A straight line in space is defined by two distinct points
A
=
(
A
0
,
A
1
,
A
2
,
A
3
)
⊤
∈
R
P
3
{\displaystyle A=\left(A_{0},A_{1},A_{2},A_{3}\right)^{\top }\in \mathbb {R} {\mathcal {P}}^{3}}
B
=
(
B
0
,
B
1
,
B
2
,
B
3
)
⊤
∈
R
P
3
{\displaystyle B=\left(B_{0},B_{1},B_{2},B_{3}\right)^{\top }\in \mathbb {R} {\mathcal {P}}^{3}}
homogeneous coordinates of the projective space . Its Plücker matrix is:
[
L
]
×
∝
A
B
⊤
−
B
A
⊤
=
(
0
−
L
01
−
L
02
−
L
03
L
01
0
−
L
12
−
L
13
L
02
L
12
0
−
L
23
L
03
L
13
L
23
0
)
{\displaystyle [\mathbf {L} ]_{\times }\propto \mathbf {A} \mathbf {B} ^{\top }-\mathbf {B} \mathbf {A} ^{\top }=\left({\begin{array}{cccc}0&-L_{01}&-L_{02}&-L_{03}\\L_{01}&0&-L_{12}&-L_{13}\\L_{02}&L_{12}&0&-L_{23}\\L_{03}&L_{13}&L_{23}&0\end{array}}\right)}
Where the skew-symmetric
4
×
4
{\displaystyle 4\times 4}
Plücker coordinates
L
∝
(
L
01
,
L
02
,
L
03
,
L
12
,
L
13
,
L
23
)
⊤
{\displaystyle \mathbf {L} \propto (L_{01},L_{02},L_{03},L_{12},L_{13},L_{23})^{\top }}
with
L
i
j
=
A
i
B
j
−
B
i
A
j
.
{\displaystyle L_{ij}=A_{i}B_{j}-B_{i}A_{j}.}
Plücker coordinates fulfill the Grassmann–Plücker relations
L
01
L
23
−
L
02
L
13
+
L
03
L
12
=
0
{\displaystyle L_{01}L_{23}-L_{02}L_{13}+L_{03}L_{12}=0}
and are defined up to scale. A Plücker matrix has only rank 2 and four degrees of freedom (just like lines in
R
3
{\displaystyle \mathbb {R} ^{3}}
A
{\displaystyle \mathbf {A} }
B
{\displaystyle \mathbf {B} }
cross product for both the intersection (meet) of two lines, as well as the joining line of two points in the projective plane.
The Plücker matrix allows us to express the following geometric operations as matrix-vector product:
Plane contains line:
0
=
[
L
]
×
E
{\displaystyle \mathbf {0} =[\mathbf {L} ]_{\times }\mathbf {E} }
X
=
[
L
]
×
E
{\displaystyle \mathbf {X} =[\mathbf {L} ]_{\times }\mathbf {E} }
L
{\displaystyle \mathbf {L} }
E
{\displaystyle \mathbf {E} }
Point lies on line:
0
=
[
L
~
]
×
X
{\displaystyle \mathbf {0} =[{\tilde {\mathbf {L} }}]_{\times }\mathbf {X} }
E
=
[
L
~
]
×
X
{\displaystyle \mathbf {E} =[{\tilde {\mathbf {L} }}]_{\times }\mathbf {X} }
E
{\displaystyle \mathbf {E} }
X
{\displaystyle \mathbf {X} }
L
{\displaystyle \mathbf {L} }
Direction of a line:
[
L
]
×
π
∞
=
[
L
]
×
(
0
,
0
,
0
,
1
)
⊤
=
(
−
L
03
,
−
L
13
,
−
L
23
,
0
)
⊤
{\displaystyle [\mathbf {L} ]_{\times }\pi ^{\infty }=[\mathbf {L} ]_{\times }(0,0,0,1)^{\top }=\left(-L_{03},-L_{13},-L_{23},0\right)^{\top }}
Closest point to the origin
X
0
≅
[
L
]
×
[
L
]
×
π
∞
.
{\displaystyle \mathbf {X} _{0}\cong [\mathbf {L} ]_{\times }[\mathbf {L} ]_{\times }\pi ^{\infty }.}
Intersection with a plane
edit
The meet of a plane and a line in projective three-space as expressed by multiplication with the Plücker matrix Let
E
=
(
E
0
,
E
1
,
E
2
,
E
3
)
⊤
∈
R
P
3
{\displaystyle \mathbf {E} =\left(E_{0},E_{1},E_{2},E_{3}\right)^{\top }\in \mathbb {R} {\mathcal {P}}^{3}}
E
0
x
+
E
1
y
+
E
2
z
+
E
3
=
0.
