Plücker matrix

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 . ${\displaystyle 4\times 4}$

definition

A straight line in space is defined by two different points and in homogeneous coordinates of the projective space . Your Plücker matrix is: ${\displaystyle A=\left(A_{0},A_{1},A_{2},A_{3}\right)^{\top }\in \mathbb {R} {\mathcal {P}}^{3}}$ ${\displaystyle B=\left(B_{0},B_{1},B_{2},B_{3}\right)^{\top }\in \mathbb {R} {\mathcal {P}}^{3}}$

${\displaystyle [\mathbf {L} ]_{\times }\propto \mathbf {A} \mathbf {B} ^{\top }-\mathbf {B} \mathbf {A} ^{\top }=\left({\begin{array}{cccc}0&-L_{01}&-L_{02}&-L_{03}\\L_{01}&0&-L_{12}&-L_{13}\\L_{02}&L_{12}&0&-L_{23}\\L_{03}&L_{13}&L_{23}&0\end{array}}\right)}$

Whereby the asymmetrical - matrix by the 6 Plücker coordinates ${\displaystyle 4\times 4}$

${\displaystyle \mathbf {L} \propto (L_{01},L_{02},L_{03},L_{12},L_{13},L_{23})^{\top }}$

With

${\displaystyle L_{ij}=A_{i}B_{j}-B_{i}A_{j}}$

is described. The Plücker coordinates satisfy the Graßmann-Plücker relation

${\displaystyle L_{01}L_{23}-L_{02}L_{13}+L_{03}L_{12}=0}$

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. ${\displaystyle \mathbb {R} ^{3}}$

properties

The Plücker matrix allows the following geometric operations to be expressed in the matrix-vector product:

• Plane contains straight lines: ${\displaystyle \mathbf {0} =[\mathbf {L} ]_{\times }\mathbf {E} }$

• ${\displaystyle \mathbf {X} =[\mathbf {L} ]_{\times }\mathbf {E} }$is the intersection of the straight line with the plane ('Meet') ${\displaystyle \mathbf {L} }$ ${\displaystyle \mathbf {E} }$

• Point lies on a straight line: ${\displaystyle \mathbf {0} =[{\tilde {\mathbf {L} }}]_{\times }\mathbf {X} }$

• ${\displaystyle \mathbf {E} =[{\tilde {\mathbf {L} }}]_{\times }\mathbf {X} }$is the common plane that contains the point and the straight line ('join'). ${\displaystyle \mathbf {E} }$ ${\displaystyle \mathbf {X} }$ ${\displaystyle \mathbf {L} }$

• Direction of a straight line: (Note: Can also be interpreted as a plane through the coordinate origin, orthogonal to the straight line) ${\displaystyle [\mathbf {L} ]_{\times }\pi ^{\infty }=[\mathbf {L} ]_{\times }(0,0,0,1)^{\top }=(-L_{03},-L_{13},-L_{23},0)^{\top }}$

• Point closest to the origin of coordinates: ${\displaystyle \mathbf {X} _{0}\cong [\mathbf {L} ]_{\times }[\mathbf {L} ]_{\times }\pi ^{\infty }.}$

Uniqueness

Any two different points on the straight line can be found by linear combinations of and : ${\displaystyle \mathbf {A} }$ ${\displaystyle \mathbf {B} }$

${\displaystyle \mathbf {A} ^{\prime }\propto \mathbf {A} \alpha +\mathbf {B} \beta {\text{ und }}\mathbf {B} ^{\prime }\propto \mathbf {A} \gamma +\mathbf {B} \delta }$.

