Talk:Projective geometry

message: We need some main concrete elements in this article:
* some mathematical content about "What" is projective geometry. i.e. perhaps the basic axioms of projective geometry, and some important theorems. In particular, the Fundamental Theorem of Projective Geometry. Possibly some words about pole and palar, pascal hexagram stuff, degauss's theorem, and the projective coordinates.
* some history of projective geometry. (i.e. its origin from artists wanting a realistic drawing, now know as 'descriptive geometry'.)
* modern projective geometry: higher dimensions and other abstract development that has nothing to do with "projections". Lattice theory...? and projective geometry and crytography...
 Xah P0lyglut 14:36, 2003 Dec 13 (UTC)
We have already projective plane, projective space, homogeneous coordinates, Desargues' theorem, Möbius transformation, incidence (mathematics), all of which relate in some way to projective geometry.
How about an article about cross-ratio?
Charles Matthews 16:08, 13 Dec 2003 (UTC)
thread_title: Main elements

message: A line in R3 can be represented in P2 space by the equation ax+by+c=0. If we treat a, b and c as the column vector l and x, y,1 as the column vector x then the equation for the line can be written in matrix form as[...].
Yes I agree, confusion, I think it is now better. I think if I was now asked in an exam "What is P2 space"? I would write "the set of subspaces that can be defined in R3", and leave it at that, I would probable fail. I think it is going to be very hard to write a good article on this topic, this is where art, engineering and maths meet. Charles Esson 00:22, 22 October 2006 (UTC)
thread_title: What is that supposed to mean?

message: I don't know, in an effort to plug all the holes mathematics has been turned into a lot of mumbo jumbo. I made an attempt to describe the line at infinity in P2 space. Perhaps this is not the place.
I see that this page has been marked as important for mathematics so I will expand my argument. I added the section 'Visualize P2' in an attempt to show what I meant by "make the subject accessible".
I will use the article "introduction to projective line" to try and show what I see as the problem. The introduction to that article reads.
"In mathematics, a projective line is a one-dimensional projective space. The projective line over a field K, denoted P1(K), may be defined as the set of one-dimensional subspaces of the two-dimensional vector space K2 (it does carry other geometric structures). The projective line may also be thought of as the line K together with an idealised point at infinity."
To break that apart 
"In mathematics, a projective line is a one-dimensional projective space." Yes you have to say that.
"The projective line over a field K" Yes and like your average reader has studied modern algebra.
"denoted P1(K)" No doubt in some texts.
"may be defined as the set of one-dimensional subspaces of the two-dimensional vector space K2" and I would have thought of K3 and K4.....,what does this add?
"(it does carry other geometric structures)." And they are? I think the one dimensional subspace bit has become a lot more important to mathematicians than any geometric interpretation.
I would argue that this page should be an introduction that stays away from material that will only be understood my someone who has done a maths major and that the sub pages ( projection line, projection plane, cross product) should go deeper. It is too an important topic to have it hide behind language only understood by a very small group.
Rant mode off. Charles Esson 23:58, 21 October 2006 (UTC)
thread_title: shouldn't some effort be made to make the subject accessible

message: Axioms are the foundationa used by mathamaticians to describe the structure, not the structure. I think they belong
in a section of their own marked axioms, not in a section called description.
 
(Veblen and Young 1938, Kasner and Newman 1989 from http://mathworld.wolfram.com/ProjectiveGeometry.html, it is interesting
to note that that wolfran chickened out and gave this set of axioms and nothing more).
*1. CA If A and B are distinct points on a plane, there is at least one line containing both A and B.
*2. CA If A and B are distinct points on a plane, there is not more than one line containing both A and B.
*3. _A Any two lines in a plane have at least one point of the plane (which may be the point at infinity) in common.
*4. _ There is at least one line on a plane.
*5. CA Every line contains at least three points of the plane.
*6. C All the points of the plane do not belong to the same line
 