{\displaystyle E_{0}x+E_{1}y+E_{2}z+E_{3}=0.}
which does not contain the line
L
{\displaystyle \mathbf {L} }
X
=
[
L
]
×
E
=
A
B
⊤
E
⏟
α
−
B
A
⊤
E
⏟
β
=
A
α
+
B
β
,
{\displaystyle \mathbf {X} =[\mathbf {L} ]_{\times }\mathbf {E} =\mathbf {A} {\underset {\alpha }{\underbrace {\mathbf {B} ^{\top }\mathbf {E} } }}-\mathbf {B} {\underset {\beta }{\underbrace {\mathbf {A} ^{\top }\mathbf {E} } }}=\mathbf {A} \alpha +\mathbf {B} \beta ,}
which lies on the line
L
{\displaystyle \mathbf {L} }
A
{\displaystyle \mathbf {A} }
B
{\displaystyle \mathbf {B} }
X
{\displaystyle \mathbf {X} }
E
{\displaystyle \mathbf {E} }
E
⊤
X
=
E
⊤
[
L
]
×
E
=
E
⊤
A
⏟
α
B
⊤
E
⏟
β
−
E
⊤
B
⏟
β
A
⊤
E
⏟
α
=
0
,
{\displaystyle \mathbf {E} ^{\top }\mathbf {X} =\mathbf {E} ^{\top }[\mathbf {L} ]_{\times }\mathbf {E} ={\underset {\alpha }{\underbrace {\mathbf {E} ^{\top }\mathbf {A} } }}{\underset {\beta }{\underbrace {\mathbf {B} ^{\top }\mathbf {E} } }}-{\underset {\beta }{\underbrace {\mathbf {E} ^{\top }\mathbf {B} } }}{\underset {\alpha }{\underbrace {\mathbf {A} ^{\top }\mathbf {E} } }}=0,}
and must therefore be their point of intersection.
In addition, the product of the Plücker matrix with a plane is the zero-vector, exactly if the line
L
{\displaystyle \mathbf {L} }
α
=
β
=
0
⟺
E
{\displaystyle \alpha =\beta =0\iff \mathbf {E} }
L
.
{\displaystyle \mathbf {L} .}
The join of a point and a line in projective three-space as expressed by multiplication with the Plücker matrix In projective three-space, both points and planes have the same representation as 4-vectors and the algebraic description of their geometric relationship (point lies on plane) is symmetric. By interchanging the terms plane and point in a theorem, one obtains a dual theorem which is also true.
In case of the Plücker matrix, there exists a dual representation of the line in space as the intersection of two planes:
E
=
(
E
0
,
E
1
,
E
2
,
E
3
)
⊤
∈
R
P
3
{\displaystyle E=\left(E_{0},E_{1},E_{2},E_{3}\right)^{\top }\in \mathbb {R} {\mathcal {P}}^{3}}
and
F
=
(
F
0
,
F
1
,
F
2
,
F
3
)
⊤
∈
R
P
3
{\displaystyle F=\left(F_{0},F_{1},F_{2},F_{3}\right)^{\top }\in \mathbb {R} {\mathcal {P}}^{3}}
in homogeneous coordinates of projective space . Their Plücker matrix is:
[
L
~
]
×
=
E
F
⊤
−
F
E
⊤
{\displaystyle \left[{\tilde {\mathbf {L} }}\right]_{\times }=\mathbf {E} \mathbf {F} ^{\top }-\mathbf {F} \mathbf {E} ^{\top }}
and
G
=
[
L
~
]
×
X
{\displaystyle \mathbf {G} =\left[{\tilde {\mathbf {L} }}\right]_{\times }\mathbf {X} }
describes the plane
G
{\displaystyle \mathbf {G} }
X
{\displaystyle \mathbf {X} }
L
{\displaystyle \mathbf {L} }
edit
As the vector
X
=
[
L
]
×
E
{\displaystyle \mathbf {X} =[\mathbf {L} ]_{\times }\mathbf {E} }
E
{\displaystyle \mathbf {E} }
∀
E
∈
R
P
3
:
X
=
[
L
]
×
E
lies on
L
⟺
[
L
~
]
×
X
=
0
.
