Plücker coordinates

Plücker coordinates are coordinates for straight lines in 3-dimensional space. They are named after Julius Plücker . They are a special case of the more general Graßmann-Plücker coordinates .

To describe a straight line in 3-dimensional projective space , we select two points lying on the straight line and with homogeneous coordinates and and define for all ${\displaystyle L}$ ${\displaystyle P^{3}}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle \left[x_{0}\colon x_{1}\colon x_{2}\colon x_{3}\right]}$ ${\displaystyle \left[y_{0}\colon y_{1}\colon y_{2}\colon y_{3}\right]}$ ${\displaystyle i<j}$

${\displaystyle p_{ij}=\det \left({\begin{array}{cc}x_{i}&y_{i}\\x_{j}&y_{j}\end{array}}\right)=x_{i}y_{j}-x_{j}y_{i}}$.

Im is the point with homogeneous coordinates ${\displaystyle P^{5}}$

${\displaystyle \left[p_{01}\colon p_{02}\colon p_{03}\colon p_{12}\colon p_{13}\colon p_{23}\right]}$

well-defined, regardless of the choice of homogeneous coordinates for and . Since the determinant is multilinear and alternating, these coordinates depend only on the straight and not of the selected points on the straight and off. ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle L}$ ${\displaystyle x}$ ${\displaystyle y}$

The six coordinates satisfy the Plücker relation

${\displaystyle p_{01}p_{23}+p_{03}p_{12}=p_{02}p_{13}}$.

This equation describes a conical quadric in the as, Klein quadric is referred to. The straight lines im are thus parameterized by the points of the Klein quadric. ${\displaystyle P^{5}}$ ${\displaystyle P^{3}}$

With the coordinates and the Plücker relation can also be formulated as. A typical application is the description of straight lines lying in one plane. If a straight line by the coordinates or and a second straight line by the coordinates or is given, then are the straight lines , and if and only in a plane when ${\displaystyle d=(p_{01},p_{02},p_{03})}$ ${\displaystyle m=(p_{23},-p_{13},p_{12})}$ ${\displaystyle d\cdot m=0}$ ${\displaystyle L}$ ${\displaystyle (p_{01},\ldots ,p_{23})}$ ${\displaystyle (d,m)}$ ${\displaystyle L^{\prime }}$ ${\displaystyle (p_{01}^{\prime },\ldots ,p_{23}^{\prime })}$ ${\displaystyle (d^{\prime },m^{\prime })}$ ${\displaystyle L}$ ${\displaystyle L^{\prime }}$

${\displaystyle d\cdot m^{\prime }+m\cdot d^{\prime }=0}$

applies.

literature

• A. Beutelspacher, U. Rosenbaum: Projective geometry Vieweg + Teubner, 2nd edition, Braunschweig et al. 2004, ISBN 9783322803290 , pp. 162-171
• Plücker coordinates . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).