H.S,M Coxter Gives eight.
*1. V There exists a point and a line that are not incident. ( same as 6)
*2. V Every line is incident with at least three distinct points. ( same as 5)
*3. V Any two distinct points are incident with just one line. (same as 1&2)
*4. _ If A,B,C,D are four distinct points such that AB meets CD, then AC meets BD. ( new)
*5. _ If ABC is a plane, there is at least one point not in the plane ABC. ( new)
*6. _ Any two distinct planes have at least two common points. (new)
*7. _ The three diagonal points of a complex quadrangle are never collinear. ( and this is self evident? )
*8. _ If a projectivity leaves invariant each of three distinct points on a line, it leaves invariarnt every point on the line.
Coxter is however honest and states "The precise choice is a mattter of taste"
This article starts with three
* V G1: Every line contains at least 3 points ( same as 5 in young)
* V G2: Every two points, A and B, lie on a unique line, AB. ( same as 1 and 2 in young)
* C G3: If lines AB and CD intersect, then so do lines AC and BD (where it is assumed that A and D are distinct from B and C). ( Coxter's 4)
Albrecht gives four.
*VCT For any two distict points P and Q there is exactly line that is incident with P and Q. ( 1 and 2 of Veblem)
*_CT  Let A,B,C and D be four points such that AB intersect CD, Tne AC also intersect the line BD. ( 4 of Coxter Albrecht attributes it to Veblen)
Albrecht says this is an ingenious way to say that any two lines in a plane meet before knowing what a plane is and that this axion is sometimes atributed to Pasch but his version is different. Albrecht replaces 2 with -any two lines have a point in common, that is 3 of Veblen. 
*VCT Any line is incident with at least three points. ( seems a common theme)
*___There are at least two lines. (Veblen 4 and 6 because 4 gives you a line and 6 says you have to have another)
Seems to me Albrecht has done well, knocking Veblen and Young from 6 back to 4. The question; what is Coxter on about? has he seen something Albrecht, Veblena nd Young missed or is he just verbose. Charles Esson 22:11, 24 October 2006 (UTC)
replies: text: Coxeter (projective geometry), Hilbert (Geometry and the imagination) and Greenberg (Euclidean and non-Euclidean geometry) all give different sets of axioms too. Apparently, Bachmann used yet another set based on incidence, reflection and orthogonality. In a mathematical sense Coxeter is correct in saying that it is a matter of taste, since all these formulations are logically equivalent. Within certain limits, one may pick and choose which propositions one wishes to state as axioms, and which one then goes on to derive from them - all the various axiom sets of PG may be derived as theorems from any other set. But in a philosophical sense Coxeter misses an important point - some ideas are more fundamental than others, usually the ones that may be expressed most simply. It is an important philosophical issue to identify the fundamental ones. AFAIK this remains an open problem. -- Steelpillow (talk) 21:56, 26 April 2008 (UTC)
replies: text: I have now split off the axioms into a separate section, and added some fragments of the above. Could probably still be improved on. -- Steelpillow (talk) 12:16, 3 May 2008 (UTC)
thread_title: Notes Axioms of projective geometry

message: Can't write an article if you don't know what it is about.
Algebraic projective geometry redirects here; maybe the headings can help us sort it all out.
Projective Geometry Albecht Beuelspacher and Ute Rosenbaun 2000 cambridge press.
For David Hibert a geometry is a collection of theorems that follow from its axiom system. Albecht side stepped the issue stating " It is an extremely good language to describe a multitude of phenomena inside and outside of mathematics".
Ok so you can abstract something to the point were it is of little value. Lets go back to Algebraic Projective Geometry by J.G. Semple first published in 1952.
A summary: A collection of axioms and theorems makes many geometries possible, we like euclidian geometry because we can use it to build things. We like projective geometry because it is more symmetrical because of duality and because homogeneous coordinates makes the show linear. Projective geometry transforms conics into conics. Projective geometry can be converted to euclidian geometry by introducing the line at infinity and the circular points.
To quote ( page 7):
"We thus have two geometries, projective geometry and euclidian geometry, which fit naturally together and between them include most of the classical geometric theorems." To complete the set you need to consider affine geometry. 
After the introduction the wikipedia article "affine geometry" has a section called "Intuitive background" perhaps that is the way to go.
Projective Geometry H.S.M. Coxeter 1974
The heading looked promising: What is Projective Geometry Summary: If you discard the compass and use the straight edge what remains; projective geometry.
Ok lets be shot of the mathamatic books. They have lost the plot.
An invitation to 3D  Yi MA etc.
"Perspective projection with its roots tracing back to ancient Greek Philosophers and Renaissance artists has been widely studied in projective geometry ( a branch of algebra in mathematics)".
So do go down that path; present projective geometry as a branch of algebra  and Perspective projection as one of the many possible applications. I think that is the right thing to do. We still have problems. You could look at projective geometry as part of modern algebra ( you have no idea how dry that is) or part of linear algebra which is a little more interesting; and much more used.
Ramblings by Charles Esson 22:12, 23 October 2006 (UTC)
thread_title: What is projective Geometry?