{\displaystyle \forall \mathbf {E} \in \mathbb {R} {\mathcal {P}}^{3}:\,\mathbf {X} =[\mathbf {L} ]_{\times }\mathbf {E} {\text{ lies on }}\mathbf {L} \iff \left[{\tilde {\mathbf {L} }}\right]_{\times }\mathbf {X} =\mathbf {0} .}
Thus:
(
[
L
~
]
×
[
L
]
×
)
⊤
=
[
L
]
×
[
L
~
]
×
=
0
∈
R
4
×
4
.
{\displaystyle \left([{\tilde {\mathbf {L} }}]_{\times }[\mathbf {L} ]_{\times }\right)^{\top }=[\mathbf {L} ]_{\times }\left[{\tilde {\mathbf {L} }}\right]_{\times }=\mathbf {0} \in \mathbb {R} ^{4\times 4}.}
The following product fulfills these properties:
(
0
L
23
−
L
13
L
12
−
L
23
0
L
03
−
L
02
L
13
−
L
03
0
L
01
−
L
12
L
02
−
L
01
0
)
(
0
−
L
01
−
L
02
−
L
03
L
01
0
−
L
12
−
L
13
L
02
L
12
0
−
L
23
L
03
L
13
L
23
0
)
=
(
L
01
L
23
−
L
02
L
13
+
L
03
L
12
)
⋅
(
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
)
=
0
,
{\displaystyle {\begin{aligned}&\left({\begin{array}{cccc}0&L_{23}&-L_{13}&L_{12}\\-L_{23}&0&L_{03}&-L_{02}\\L_{13}&-L_{03}&0&L_{01}\\-L_{12}&L_{02}&-L_{01}&0\end{array}}\right)\left({\begin{array}{cccc}0&-L_{01}&-L_{02}&-L_{03}\\L_{01}&0&-L_{12}&-L_{13}\\L_{02}&L_{12}&0&-L_{23}\\L_{03}&L_{13}&L_{23}&0\end{array}}\right)\\[10pt]={}&\left(L_{01}L_{23}-L_{02}L_{13}+L_{03}L_{12}\right)\cdot \left({\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{array}}\right)=\mathbf {0} ,\end{aligned}}}
due to the Grassmann–Plücker relation . With the uniqueness of Plücker matrices up to scalar multiples, for the primal Plücker coordinates
L
=
(
L
01
,
L
02
,
L
03
,
L
12
,
L
13
,
L
23
)
⊤
{\displaystyle \mathbf {L} =\left(L_{01},\,L_{02},\,L_{03},\,L_{12},\,L_{13},\,L_{23}\right)^{\top }}
we obtain the following dual Plücker coordinates:
L
~
=
(
L
23
,
−
L
13
,
L
12
,
L
03
,
−
L
02
,
L
01
)
⊤
.
{\displaystyle {\tilde {\mathbf {L} }}=\left(L_{23},\,-L_{13},\,L_{12},\,L_{03},\,-L_{02},\,L_{01}\right)^{\top }.}
In the projective plane
edit
Duality of join and meet operations in two-space. The 'join' of two points in the projective plane is the operation of connecting two points with a straight line. Its line equation can be computed using the cross product :
l
∝
a
×
b
=
(
a
1
b
2
−
b
1
a
2
b
0
a
2
−
a
0
b
2
a
0
b
1
−
a
1
b
0
)
=
(
l
0
l
1
l
2
)
.
