Talk:Plücker coordinates

Cleanup needed?

I don't see why this article, which wasn't that bad, rates a cleanup tag. I've put the tag below:

Gene Ward Smith 20:35, 12 November 2005 (UTC)

I initially added the cleanup tag because I thought the article was too difficult to read (too many inline PNGs), but perhaps that is subjective. There is also a weird heading structure (level 3s before 2s). I still think the article needs cleanup. I'll add it to my todo list. -- Fropuff 16:48, 13 January 2006 (UTC)

RM tag. Still needs expert attention IMO. Kaisershatner 22:12, 19 April 2006 (UTC)

Some remarks

I have some questions/remarks about the article

When proving that the plucker coordinates are independent of the chosen couple of points, the article now considers X+rY and X+sY i would consider u*X+r*Y and v*X+s*Y it is just as easy and avoids irritating cases

The convention that I know for the Plucker coordinates is (p_{01},p_{02},p_{03},p_{23},p_{31},p_{12}) all the formula become nicer know, the equation of the quadric becomes : p_{01}p_{23} +p_{02}_{31}+p_{03}p_{12}

In my course, we spoke of Klein quadric all the time, and I find Klein quadric all over the net, while I haven't found any mention of a Plucker quadric.

So what are your thoughts? Just change it myself (be bold policy?)?Evilbu 11:29, 11 February 2006 (UTC)

I have changed it. (You can sign with ~~~~.) Charles Matthews 11:12, 11 February 2006 (UTC)

I see. So i'll just go ahead and change the entire convention all over the article (the article does request cleanup)? Sorry about the signing. I will remember eventually. Evilbu 11:29, 11 February 2006 (UTC)

It's a good idea. There are too many symbols in the article, though. The thing about 'lines' is strange: if you use the minor determinant definition, this is simply explaining that row operations change nothing, which is not worth doing as algebra. But go ahead anyway. Charles Matthews 11:40, 11 February 2006 (UTC)

I heavily changed it all, and it's probably not gonna stop here. I hope the creator will not be upset. I disagree with seeing it as a map from couples to points, I rather see it as a map directly from lines to points, thus requiring a little checking that it is well defined. More elaborations about injectivity and surjectivity can be provided in matrix language. One thing that stops me know is that I don't know the english words for these things, anyone who can tell me english word is welcome. I'll give the Dutch words with description :

stralenschoof (ray-sheef?) : all lines in 3d through a specific point stralenveld (ray field?) : all lines in 3d in a specific plane stralenwaaier(ray waver..?) : all lines in 3d through a specific point and in a specific plane through that point. Yup, waaier is Dutch for 'fan'. I'd spell 'sheef' as 'sheaf' as in 'sheaf of wheat', probably the cone of lines through a point, but what do I know? I'm a Dutch-speaking layman :\ 87.211.115.30 (talk) 01:58, 26 March 2015 (UTC)

Evilbu 15:56, 11 February 2006 (UTC)

Nobody? Come on guys, I am not a native speaker of English, I could use some help here! Pencil, could that be one of the answers? Evilbu 09:39, 11 June 2006 (UTC)

I'm not an algebraic geometer, but I think pencil is on the right track. I think I've heard it used for "all lines through a given point", but I suspect that the meaning is far more general. You could try asking at Wikipedia_talk:WikiProject_Mathematics, that's usually a good place for questions like this. There's at least one dutchman there too :-) Dmharvey 11:46, 11 June 2006 (UTC)

Original sin

Since the first instantiation of this article, by Beamishboy (contributions) in October of 2004, it has contained two statements concerning uses: one about lines through a point and one about lines in a plane. Both are demonstrably wrong!

• For the point (x₀:x₁:x₂:x₃) = (1:0:0:1), the left-hand side of the given equation is identically zero, no matter what the Plücker coordinates.

  $x_{0}x_{2}p_{13}-x_{0}x_{1}p_{23}-x_{1}x_{3}p_{02}=0.\,\!$

• For the plane (a₀:a₁:a₂:a₃) = (1:0:0:0), we have the same problem with its equation.

  $a_{1}p_{01}+a_{2}p_{02}+a_{3}p_{03}=0.\,\!$

Furthermore, both claims seem prima facie impossible, because in each case the required set of lines has two degrees of freedom, two less than the dimension of the Klein quadric, née "Plücker quadric". (Lines through a point are equivalent to P², points on a unit sphere with opposite points identified; lines in a plane are equivalent to R², by duality with points.)

