The Plucker matrix is a special skew-symmetric - matrix , the straight one in the projective space characterized. The matrix is described by the 6 Plücker coordinates with 4 degrees of freedom. They are named after the German mathematician Julius Plücker .
![4 times 4](https://wikimedia.org/api/rest_v1/media/math/render/svg/89eb2e0f4ddfe5f30c8016a0f2aa1fb5ecedfe20)
definition
A straight line in space is defined by two different points
and
in homogeneous coordinates of the projective space . Your Plücker matrix is:
![A = \ left (A_0, A_1, A_2, A_3 \ right) ^ \ top \ in \ mathbb {R} \ mathcal {P} ^ 3](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c74e3c2e857dfac3215e4b7e72c790339a86f59)
![B = \ left (B_0, B_1, B_2, B_3 \ right) ^ \ top \ in \ mathbb {R} \ mathcal {P} ^ 3](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ff7beff6dc53068033b4686dacb627a53d8c16)
![{\ 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)
Whereby the asymmetrical - matrix by the 6 Plücker coordinates
![\ mathbf {L} \ propto (L_ {01}, L_ {02}, L_ {03}, L_ {12}, L_ {13}, L_ {23}) ^ \ top](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ec5ef849b37892d96b17cb579399a4973f32e87)
With
![L_ {ij} = A_iB_j - B_iA_j](https://wikimedia.org/api/rest_v1/media/math/render/svg/08ebcdafb6484d9fe4372ca45f4a22aa7534fc34)
is described. The Plücker coordinates satisfy the Graßmann-Plücker relation
![{\ displaystyle L_ {01} L_ {23} -L_ {02} L_ {13} + L_ {03} L_ {12} = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b905cb480048c0f49ec3870b73a8e91c9aadc44)
and are defined up to scalar multiples. Every Plücker matrix has only rank 2 and four degrees of freedom (like every straight line in ). It is independent of the choice of points A and B and is also a generalization of the straight line equation or the cross product for both the intersection of two straight lines and the straight connecting line through two points in the projective plane.
![\ mathbb {R} ^ {3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5)
properties
The Plücker matrix allows the following geometric operations to be expressed in the matrix-vector product:
- Plane contains straight lines:
-
is the intersection of the straight line with the plane ('Meet')![{\ mathbf {L}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f5f750865376a1a4ae2b15a00b4ff9c75a66630)
![\ mathbf {E}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7f22b39d51f780fc02859059c1757c606b9de2)
- Point lies on a straight line:
-
is the common plane that contains the point and the straight line ('join').![\ mathbf {E}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7f22b39d51f780fc02859059c1757c606b9de2)
![{\ mathbf {X}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd)
![{\ mathbf {L}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f5f750865376a1a4ae2b15a00b4ff9c75a66630)
- Direction of a straight line: (Note: Can also be interpreted as a plane through the coordinate origin, orthogonal to the straight line)
![{\ displaystyle [\ mathbf {L}] _ {\ times} \ pi ^ {\ infty} = [\ mathbf {L}] _ {\ times} (0,0,0,1) ^ {\ top} = (-L_ {03}, - L_ {13}, - L_ {23}, 0) ^ {\ top}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfa84a4bb24e69c209748ec91c55c67982aea9a0)
- Point closest to the origin of coordinates:
Uniqueness
Any two different points on the straight line can be found by linear combinations of and :
![\ mathbf {A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1)
![\ mathbf {B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cafb0ef39b0f5ffa23c170aa7f7b4e718327c4d1)
-
.
Your Plücker matrix is then:
![{\ displaystyle [\ mathbf {L} ^ {\ prime}] _ {\ times} = \ mathbf {A} ^ {\ prime} \ mathbf {B} ^ {\ prime \ top} - \ mathbf {B} ^ {\ prime} \ mathbf {A} ^ {\ prime \ top} = (\ 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} = {\ underset {\ lambda} {\ underbrace {(\ alpha \ delta - \ beta \ gamma)}}} [\ mathbf {L}] _ {\ times},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78eecfc7dcd72cae6de8f6818e8afe0a6d5e85df)
thus identical to up to a scalar multiple .
![[\ mathbf {L}] _ {\ times}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6aa47c15610a4b62904f15880e96fb2842a33305)
Intersection with plane
The intersection of a straight line in space L with a plane E as a multiplication with the Plücker matrix
Let it be a plane with the equation
![\ mathbf {E} = (E_ {0}, E_ {1}, E_ {2}, E_ {3}) ^ {\ top} \ in \ mathbb {R} \ mathcal {P} ^ {3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/638d3131c5769cfc8191bfbdec59e6560a5ac5c0)
![E_ {0} x + E_ {1} y + E_ {2} z + E_ {3} = 0.](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f468620030b32f4f94fbde6e1401ba3b6499aa7)
which does not contain the straight line . Then the matrix-vector product of the Plücker matrix with the plane describes a point
![{\ mathbf {L}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f5f750865376a1a4ae2b15a00b4ff9c75a66630)
![\ 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)
which lies on the straight line because it is a linear combination of and .
is also on the plane![{\ mathbf {L}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f5f750865376a1a4ae2b15a00b4ff9c75a66630)
![\ mathbf {A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1)
![\ mathbf {B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cafb0ef39b0f5ffa23c170aa7f7b4e718327c4d1)
![{\ mathbf {X}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd)
![\ 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)
and must therefore be the intersection of the straight line and the plane.