{\displaystyle \mathbf {l} \propto \mathbf {a} \times \mathbf {b} =\left({\begin{array}{c}a_{1}b_{2}-b_{1}a_{2}\\b_{0}a_{2}-a_{0}b_{2}\\a_{0}b_{1}-a_{1}b_{0}\end{array}}\right)=\left({\begin{array}{c}l_{0}\\l_{1}\\l_{2}\end{array}}\right).}
Dually, one can express the 'meet', or intersection of two straight lines by the cross-product:
x
∝
l
×
m
{\displaystyle \mathbf {x} \propto \mathbf {l} \times \mathbf {m} }
The relationship to Plücker matrices becomes evident, if one writes the cross product as a matrix-vector product with a skew-symmetric matrix:
[
l
]
×
=
a
b
⊤
−
b
a
⊤
=
(
0
l
2
−
l
1
−
l
2
0
l
0
l
1
−
l
0
0
)
{\displaystyle [\mathbf {l} ]_{\times }=\mathbf {a} \mathbf {b} ^{\top }-\mathbf {b} \mathbf {a} ^{\top }=\left({\begin{array}{ccc}0&l_{2}&-l_{1}\\-l_{2}&0&l_{0}\\l_{1}&-l_{0}&0\end{array}}\right)}
and analogously
[
x
]
×
=
l
m
⊤
−
m
l
⊤
{\displaystyle [\mathbf {x} ]_{\times }=\mathbf {l} \mathbf {m} ^{\top }-\mathbf {m} \mathbf {l} ^{\top }}
Geometric interpretation
edit
Let
d
=
(
−
L
03
,
−
L
13
,
−
L
23
)
⊤
{\displaystyle \mathbf {d} =\left(-L_{03},\,-L_{13},\,-L_{23}\right)^{\top }}
m
=
(
L
12
,
−
L
02
,
L
01
)
⊤
{\displaystyle \mathbf {m} =\left(L_{12},\,-L_{02},\,L_{01}\right)^{\top }}
[
L
]
×
=
(
[
m
]
×
d
−
d
0
)
{\displaystyle [\mathbf {L} ]_{\times }=\left({\begin{array}{cc}[\mathbf {m} ]_{\times }&\mathbf {d} \\-\mathbf {d} &0\end{array}}\right)}
and
[
L
~
]
×
=
(
[
−
d
]
×
m
−
m
0
)
,
{\displaystyle [{\tilde {\mathbf {L} }}]_{\times }=\left({\begin{array}{cc}[-\mathbf {d} ]_{\times }&\mathbf {m} \\-\mathbf {m} &0\end{array}}\right),}
[citation needed where
d
{\displaystyle \mathbf {d} }
m
{\displaystyle \mathbf {m} }
geometric intuition of Plücker coordinates .
![{\displaystyle [\mathbf {L} ]_{\times }\propto \mathbf {A} \mathbf {B} ^{\top }-\mathbf {B} \mathbf {A} ^{\top }=\left({\begin{array}{cccc}0&-L_{01}&-L_{02}&-L_{03}\\L_{01}&0&-L_{12}&-L_{13}\\L_{02}&L_{12}&0&-L_{23}\\L_{03}&L_{13}&L_{23}&0\end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37df98efce8bfc41093b5727a00b03d516590648)
![{\displaystyle \mathbf {0} =[\mathbf {L} ]_{\times }\mathbf {E} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e45a31244b21858f4f70ead8a07adc49666d7a0)
![{\displaystyle \mathbf {X} =[\mathbf {L} ]_{\times }\mathbf {E} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c35a7875bb2905504b075e52442684bd9a84b043)
![{\displaystyle \mathbf {0} =[{\tilde {\mathbf {L} }}]_{\times }\mathbf {X} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/67c15cf28f43843b1232daded3566c7b05923e7d)
![{\displaystyle \mathbf {E} =[{\tilde {\mathbf {L} }}]_{\times }\mathbf {X} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d3bd7ce469f2778bfc4bf64979ca4f2c09b5e01)
![{\displaystyle [\mathbf {L} ]_{\times }\pi ^{\infty }=[\mathbf {L} ]_{\times }(0,0,0,1)^{\top }=\left(-L_{03},-L_{13},-L_{23},0\right)^{\top }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00f34027ea1029c9e8bb85c61ef940fb3cb7651e)
![{\displaystyle \mathbf {X} _{0}\cong [\mathbf {L} ]_{\times }[\mathbf {L} ]_{\times }\pi ^{\infty }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fdaea009ba2aa0af4039fc845fac55fd9e26e95)
![{\displaystyle {\begin{aligned}{[}\mathbf {L} ^{\prime }{]}_{\times }&=\mathbf {A} ^{\prime }\mathbf {B} ^{\prime }-\mathbf {B} ^{\prime }\mathbf {A} ^{\prime }\\[6pt]&=(\mathbf {A} \alpha +\mathbf {B} \beta )(\mathbf {A} \gamma +\mathbf {B} \delta )^{\top }-(\mathbf {A} \gamma +\mathbf {B} \delta )(\mathbf {A} \alpha +\mathbf {B} \beta )^{\top }\\[6pt]&=\underbrace {(\alpha \delta -\beta \gamma )} _{\lambda }[\mathbf {L} ]_{\times },\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/896c23741f1d55ba3c54bddb7b495aa9eab97639)