I have been giving this article a major facelift. This section will be getting a face transplant. --KSmrq^T 23:29, 12 June 2006 (UTC)

Also, can anyone confirm the 1844 date for the introduction of Plücker coordinates? I would like to cite the original work, but have been unable to locate anything with that date. --KSmrq^T 12:31, 13 June 2006 (UTC)

Grassmann coordinates

We should have a full article on Grassmann coordinates. I have removed the section discussing these and placed it here to assist that effort.

---

Other formulations of Plücker coordinates

Plucker coordinates can be formulated more generally using the ideas of multilinear algebra and the wedge product. This leads to a simple formulation of the generalisation of the Plücker mapping.

A Plücker map is a map, π,

${\begin{matrix}\pi :&\mathrm {G} _{k,n}(\mathbb {C} )&\rightarrow &P(\wedge ^{k}\mathbb {C} ^{n})\\&\operatorname {span} (v_{1},\ldots ,v_{k})&\mapsto &\mathbb {C} (v_{1}\wedge \ldots \wedge v_{k})\\\end{matrix}}$

where G_k,n(C) is the Grassmannian, i.e. the space of all k-dimensional subspaces of an n-dimensional complex vector space, Cⁿ. The map described above is the particular case where k = 2, n = 4.

In the general context, the Grassmannian can be completely characterised as an intersection of quadrics, each coming from a relation on the Plücker coordinates that derives from linear algebra.

---

With that, my editorial sweep through this article is complete. I trust I have left it better than I found it. --KSmrq^T 03:54, 13 June 2006 (UTC)

Yeah, I was just gonna say, I thought Plucker coords were more general than just this 2-5 dimensional case discussed in this article. Anyone want to write an article on the Grassmannian variety which would be more specific on the projective variety, as Grassmannian is rather general ?? —Preceding unsigned comment added by Linas (talk • contribs) 04:41, 2006 June 30

I'm following the convention of Hodge and Pedoe in calling the general case Grassmann coordinates, though I understand that sometimes the Plücker name is used instead. Currently I've left a red link, not even a stub, for these. They also call the quadric in P⁵ the Plücker quadric, which I suspect has more historical support. Klein was Plücker's student (we should all be so lucky). It turns out that I'm still tinkering with this article, and when I'm done it should be a helpful stepping stone to the general case. (And a good warmup for writing it. One of the challenges is to avoid indigestible index soup.)

We have several topics to play with in other articles. First, there is the notation and theory of Grassmann (and dual Grassmann) coordinates, with their attendent Plücker relations. I'll probably feel guilty until I at least make a stub attempt. Then we've got the whole "flag" and Schubert cell bit, and the Schubert calculus built on it. This material is more important than our current tiny article (which really should be tagged a stub) gives any hint of, and needs a more expansive treatment.

I was concentrating on this line of development, including making a 3D rendering or two, when my attention was diverted by some silliness elsewhere.

My leaning would be to separate an article on general Grassmann coordinates from a variety discussion. If that proves to be too unnatural a distinction, then we'll find out as we go.

Anyway, if that "anyone" turns out to be me, it'll probably be a little while before I get to it. Any bits you would particularly like to see included in such an article? --KSmrq^T 05:46, 30 June 2006 (UTC)

Following recent discussions at User talk:KSmrq, I would like to propose that the material excised from the article and placed above has a natural home in the article Plücker embedding (which currently redirects here). Geometry guy 18:26, 27 March 2007 (UTC)

I've placed disambiguation between Plücker coordinates and Grassmann coordinates/Plücker embedding, so the scope of the articles should be clear. The Grassmann coordinates article is currently very brief, and could use elaboration. Nbarth (talk) 22:36, 18 November 2007 (UTC)

"Mathematically unlikely"

"In the (mathematically unlikely) event that two lines are coplanar but not parallel[...]"

What is "mathematically unlikely" supposed to mean? 134.169.128.67 17:36, 27 September 2006 (UTC)

Two lines in 3D chosen at random have a vanishingly small probability of being coplanar. It is common in geometry to stipulate "general position" to rule out such unlikely coincidences. However, lines in applications are not chosen at random, and it is often necessary to deal with such a coincidence. --KSmrq^T 19:30, 27 September 2006 (UTC)

I see. May it may be removed then, as it is not necessary and confusing?

