more could be done to improve the article, but as with everyone else, my time is limited.
Moreover, much of the writing in the original article is very hard to understand. I tried to preserve those editor's ideas while changing the words to be more clear. To the extent that more needs to be done, I invite others to help.
In addition, my edits have added axioms which a previous commenter indicated were missing. Finally, I hope my changes, will address the criticism of the page as made by User:IvorJ
Definition of affine space in terms of its operations
It seems to me that in the sentence "Affine space can also be viewed as a vector space with the subtraction and scalar multiplication operations" we should replace "with" with "without". However, I'm new to the subject, so I would like someone more knowledgeable to concur. Thanks! --JohnFries (talk) 19:45, 13 March 2008 (UTC)
- It has to be "with" because without those operations you wouldn't have any structure, just a set of points. However I don't see how replacing addition by subtraction makes any difference. Using subtraction alone one can define addition as x + y = x − ((x − x) − y). Using scalar multiplication it can be defined even faster as x + y = x − (-1)×y. What was the editor trying to say here? --Vaughan Pratt (talk) 17:11, 24 August 2008 (UTC)
I replaced the above-mentioned incorrect basis with one that does the job. (In fact this is the canonical such, being the clone or closed nest of operations of any complete basis for the affine sublanguage of linear algebra.) --Vaughan Pratt (talk) 00:31, 13 October 2008 (UTC)
I just made a change to the wording of the section on axioms. I'm not an expert on axioms of affine geometry, but those given here could not possibly describe anything but an affine plane, and only over an ordered field like the reals; therefore I restricted the scope of the claim. I'm all for a better formulation than the one I gave, but not the old one. So I think one should mention other ways of setting up axioms that would encompass affine geometry of arbitrary dimension over arbitrary fields. The easiest solution is to assume at the outset a vector space (which hides their axioms, and those of the base field) and just enumerate the basic properties of forming vectors between two points, and of adding vectors to points. That is really easy, but maybe not very satisfactory from a purely geometric point of view. With some effort one can probably replace all vector axioms by properties mentioning only points, but this might give a lot of mess. Anyway there must be books that give an axiomatic approach to general affine geometry. Marc van Leeuwen (talk) 14:50, 20 February 2009 (UTC)
- This axiomatization actually appears to have been copied verbatim from Coxeter, Introduction to Geometry, p. 192. I'll add a reference to Coxeter. It does imply the reals, but only because the axioms of ordered geometry stuff like Dedekind's axiom. I think it would be extremely ugly to replace the present axiomatization with a development in terms of a vector space. First off, Coxeter's axiomatization doesn't require any preexisting field. In fact, these axioms allow an affine parameter to be constructed from scratch. IMO it would be ugly and counter to the geometrical spirit to derive the geometry from the field rather than the other way around. Coxeter's axiomatization is very elegant, and has a minimal number of primitive concepts (basically just betweenness). The other reason I think your proposal would be ugly is that it takes a less general object (a vector space) and then makes it more general by erasing the singling out of the origin as a special point. The essential logic behind Coxeter's axiomatization is to show that affine geometry like projective geometry, but with a particular line removed (the line at infinity).--188.8.131.52 (talk) 03:07, 29 August 2009 (UTC)
- According to the Oxford English Dictionary, second supplement, affine in this sense comes from preserving "finiteness" (respecting the points at infinity in projective space, therefore) and can be found in Veblen & Young, 1918. Charles Matthews (talk) 08:11, 3 February 2010 (UTC)
This article should not rely on linear algebra for its description. Today a new section on "Projective view" was added to indicate the context of affine geometry. The lede will have to be reworked to make the article accurately reflect the place of affine geometry in mathematics.Rgdboer (talk) 20:23, 18 October 2011 (UTC)
- Please be more specific. The current lead says "Affine geometry can be developed on the basis of linear algebra" which is perfectly accurate. It does not have to be developed like that: there are axiomatic descriptions of affine geometry that never mention a field. Yet the translations in an affine space turn out to always be a vector space over some field, so in the end one arrives at the same resulting theory. The axiomatic approach can be considered more "purely geometric", but whether that is important is a matter of taste and pedagogical preferences. From a pragmatic point of view, since students are likely to know linear algebra anyway, the approach saying from the start that an affine space has a vector space attached to it can save a lot of time, and in fact manifests strong parallels between linear algebra an affine geometry. What should not be done, and it is done in various geometry related articles, is to assume implicitly that every affine space is coordinatized, and particular that there is an origin. In the current lead the phrase "affine transformations, i.e. non-singular linear transformations and translations" does that, and I fully agree that it should be changed (there is no such thing as a linear transformation of an affine space, and by the way for an affine transformation there is no requirement of non-singularity). But please don't start transforming the whole article/lead according to one particular approach; we should present a balanced view. Marc van Leeuwen (talk) 08:20, 19 October 2011 (UTC)
Thank you for the thoughful response. Yes, too much abstraction can make a subject intangible. The variety and facility of linear algebra must be acknowledged. "the translations in an affine space turn out to always be a vector space over some field", I contest. For instance, given a ring (mathematics) A and its group of units U there is an affine group of mappings on A generated by additions and multiplications. The case where A is a matrix ring is the linear algebra situation.Rgdboer (talk) 01:45, 18 November 2011 (UTC)
Investigating this reliance on linear algebra, there is the book by Daniel Hughes and Fred Piper, Projective Planes (1973) which is cited in the German version of "Affine geometry". On page 18 they state "studying projective geometry with linear algebra is much the easiest and most natural way to grasp the subject". This statement supports the approach taken. However, one may note the the German article is in Category:Synthetic geometry. In 1973 Hughes and Piper developed the subject in a rigorous fashion that recalls the reasoned nature of earlier, more geometric texts. The treatment of affine geometry by Hughes and Piper is integrated with the larger subject.Rgdboer (talk) 20:30, 2 December 2011 (UTC)
The analytic versus the synthetic approach
An encyclopedia article needs to be useful for the beginner as well as the expert. It seems to me that those who understand this article will not need it, and those who have studied only Euclidean geometry in school and simply want to know what 'affine geometry' is, will not understand it. A concise definition is that affine geometry is the study of parallelism: its axioms are Euclid's first second and fifth postulates. That should be stated clearly straightaway. In this sense, affine geometry is a subset of Euclidean geometry, not a "generalization".
There are two approaches to geometry: the analytic and the axiomatic. The emphasis in the article is too much biased toward the analytic approach. I think that is the problem.Ericlord (talk) 16:14, 7 February 2012 (UTC)
- I agree and have modified the introduction accordingly today.Rgdboer (talk) 23:34, 21 August 2012 (UTC)
The article affine space has been developed more recently and overlaps significantly with this article. Editors may want to compare the developments and someone may suggest a WP:Merger if appropriate. Signed Rgdboer (talk) 22:35, 30 June 2015 (UTC)
The comment(s) below were originally left at several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section., and are posted here for posterity. Following
|All sections need some expansion. More material needed on examples and applications. Differences between Euclidean and affine geometry should be mentioned. Tompw 18:34, 5 October 2006 (UTC)|
Last edited at 18:50, 14 April 2007 (UTC). Substituted at 01:43, 5 May 2016 (UTC)