Your Plücker matrix is ​​then:

${\displaystyle [\mathbf {L} ^{\prime }]_{\times }=\mathbf {A} ^{\prime }\mathbf {B} ^{\prime \top }-\mathbf {B} ^{\prime }\mathbf {A} ^{\prime \top }=(\mathbf {A} \alpha +\mathbf {B} \beta )(\mathbf {A} \gamma +\mathbf {B} \delta )^{\top }-(\mathbf {A} \gamma +\mathbf {B} \delta )(\mathbf {A} \alpha +\mathbf {B} \beta )^{\top }={\underset {\lambda }{\underbrace {(\alpha \delta -\beta \gamma )} }}[\mathbf {L} ]_{\times },}$

thus identical to up to a scalar multiple . ${\displaystyle [\mathbf {L} ]_{\times }}$

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 ${\displaystyle \mathbf {E} =(E_{0},E_{1},E_{2},E_{3})^{\top }\in \mathbb {R} {\mathcal {P}}^{3}}$

${\displaystyle E_{0}x+E_{1}y+E_{2}z+E_{3}=0.}$

which does not contain the straight line . Then the matrix-vector product of the Plücker matrix with the plane describes a point ${\displaystyle \mathbf {L} }$

${\displaystyle \mathbf {X} =[\mathbf {L} ]_{\times }\mathbf {E} =\mathbf {A} {\underset {\alpha }{\underbrace {\mathbf {B} ^{\top }\mathbf {E} } }}-\mathbf {B} {\underset {\beta }{\underbrace {\mathbf {A} ^{\top }\mathbf {E} } }}=\mathbf {A} \alpha +\mathbf {B} \beta ,}$

which lies on the straight line because it is a linear combination of and . is also on the plane ${\displaystyle \mathbf {L} }$ ${\displaystyle \mathbf {A} }$ ${\displaystyle \mathbf {B} }$ ${\displaystyle \mathbf {X} }$ ${\displaystyle \mathbf {E} }$

${\displaystyle \mathbf {E} ^{\top }\mathbf {X} =\mathbf {E} ^{\top }[\mathbf {L} ]_{\times }\mathbf {E} ={\underset {\alpha }{\underbrace {\mathbf {E} ^{\top }\mathbf {A} } }}{\underset {\beta }{\underbrace {\mathbf {B} ^{\top }\mathbf {E} } }}-{\underset {\beta }{\underbrace {\mathbf {E} ^{\top }\mathbf {B} } }}{\underset {\alpha }{\underbrace {\mathbf {A} ^{\top }\mathbf {E} } }}=0,}$

and must therefore be 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: ${\displaystyle \mathbf {L} }$

${\displaystyle \alpha =\beta =0\iff \mathbf {E} }$ contains ${\displaystyle \mathbf {L} .}$

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

${\displaystyle E=\left(E_{0},E_{1},E_{2},E_{3}\right)^{\top }\in \mathbb {R} {\mathcal {P}}^{3}}$

and

${\displaystyle F=\left(F_{0},F_{1},F_{2},F_{3}\right)^{\top }\in \mathbb {R} {\mathcal {P}}^{3}}$

in homogeneous coordinates of the projective space . Your Plücker matrix is:

${\displaystyle [{\tilde {\mathbf {L} }}]_{\times }=\mathbf {E} \mathbf {F} ^{\top }-\mathbf {F} \mathbf {E} ^{\top }}$

and

${\displaystyle \mathbf {G} =[{\tilde {\mathbf {L} }}]_{\times }\mathbf {X} }$

describes a plane that contains both the point and the straight line . ${\displaystyle \mathbf {G} }$ ${\displaystyle \mathbf {X} }$ ${\displaystyle \mathbf {L} }$

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 ${\displaystyle \mathbf {X} =[\mathbf {L} ]_{\times }\mathbf {E} }$ ${\displaystyle \mathbf {E} }$

${\displaystyle \forall \mathbf {E} \in \mathbb {R} {\mathcal {P}}^{3}:\,\mathbf {X} =[\mathbf {L} ]_{\times }\mathbf {E} {\text{ lies on }}\mathbf {L} \iff [{\tilde {\mathbf {L} }}]_{\times }\mathbf {X} =\mathbf {0} .}$

So:

${\displaystyle \left([{\tilde {\mathbf {L} }}]_{\times }[\mathbf {L} ]_{\times }\right)^{\top }=[\mathbf {L} ]_{\times }[{\tilde {\mathbf {L} }}]_{\times }=\mathbf {0} \in \mathbb {R} ^{4\times 4}.}$

