Talk:Plücker coordinates

message: 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)
replies: text: 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)

text: RM tag.  Still needs expert attention IMO.  Kaisershatner 22:12, 19 April 2006 (UTC)
thread_title: Cleanup needed?

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

text: 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)

text: 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)
thread_title:  Some remarks 

message: 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_0x_2p_{13} - x_0x_1p_{23} - x_1x_3p_{02} = 0 . \,\!
* For the plane (a0:a1:a2:a3) = (1:0:0:0), we have the same problem with its equation.
*: a_1p_{01} + a_2p_{02} + a_3p_{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. --KSmrqT 23:29, 12 June 2006 (UTC)
replies: text: 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. --KSmrqT 12:31, 13 June 2006 (UTC)
thread_title:  Original sin 

message: 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)
thread_title:  2-planes in 4-space 

message: 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)
thread_title:  Plücker Matrix 

message: 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)
thread_title:  Incorrect coplanar equation ? 

message: == "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)
thread_title:  Line-line crossing 

message: In the section "Primal coordinates" it says that the homogenous coordinates are , 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)
thread_title:  A confusion?