One other request: in "Line-line join" a description of the Intersection point would be nice. 134.169.128.67 11:24, 28 September 2006 (UTC)

I'm sorry if the wording confuses you. However, it conveys information important in applications. The slightest numerical perturbation, such as that introduced by measurement or by roundoff, will destroy the coincidence. Robust software must take that into account, and the parenthetical remark acts as a "word to the wise". Contrast this case with other calculations presented in the article, which are not so delicate. I will attempt a more expanded comment in the article.

A description of the intersection point is not pretty. In the article by Shoemake in Ray Tracing News (RTN) we find a formula that (in our notation) reads as follows:

“Let n be a unit vector along a coordinate axis, with (d×d′)·n non-zero.

• ((d×d′)·n:(m·n)d′−(m′·n)d−(m·d′)n) is the point of intersection, if any.”

It is difficult to see the mathematical interest, and those with a practical bent will probably consult the RTN article for themselves. There is a more general approach to meets and joins that accomodates common subspaces, but it properly belongs in the (as-yet-unwritten) article on Grassmann coordinates, and would be far too long for one minor result in the present article. The Shoemake article suggests that (in our notation):

“When neither m nor m′ is null, the far simpler formula (d·m′:m×m′) will suffice; it is the dual of the common plane (m·d′:d×d′).

Robust software for geometric computations may very well devote a disproportionately large amount of code to handling rare cases; I'm reluctant to make that same investment in the article, especially since we already cite ample specialized literature to consult. --KSmrq^T 18:06, 28 September 2006 (UTC)

This formula for the intersection of two lines doesn't work. I have to say I don't know if I'm more embarrassed that we the human race haven't worked out the correct formulas for line and plane intersections yet, or that so many people think we have.

-Don't say "blade" if you mean an alternating tensor, or if you mean an oriented affine subspace of a vector space, or if you mean an area form on a plane in Euclidean 3-space.

-The vector cross product is no more `geometric' than any other algebraic operation which may have a geometric interpretation. The ability to form an orthonormal frame with your right hand while visualizing the symbols A, B, and A x B on your fingertips is fantastic, but so is imagining a grid of apples in rows and columns when you see 3*5. If you think every multiplication of two quantities is going to be visualizable by an apple grid then you're in trouble. If you think every multiplication of two quantities is going to be expressible by vector cross product then you're in trouble; you start writing non-sense formulas that are "easier to use" instead of doing the work of learning commutative algebra from a book.

-All of this "geometric algebra" stuff is just "vector calculus". I never want to hear the phrase "Gibbs' vector calculus" again. You can use whatever notation you want, modern mathematics has arrived: If you're using a combinatorial identity, do it. If you're using a tensor operation, do it. If you're using linear algebra, do it. Some part of the world has for some reason become fixated on notation as theory. Give that up and you will succeed.

24.188.36.136 (talk) 04:23, 9 October 2014 (UTC)Jimmy

2-planes in 4-space

In the article for Grassmannians, this article is cited as a link to explain the structure of the Grassmann manifold consisting of 2-planes in 4-space (the simplest Grassmannian that isn't projective space). Yet the article itself doesn't say anything about that, and the connection isn't clear to me based on what's written. Sxp151 (talk) 02:52, 25 September 2009 (UTC)

Plücker Matrix

This article fails completely to mention the relation to anti-symmetric matrices. I suggest adding the following important derivation:

... Please help with the correct notation etc. I have never written a Wikipedia article. $\cong$ means up to scale in this paragraph:

We can describe a line in projective 3-space using a linear combination of two points $\mathbf {a}$ and $\mathbf {b}$ using Plücker's $\mu :\mathbf {l} (\lambda )=\lambda \mathbf {a} +\mathbf {b}$ The anti-symmetric matrix $L\cong \mathbf {a} \mathbf {b} ^{T}-\mathbf {b} \mathbf {a} ^{T}$ describes the line.

We see that multiplication by a plane $L\mathbf {P}$ yields a point on the line $L\mathbf {P} \cong \mathbf {a} \mathbf {b} ^{T}\mathbf {P} -\mathbf {b} \mathbf {a} ^{T}\mathbf {P} =\mathbf {a} \alpha +\mathbf {b} \beta \cong \mathbf {x}$. The same point is also contained in the plane because $\mathbf {P} ^{T}x=\mathbf {P} ^{T}L\mathbf {P} =\mathbf {P} ^{T}\mathbf {a} \mathbf {b} ^{T}\mathbf {P} -\mathbf {P} ^{T}\mathbf {b} \mathbf {a} ^{T}\mathbf {P} =0$

Since the point x is on the line and on the plane it must be the intersection of the two.