message: Ideas welcome
Describe projective space as per Geometry of multiple Images Oliver Faugeras page 78.
State that projective geometry can be built up  Syntheticly or Algebraicly
Synthetic long histroy, couple of milestone, link to perspective art.
Algebraic Homogeneous coordinates, who when. 
*projective plane
*Axiomatic projective space
*Real projective plane
*Complex projective plane
* Erlangen program
Charles Esson 12:40, 27 October 2006 (UTC)
replies: text: I think that the rigorous synthetic, algebraic, and group-theoretic approaches are best for later. I'd prefer that the introduction focus less on rigor and more on intuition. Specifically, that the points and lines of projective geometry can be modeled as lines and planes through the origin in R3, that this matches human visual perception (with the eye at the origin), that the points and lines of the usual Euclidean plane can be embedded into the projective plane by placing the Euclidean plane as a plane in R3 that does not go through the origin and then looking at it from the origin, and that projective geometry models the familiar visual phenomena that the horizon looks like a line and that lines that are parallel in R3 are seen to meet at a vanishing point on the horizon. —David Eppstein 00:28, 28 October 2006 (UTC)
thread_title: Proposed intoduction

message: There is no wikipedia article on planar geometry. That would help with sorting it all out.
—The preceding unsigned comment was added by Charles Esson (talk • contribs) 21:54, 23 October 2006 (UTC)
replies: text: There isn't? What's wrong with Plane (mathematics) or Euclidean plane? —David Eppstein 22:09, 23 October 2006 (UTC)
replies: text: Don't think they really cover the topic, Planer Geometry is about transformations of the plane; the reason why I think it would help here is you could move the P2 stuff I wrote ( which is an example of a Planer Geometry ) to Planar Geometry reference it and aim this article directly at the mathematics.Charles Esson 22:22, 23 October 2006 (UTC)
replies: text: There is a lot more to geometry than transformations. But if you want transformations of the plane, there's Euclidean plane isometry, Projective transformation, Möbius transformation, Inversion (geometry), or more generally Transformation (mathematics). —David Eppstein 22:37, 23 October 2006 (UTC)
replies: text: Yes I see your point. Also found 3D projection. My problem; if you look at Perspective (graphical) no mention is made of the mathematics. If Projective geometry is left to the mathematicians you get the description this article enjoys. The trouble is I am no computer science professor, but I note you are -:) Charles Esson 12:57, 24 October 2006 (UTC)
thread_title: Planar Geometry

message: Is this correct: "P2 is used to map a plane into a plane" ? AFAK, P2 is the projective plane (in contrast to the real Euclidean plane R2). Am I wrong? [sorry about the lazy font] -- Steelpillow (talk) 12:27, 2 May 2008 (UTC)
replies: text: Changed it to what I believe to be correct. Feel free to educate me. -- Steelpillow (talk) 21:24, 9 May 2008 (UTC)
thread_title: Visualising P<sup>2</sup>

message: Discussion moved to Talk:Projective_space#Projective_geometry. -- Steelpillow (talk) 21:39, 9 May 2008 (UTC)
thread_title: Axioms of projective geometry