The following product fulfills these properties:

${\displaystyle \left({\begin{array}{cccc}0&L_{23}&-L_{13}&L_{12}\\-L_{23}&0&L_{03}&-L_{02}\\L_{13}&-L_{03}&0&L_{01}\\-L_{12}&L_{02}&-L_{01}&0\end{array}}\right)\left({\begin{array}{cccc}0&-L_{01}&-L_{02}&-L_{03}\\L_{01}&0&-L_{12}&-L_{13}\\L_{02}&L_{12}&0&-L_{23}\\L_{03}&L_{13}&L_{23}&0\end{array}}\right)=\left(L_{01}L_{23}-L_{02}L_{13}+L_{03}L_{12}\right)\cdot \left({\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{array}}\right)=\mathbf {0} ,}$

due to the Graßmann-Plücker relation . With the uniqueness of the Plücker matrix down to scalar multiples, the primal Plücker coordinates are obtained

${\displaystyle \mathbf {L} =\left(L_{01},\,L_{02},\,L_{03},\,L_{12},\,L_{31},\,L_{23}\right)^{\top }}$

the following dual coordinates:

${\displaystyle {\tilde {\mathbf {L} }}=\left(L_{23},\,-L_{13},\,L_{12},\,L_{03},\,-L_{02},\,L_{01}\right)^{\top }.}$

example

The projective straight line im corresponding to the - plane im can be represented by ${\displaystyle x_{1}}$ ${\displaystyle x_{4}}$ ${\displaystyle \mathbb {R} ^{4}}$ ${\displaystyle \mathbb {R} P^{3}}$

${\displaystyle {\begin{pmatrix}0\\0\\0\\1\end{pmatrix}}{\begin{pmatrix}1&0&0&0\end{pmatrix}}-{\begin{pmatrix}1\\0\\0\\0\end{pmatrix}}{\begin{pmatrix}0&0&0&1\end{pmatrix}}={\begin{pmatrix}0&0&0&-1\\0&0&0&0\\0&0&0&0\\1&0&0&0\end{pmatrix}}}$

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 :

${\displaystyle \mathbf {l} \propto \mathbf {a} \times \mathbf {b} =\left({\begin{array}{c}a_{1}b_{2}-b_{1}a_{2}\\b_{0}a_{2}-a_{0}b_{2}\\a_{0}b_{1}-a_{1}b_{0}\end{array}}\right)=\left({\begin{array}{c}l_{0}\\l_{1}\\l_{2}\end{array}}\right).}$

Dual , one may express the 'meet', ie the intersection of two lines by the cross product:

${\displaystyle \mathbf {x} \propto \mathbf {l} \times \mathbf {m} }$

If you now write the cross product as a multiplication with a skew-symmetrical matrix, the connection to the Plücker matrix becomes obvious:

${\displaystyle [\mathbf {l} ]_{\times }=\mathbf {a} \mathbf {b} ^{\top }-\mathbf {b} \mathbf {a} ^{\top }=\left({\begin{array}{ccc}0&-l_{2}&l_{1}\\l_{2}&0&-l_{0}\\-l_{1}&l_{0}&0\end{array}}\right)}$

and analog ${\displaystyle [\mathbf {x} ]_{\times }=\mathbf {l} \mathbf {m} ^{\top }-\mathbf {m} \mathbf {l} ^{\top }}$

literature

• Jürgen Richter-Gebert : Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Projective Geometry . Springer Science & Business Media, 2011, ISBN 978-3-642-17286-1 .

• Jorge Stolfi : Oriented Projective Geometry: A Framework for Geometric Computations . Academic Press , 1991, ISBN 978-1-4832-4704-5 . From original Stanford Ph.D. dissertation, Primitives for Computational Geometry , available as [1] .
  

• 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

• Presentation on Projective Space in Computer Vision (PDF, 0.35 MB)