Definition
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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.
Properties
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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 }.}
Uniqueness
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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} .}
Dual Plücker matrix
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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} }
Relationship between primal and dual Plücker matrices
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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 .
References
edit
![{\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)