message: The introduction says that "Projective geometry grew out of the principles of perspective art established during the Renaissance period, and was first systematically developed by Desargues in the 17th century..." This has no references or citations. Although it is a very nice sentence, without being verified it needs to be removed. Unless someone can come up with a verifiable source for this quote in the next 21 days then I propose that it be removed. Any objections?  Δεκλαν Δαφισ   (talk)  23:57, 16 October 2008 (UTC)
replies: text: I have waited almost two months and there were no objections so I have made the edit.  Δεκλαν Δαφισ   (talk)  20:04, 11 December 2008 (UTC)
replies: text: Well, you shouldn't make it a minor edit, anyway. And it didn't need to be removed. The history is not that controversial, and could certainly help someone locate the subject, knowing nothing about it. Charles Matthews (talk) 21:59, 11 December 2008 (UTC)
replies: text:  Wikipedia is supposed to contain verifiable fact from reliable sources, and not supposition. Since no reference was given for the statement it is not verifiable. In fact, as a mathematician working in affine and projective geometry I have never heard that "Projective geometry grew out of the principles of perspective art established during the Renaissance period", so it isn't even common knowledge. Would you prefer that Wikipedia's pages be filled with unverifiable tit-bits? If you can find a reference for the statement then please feel free to re-introduce the statement along with its verifiable source.  Δεκλαν Δαφισ   (talk)  11:13, 12 December 2008 (UTC)
replies: text:  p.s. You are right: my last edit of the article did not constitute a minor edit. I have re-read the guidelines and shall follow them more closely in the future.  Δεκλαν Δαφισ   (talk)  11:16, 12 December 2008 (UTC)
replies: text: You were justified in removing this after a citation needed template had been sitting there for quite a while. I have improved the wording and given a reference. The first book on perspective geometry, by Desargues, was actually called La Perspectiva, by the way!hgilbert (talk) 16:26, 12 December 2008 (UTC)
thread_title: Nice and flowery, but where are the facts?

message: With regard to the edit  which removed some unferenced claims, does anyone actually have any verifyable sources for this information? It would be quite nice if someone did.  Δεκλαν Δαφισ   (talk)  12:04, 2 February 2009 (UTC)
thread_title:  Unferenced details 

message: The discussion of angles in the introduction is misleading.  The claim that angles are not preserved under projective transformations is incorrect in a setting where angles make sense, for instance in real projective geometry.  The reference to perspective drawing allegedly "proving" this is confusing.  It would be much better to mention length as a basic Euclidean invariant not preserved by projective transformations. Tkuvho (talk) 04:59, 21 June 2010 (UTC)
replies: text: Well, the discussion above shows that it was thought to be an improvement to include a discussion of non-invariance early on, to reduce the abstraction. Since angles aren't even invariant under affine transformations, in a clear-cut sense, any changes ought to be well explained. Charles Matthews (talk) 08:16, 21 June 2010 (UTC)
replies: text: You are right, I was thinking of Mobius transformations as operating on the upperhalf plane, when the angles are preserved.  Never mind. Tkuvho (talk) 12:19, 21 June 2010 (UTC)
thread_title:  angle 

message: Substituted at 02:30, 5 May 2016 (UTC)
thread_title: Assessment comment

message: I'm unsure if this merits inclusion somewhere in the article but I first came across projective Geometry while at a Steiner School at about age 12-14 (don't remember exactly).
It was taught initially from the perspective (ho ho) of artists, vanishing points at infinity etc then proceeded to delve deeper into the mathematics.
I still have some gloriously drawn and coloured images from my excercise books. No protractors or any of that Euclidian nonsense; HB and coloured pencils with a straight edge were all that was required.
Some of the theoroms we proved were apparently quite complex & advanced, according to a friend with a maths doctorate I met in adult life.
My main point in making this comment is that there may be some way to write an easier to read but still rigourously truthful introduction by bowldlerising some of the Steiner / Waldorf School's teaching materials for the subject.
I am an electronics engineer and lack the expertise to make any coherent edits, just thought I'd point the way for someone who does.  — Preceding unsigned comment added by 82.30.88.119 (talk) 16:36, 10 June 2017 (UTC)
thread_title:  Steiner / Waldorf Schools 

message: The article mentions Desargues, but omits earlier figures who did important work, e.g., Dürer, Pappus. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:24, 12 March 2024 (UTC)
thread_title:  Antecedents