![{\displaystyle [\mathbf {L} ]_{\times }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6aa47c15610a4b62904f15880e96fb2842a33305)
![{\displaystyle \mathbf {X} =[\mathbf {L} ]_{\times }\mathbf {E} =\mathbf {A} {\underset {\alpha }{\underbrace {\mathbf {B} ^{\top }\mathbf {E} } }}-\mathbf {B} {\underset {\beta }{\underbrace {\mathbf {A} ^{\top }\mathbf {E} } }}=\mathbf {A} \alpha +\mathbf {B} \beta ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df7b077530cc418d063c2bfd63ccf391e92501ae)
![{\displaystyle \mathbf {E} ^{\top }\mathbf {X} =\mathbf {E} ^{\top }[\mathbf {L} ]_{\times }\mathbf {E} ={\underset {\alpha }{\underbrace {\mathbf {E} ^{\top }\mathbf {A} } }}{\underset {\beta }{\underbrace {\mathbf {B} ^{\top }\mathbf {E} } }}-{\underset {\beta }{\underbrace {\mathbf {E} ^{\top }\mathbf {B} } }}{\underset {\alpha }{\underbrace {\mathbf {A} ^{\top }\mathbf {E} } }}=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbb6377f5565183125d068f3cd570b420fbc70bb)
![{\displaystyle \left[{\tilde {\mathbf {L} }}\right]_{\times }=\mathbf {E} \mathbf {F} ^{\top }-\mathbf {F} \mathbf {E} ^{\top }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f7f88cf57ecb661521ec1166fdb5f0f36e4bac6)
![{\displaystyle \mathbf {G} =\left[{\tilde {\mathbf {L} }}\right]_{\times }\mathbf {X} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f568b106a9fb4a963018b2fa99e2c12978fc7d3)
![{\displaystyle \forall \mathbf {E} \in \mathbb {R} {\mathcal {P}}^{3}:\,\mathbf {X} =[\mathbf {L} ]_{\times }\mathbf {E} {\text{ lies on }}\mathbf {L} \iff \left[{\tilde {\mathbf {L} }}\right]_{\times }\mathbf {X} =\mathbf {0} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef77a2b0bcebba53e87c7f6d37ce0e216a511c01)
![{\displaystyle \left([{\tilde {\mathbf {L} }}]_{\times }[\mathbf {L} ]_{\times }\right)^{\top }=[\mathbf {L} ]_{\times }\left[{\tilde {\mathbf {L} }}\right]_{\times }=\mathbf {0} \in \mathbb {R} ^{4\times 4}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b650289f2cd0b349ec5b7b5bd53445e51285ec09)
![{\displaystyle {\begin{aligned}&\left({\begin{array}{cccc}0&L_{23}&-L_{13}&L_{12}\\-L_{23}&0&L_{03}&-L_{02}\\L_{13}&-L_{03}&0&L_{01}\\-L_{12}&L_{02}&-L_{01}&0\end{array}}\right)\left({\begin{array}{cccc}0&-L_{01}&-L_{02}&-L_{03}\\L_{01}&0&-L_{12}&-L_{13}\\L_{02}&L_{12}&0&-L_{23}\\L_{03}&L_{13}&L_{23}&0\end{array}}\right)\\[10pt]={}&\left(L_{01}L_{23}-L_{02}L_{13}+L_{03}L_{12}\right)\cdot \left({\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{array}}\right)=\mathbf {0} ,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8358abf0b478ff06560e2ef862c8d8da4c8b7a73)
![{\displaystyle [\mathbf {l} ]_{\times }=\mathbf {a} \mathbf {b} ^{\top }-\mathbf {b} \mathbf {a} ^{\top }=\left({\begin{array}{ccc}0&l_{2}&-l_{1}\\-l_{2}&0&l_{0}\\l_{1}&-l_{0}&0\end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de0cddf32a5672abd8a55046db0bbc580b2f415e)
![{\displaystyle [\mathbf {x} ]_{\times }=\mathbf {l} \mathbf {m} ^{\top }-\mathbf {m} \mathbf {l} ^{\top }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efc6323f4344ba2d2e77a45632ace96352c57e02)
![{\displaystyle [\mathbf {L} ]_{\times }=\left({\begin{array}{cc}[\mathbf {m} ]_{\times }&\mathbf {d} \\-\mathbf {d} &0\end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/badba1f4f90d1116cb6bf2d42ed40d8ce6c27603)
![{\displaystyle [{\tilde {\mathbf {L} }}]_{\times }=\left({\begin{array}{cc}[-\mathbf {d} ]_{\times }&\mathbf {m} \\-\mathbf {m} &0\end{array}}\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fdf703ca2a6bd5db00603fe6f94df26ed8320d3)