Furthermore, the product of the Plücker matrix with a plane results in the zero vector if and only if the straight line is contained in the plane:
![{\ mathbf {L}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f5f750865376a1a4ae2b15a00b4ff9c75a66630)
-
contains
Dual Plücker matrix
The common plane G of a point X with a straight line in the space L as a multiplication with the dual Plücker matrix
In real projective space, points and planes have the same representation as homogeneous 4-vectors and the algebraic description of their relationship (point lies on plane) is symmetrical. By swapping the meaning of points and levels of a statement, you get a dual statement that is also true.
In the case of the Plücker matrix, the dual representation of a straight line in space exists as the intersection of two planes
![E = \ left (E_0, E_1, E_2, E_3 \ right) ^ \ top \ in \ mathbb {R} \ mathcal {P} ^ 3](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed38731d1456dc04bcf520faa7d90d8c2fb9f3ac)
and
![F = \ left (F_0, F_1, F_2, F_3 \ right) ^ \ top \ in \ mathbb {R} \ mathcal {P} ^ 3](https://wikimedia.org/api/rest_v1/media/math/render/svg/c343e80bbedcdb7a83a047fc1a4a91a7383ba3f5)
in homogeneous coordinates of the projective space . Your Plücker matrix is:
![[\ tilde {\ mathbf {L}}] _ {\ times} = \ mathbf {E} \ mathbf {F} ^ {\ top} - \ mathbf {F} \ mathbf {E} ^ {\ top}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4aed4317a50c489acfe4500e5a30c823320caad)
and
![\ mathbf {G} = [\ tilde {\ mathbf {L}}] _ {\ times} \ mathbf {X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ff731aeddec41b05e54cfab69a2750668a385d3)
describes a plane that contains both the point and the straight line .
![\ mathbf {G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6d9c60d3cf462a9812e9a9d021d17c7bc272a5a)
![{\ mathbf {X}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd)
![{\ mathbf {L}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f5f750865376a1a4ae2b15a00b4ff9c75a66630)
Relationship between primal and dual Plücker matrices
So if the vector for any plane is either the zero vector or represents a point on the straight line, then it must apply
![\ mathbf {X} = [\ mathbf {L}] _ {\ times} \ mathbf {E}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c35a7875bb2905504b075e52442684bd9a84b043)
![\ mathbf {E}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7f22b39d51f780fc02859059c1757c606b9de2)
![\ forall \ mathbf {E} \ in \ mathbb {R} \ mathcal {P} ^ {3}: \, \ mathbf {X} = [\ mathbf {L}] _ {\ times} \ mathbf {E} \ text {lies on} \ mathbf {L} \ iff [\ tilde {\ mathbf {L}}] _ {\ times} \ mathbf {X} = \ mathbf {0}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/56f85e443cc6e43bed608dadd6f9306d83ecd8ab)
So:
![\ left ([\ tilde {\ mathbf {L}}] _ {\ times} [\ mathbf {L}] _ {\ times} \ right) ^ {\ top} = [\ mathbf {L}] _ {\ times} [\ tilde {\ mathbf {L}}] _ {\ times} = \ mathbf {0} \ in \ mathbb {R} ^ {4 \ times4}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/498f185932a71b4fcc3b5729ceeb5edf59678cc4)
The following product fulfills these properties:
![{\ displaystyle \ 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) = \ 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}, }](https://wikimedia.org/api/rest_v1/media/math/render/svg/89c0e729843adf99eae30179abac20880db2dae0)
due to the Graßmann-Plücker relation . With the uniqueness of the Plücker matrix down to scalar multiples, the primal Plücker coordinates are obtained
![\ mathbf {L} = \ left (L_ {01}, \, L_ {02}, \, L_ {03}, \, L_ {12}, \, L_ {31}, \, L_ {23} \ right ) ^ {\ top}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0cc237184324f3ecc2361f505b5d0be0ee4d8b43)
the following dual coordinates:
![\ tilde {\ mathbf {L}} = \ left (L_ {23}, \, - L_ {13}, \, L_ {12}, \, L_ {03}, \, - L_ {02}, \, L_ {01} \ right) ^ {\ top}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/25734788c07ae3ba8e8fbed24b861c0bce93efd2)
example
The projective straight line im corresponding to the - plane im can be represented by
![x_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308)
![x_4](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb828766e82e496666b179ff70d8e2fd24a79e5f)
![\ mathbb {R} ^ {4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4abb9b9dab94f7b25a4210364f0f9032704bfb9)
![\ mathbb {R} P ^ {3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75f9414aee8c40c1dfe131f562d6ca04312d298a)
In the projective plane
The 'join' of two points in the projective plane is the operation of connecting two points with a straight line. The straight line equation can be determined by the cross product :
![\ 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).](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a7b4cb673585b137e9bc92a1ddc4c84f3ed1677)
Dual , one may express the 'meet', ie the intersection of two lines by the cross product:
![\ mathbf {x} \ propto \ mathbf {l} \ times \ mathbf {m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14d775c12b39395cd5553b400deb0e877f13d80c)
If you now write the cross product as a multiplication with a skew-symmetrical matrix, the connection to the Plücker matrix becomes obvious:
![{\ 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/8be2fbe3f96438ab9b70fb855f312da37e3b1f93)
and analog
literature
- James F. Blinn: A Homogeneous Formulation for Lines in 3 Space . In: Proceedings of the 4th Annual Conference on Computer Graphics and Interactive Techniques (= SIGGRAPH '77 ). ACM, New York, NY, USA January 1, 1977, p. 237-241 , doi : 10.1145 / 563858.563900 ( acm.org [accessed August 4, 2016]).
Web links