By the concept of Duality, we know that we can also multiply with a third point c and get the plane containing the line (a,b) and the third point c. (This would have to point to an article about Duality)

Note that an anti-symmetric 4 by 4 matrix is defined by 6 values. The matrix L is up to scale and up to the so-called Grassmann-Plücker relation (Point to that). A line can thus be represented as the 6-vector with 4 DOF and it is easily verified by expansion that this is equivalent to the following 6 determinants: $\mathbf {L} \cong \left({\begin{array}{c}l_{01}\\l_{02}\\l_{03}\\l_{12}\\l_{13}\\l_{23}\end{array}}\right)=\left({\begin{array}{c}a_{0}b_{1}-a_{1}b_{0}\\a_{0}b_{2}-a_{2}b_{0}\\a_{0}b_{3}-a_{3}b_{0}\\a_{1}b_{2}-a_{2}b_{1}\\a_{1}b_{3}-a_{3}b_{1}\\a_{2}b_{3}-a_{3}b_{2}\end{array}}\right)=\left(\left|{\begin{array}{cc}a_{0}&b_{0}\\a_{1}&b_{1}\end{array}}\right|,\,\left|{\begin{array}{cc}a_{0}&b_{0}\\a_{2}&b_{2}\end{array}}\right|,\,\left|{\begin{array}{cc}a_{0}&b_{0}\\a_{3}&b_{3}\end{array}}\right|,\,\left|{\begin{array}{cc}a_{1}&b_{1}\\a_{2}&b_{2}\end{array}}\right|,\,\left|{\begin{array}{cc}a_{3}&b_{3}\\a_{1}&b_{1}\end{array}}\right|,\,\left|{\begin{array}{cc}a_{2}&b_{2}\\a_{3}&b_{3}\end{array}}\right|\right)^{T}$

Geometric interpretation: Since we can pick any scalar multiple of the points a and b w.l.o.g. assume $b_{3}=a_{3}=1$ . Then are a_i and b_i are simply the x,y,z coordinates in $\mathbb {R} ^{3}$ We have

$\left({\begin{array}{c}l_{12}\\-l_{02}\\l_{12}\end{array}}\right)=\left({\begin{array}{c}a_{1}b_{2}-a_{2}b_{1}\\-a_{0}b_{2}+a_{2}b_{0}\\a_{1}b_{2}-a_{2}b_{1}\end{array}}\right)=\mathbf {a} \times \mathbf {b} \,{\text{ and }}\,\left({\begin{array}{c}l_{03}\\l_{13}\\l_{23}\\0\end{array}}\right)=\left({\begin{array}{c}a_{0}-b_{0}\\a_{1}-b_{1}\\a_{2}-b_{2}\\0\end{array}}\right)=\mathbf {a} -\mathbf {b}$

Which is the moment and displacement formulation. To me the latter is merely a possible interpretation of a much more general concept. — Preceding unsigned comment added by Aaichert (talk • contribs) 07:29, 19 August 2014 (UTC)

87.211.115.30 (talk) 01:49, 26 March 2015 (UTC) the link for 'moment in "it would have a moment about the origin" points to the physics moment, nto the mathematical one.

Incorrect coplanar equation ?

Where was:

Line-line crossing

== "Two lines in P3 are either skew or coplanar, and in the latter case they are either coincident or intersect in a unique point. If pij and p′ij are the Plücker coordinates of two lines, then they are coplanar precisely when d-m′+m-d′ = 0, as shown by ...." — i think the sign - (minus) is incorrect. This equation is about 2 dot products that should be zero, so the result should be zero also: d⋅m′+m⋅d = 0; as a confirmation for this, the next equation states correctly the same coplanar test, but replacing the symbols d and m, with p and indexes taken from the plucker coordinates.

Woreno (talk) 21:13, 21 August 2019 (UTC)

After checking almost all the formulas in this section with matlab and with the , i changed the equation to the dot product.

References

References

A confusion?

In the section "Primal coordinates" it says that the homogenous coordinates are (x₀ : x₁ : x₂ : x₃), thus placing the "scaling factor" $x_{0}$ first among the four coordinates, while it is placed last according to the article homogenous coordinates. So, either the articles are contradictorory or else, I am an idiot. Please explain why I am an idiot. A lot of Thanks! in advance. Episcophagus (talk) 14:29, 20 April 2023 (UTC)