Talk:Projective geometry

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Field: Geometry

Main elements

• 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.

Charles Matthews 16:08, 13 Dec 2003 (UTC)

This Entry Needs Work

Most Wiki articles on bits of mathematics are good; this one is exceptionally bad. I am especially dismayed, given how much good material there is in print on geometry and its history. Starting with the work of that wonderful writer, Howard Eves. Much of this entry rambles and is unclear. One could conclude from reading it that geometry is not a precise subject, when in fact it is the Mother of All Precision. The entry is also curiously silent about the duality pervading so much of projective geometry. Too bad Eves turned 90 the year Wikipedia came on line; he would have made a wonderful contributor!

I will fix this up a little, adding in a formal axiomatization and, given time, some details on the geometric construction of fields. The axioms currently on the page are apparently specific to projective planes, whereas the field is far more general (and axioms simpler). -- Mark, 19 May 2006

I have written the first paragraph under "Description." While the axioms stated there are taken from a problem in Eves (1997), the fastidious first order theory business, very much in the spirit of Tarski's approach to geometry, is my own. The bit about Lawrence Edwards's The Vortex of Life fascinates me, because I am a fan of D'Arcy Thompson (he and A. N. Whitehead were friendly at Cambridge, by the way). But the summary description of Edwards's thinking needs work.202.36.179.65 17:23, 8 March 2006 (UTC)

I think all that takes the article off topic. This is a classical branch of geometry, not an excursion into morphogenesis. Charles Matthews 21:11, 8 March 2006 (UTC)

Classical geometry and morphogenesis are closely related; from taking the measure of the earth (Geo-metry) to Desargues' particularly artistic orientation, geometry has always been about the forms of the world we live in. The wonder is that the real, malleable, complex forms of nature are describable geometrically - whether this is the path of the planets or the shape of a shell. Would you not agree? Hgilbert 14:13, 9 March 2006 (UTC)

Whether it is wondrous or not (and I lean toward not) is irrelevant; what is relevant is whether the section has a place in this article. Since the section is not about projective geometry, but is rather about some attempts to "apply" (in a loose sense) projective geometry to something outside of mathematics, I think it has no place here. To be honest, I am surprised that the section has survived here since December; its style doesn't even conform to that of other articles in applied mathematics. If you really think that Edwards was an important thinker, write an article about him and his book, remove the present section, and then include links to your Edwards article both from here and from the Morphogenesis page. Michael Kinyon 00:15, 12 March 2006 (UTC)

The work of Edwards is 100% projective geometry, not loose in any way. Some of it belongs to pure mathematics (see his Projective Geometry) and some to applied mathematics (which is actually also mathematics - I hope you agree). If the discovery of a mathematical description of a conchoid or hyperbolic space is mathematics, so is the discovery of a mathematical description of an egg or (conifer) cone, shapes that are exceptionally difficult to analyze without projective geometry (and in particular without Klein path curves). The article is probably in need of much supplementary work - there should be mention of Poncelet, of Klein's work in more detail, of Pascal - and if it was more complete, this section would not stand out. I would suggest balancing the article by filling out the missing pieces rather than eliminating the one area that is reasonably fully treated.

An article on Edwards' work is a good idea; someday when someone (perhaps I) has time it should be done. Separate issue, however. Hgilbert 01:03, 12 March 2006 (UTC)

The section about "Forms of the Living World" sounds kind of like mystical pseudo-science stuff. Maybe it should be replaced with a section about how projective geometry is used in Multiple View Geometry and Computer Vision.

The important thing here about Edwards' work is that he discovered his curves as an exercise in projective geometry, before applying them to natural objects. His book implies that his work represents a "rounding-off" of earlier worker's discoveries in projective geometry. From this, his contribution to PG does appear to be significant. But it should focus on the importance of his curves (which I am not competent to judge). -- Steelpillow (talk) 21:35, 26 April 2008 (UTC)

Does this make any sense?

There is a sentence in this entry that puzzles me: The whole family of circles can be seen as the conics passing through two given points on the line at infinity - at the cost of requiring complex number co-ordinates. Does this make sense to anyone? Hgilbert 02:15, 13 March 2006 (UTC)

I understand it to be true. Apparently these two points have even been given pet names by some mathematicians. Broadly, if you consider the equation for a circle, for example ax2 + by2 +c2 = 0, and interpret a, b ad c as complex numbers, then the plane becomes kind of four-dimensional and is called the unitary plane. In here, weird things happen and a conic becomes a kind of imaginary surface and the line at infinity becomes a kind of imaginary plane. The conic "surface" meets the real plane in a real conic curve and the imaginary plane in two imaginary points. Sorry if I may not be 100% accurate - I don't fully understand it myself, either. -- Steelpillow (talk) 21:45, 26 April 2008 (UTC)
Simple: a circle is (x-x0)2+(y-y0)2=R2.
In homogeneous coordinates (X-x0Z)2 + (Y-y0Z)2 = R2Z2.
Now the points (1,i,0) and (1,-i,0) lie on all circles. Pyth (talk) 18:14, 8 July 2008 (UTC)

What is that supposed to mean?

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)

Could someone rewrite this article with Latex-commands? 08:58, 27 September 2006 (UTC) Done the bit I wrote, I think the section called "description" is too abstract, the rest contains no equations.Charles Esson 03:51, 22 October 2006 (UTC)

shouldn't some effort be made to make the subject accessible

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)

Notes Axioms of projective geometry

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"

• 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)

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)
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)

What is projective Geometry?

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)

Proposed intoduction

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.

Charles Esson 12:40, 27 October 2006 (UTC)

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)

Planar Geometry

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 (talkcontribs) 21:54, 23 October 2006 (UTC).

Found it I think Plane geometry Charles Esson 12:59, 27 October 2006 (UTC)

There isn't? What's wrong with Plane (mathematics) or Euclidean plane? —David Eppstein 22:09, 23 October 2006 (UTC)
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)
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)
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)

False irony

Someone has written, in the context of the projective variation on Euclid's 5th postulate, "which makes the designation of Projective geometry as non-Euclidean ironic". I disagree. Projective, affine (including Euclidean) and hyperbolic geometries each have different forms of the parallel axiom. It is precisely this difference which makes PG "non-Euclidean", just as Euclidean geometry is "non-projective". There is no irony. -- Steelpillow (talk) 21:25, 26 April 2008 (UTC)

Taking a deeper look at this bit:
"G2: Every two points, A and B, lie on a unique line, AB. [Axiom 2]
...
"Axiom 2 is thus seen to embody a form of Euclid's 5th postulate (which makes the designation of Projective geometry as non-Euclidean ironic): given a point and a direction, there is a unique line containing the point lying in the given direction."
This last statement, after the colon, says nothing about parallelism (Euclid's 5th postulate) - it mentions only one line. Nor does axiom 2 embody parallelism, it embodies its dual - and the duality of PG derives as a consequence of several axioms. In other words, the whole paragraph is quite wrong from beginning to end. I'll just delete it, unless I can think of something useful to replace it with. -- Steelpillow (talk) 11:41, 2 May 2008 (UTC)

Visualising P2

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)

Changed it to what I believe to be correct. Feel free to educate me. -- Steelpillow (talk) 21:24, 9 May 2008 (UTC)

Axioms of projective geometry

Discussion moved to Talk:Projective_space#Projective_geometry. -- Steelpillow (talk) 21:39, 9 May 2008 (UTC)

Nice and flowery, but where are the facts?

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)

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)
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)
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)
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)
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)

Unferenced details

With regard to the edit [1] 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)

Clarity and accessibility

But put yourself in the position of a random, reasonably intelligent reader, who last took math in high-school. Having heard or read somewhere of Pascal's Hexagon Theorem, said reader looks that up, and discovers that the body of knowledge that theorem belongs to is called "projective geometry". Intrigued, the reader clicks through, only to find: In mathematics, projective geometry is the study of geometric properties which are invariant under projective transformations. This is OK, if a little bombastic, up until we get to which are invariant. To even a very educated layman, this clause is likely to be gibberish.

I'm not sure I have a solution: the given definition is, after all, 100% true, and does have a link to an article about projective transformations. (I fear a rabbit-hole there, but have not checked). I admit that my desire to serve a less-informed reader is a policy and style preference with which others might reasonably disagree.

The "Overview" section is even worse from this point of view. I would be deeply impressed if any less-mathematical reader made it as far as the last paragraph of the overview, which is the first paragraph providing any accessible information to the lay reader.

Do any other editors share these concerns? A repair would require a fairly intensive rewrite, and a lot of good work has been put in here. ACW (talk) 01:21, 14 November 2009 (UTC)

It is certainly unfair to characterise as "gibberish" something that at worst is "jargon". The lead section currently concentrates on one approach, namely that derived from the Erlangen programme. It is a mistake in Wikipedia style to do that, but those who know the subject would easily see that it comes from a respectable textbook tradition (Coxeter, I suspect) and therefore the article does start with a verifiable definition. I'll do some editorial work. What people sometimes mean by "accessibility" is a smudging of actual clarity, you know. Charles Matthews (talk) 09:25, 14 November 2009 (UTC)
I have done what I can with the early parts of the article. Readers quite unfamiliar with the subject should really take the hint at the end of the first para of the lead, and look at projective plane. It would be unreasonable to study n-dimensional Euclidean geometry without knowledge of the plane, also. Charles Matthews (talk) 10:18, 14 November 2009 (UTC)
Charles Matthews has improved the lede enormously. Remaining concerns (and I do have some) are comparative quibbles. My use of the term "gibberish" was hedged by "to even a very educated layman". In that context I stand by the claim; I agree that if I had simply said it was gibberish, that would have been very unfair indeed.
So the speed-bumps here are now much smoothed, and the following is a quibble. I am considering whether there would be a gain in moving the characterization by invariants down a couple of paragraphs. The crucial analogy is that ordinary geometry is to the Euclidean isometry group as projective geometry is to the group of projective transformations, and while that's absolutely central in a mathematical sense, there may be something more welcoming to put in the first paragraph. Perhaps, "In mathematics, projective geometry is a kind of geometry which considers only the way points and lines intersect, neglecting angles and distances. It makes fewer assumptions than classical geometry..." Well, OK, that's headed for a train-wreck (which is why I didn't just edit it in). But perhaps you see the kind of thing I mean. ACW (talk) 21:43, 14 November 2009 (UTC)
I have added more words to the initial para. Charles Matthews (talk) 17:12, 15 November 2009 (UTC)
In my opinion, Charles Matthews has remedied 95% of my original concerns. All I have left are minor quibbles, with which I decline to dilute my praise and thanks. Nicely done, sir. ACW (talk) 17:49, 29 December 2009 (UTC)

angle

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)

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)
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)