普吕克坐标

数学上，普吕克坐标是将射影三维空间中的每条线给予6个齐次坐标，也就是一个射影5维空间中的一点。普吕克坐标由朱利叶斯·普吕克于1844年给出。

定义

令L为一直线，穿过点$p(x_{0},x_{1},x_{2},x_{3})$和点$q(y_{0},y_{1},y_{2},y_{3})$。

定义$p_{ij}$为${\begin{pmatrix}x_{i}&x_{j}\\y_{i}&y_{j}\end{pmatrix}}=x_{i}y_{j}-x_{j}y_{i}$的行列式。

这蕴涵着$p_{ii}=0$和$p_{ij}=-p_{ji}$.

考虑六元组$(p_{01},p_{02},p_{03},p_{23},p_{31},p_{12})$。不是所有6个都可以同时为0，因为如果是的话，所有${\begin{pmatrix}x_{0}&x_{1}&x_{2}&x_{3}\\y_{0}&y_{1}&y_{2}&y_{3}\end{pmatrix}}$的$2\times 2$子矩阵都是零，则该矩阵最多秩为1，这个p及q为不同点的假设不符。

p和q的选取对于6元组的影响只是一个非零因子，如下所示：

考虑$p'(x'_{0},x'_{1},x'_{2},x'_{3})$和$q'(y'_{0},y'_{1},y'_{2},y'_{3})$为L上不同点，其中$x'_{i}=k_{1}x_{i}+l_{1}y_{i}$而$y'_{i}=k_{2}x_{i}+l_{2}y_{i}$。 p'和q'不同的假设归结为$k_{1}l_{2}-k_{2}l_{1}\neq 0$。 可以检验：${\begin{pmatrix}x'_{i}&x'_{j}\\y'_{i}&y'_{j}\end{pmatrix}}={\begin{pmatrix}k_{1}&l_{1}\\k_{2}&l_{2}\end{pmatrix}}{\begin{pmatrix}x_{i}&x_{j}\\y_{i}&y_{j}\end{pmatrix}}$ 这样，$(p'_{01},p'_{02},p'_{03},p'_{23},p'_{31},p'_{12})=(k_{1}l_{2}-k_{2}l_{1})(p_{01},p_{02},p_{03},p_{23},p_{31},p_{12})$

称W为所有PG(3,K)中的直线的集合。我们现在恰当地定义一个映射$\alpha$：从W到一个K上的5维摄影空间： $\alpha :W\rightarrow PG(5,K):L\rightarrow L^{\alpha }=(p_{01},p_{02},p_{03},p_{23},p_{31},p_{12})$

到克莱因二次曲面的单射性和满射性

已隱藏部分未翻譯内容，歡迎參與翻譯。

As not all Plücker coordinates are 0, suppose $p_{ij}\neq 0$. This means that the line L has no point in common with the line through the points represented by basisvectors $e_{i}$ and $e_{j}$. The equations for the hyperplane through L and $e_{i}$, and the hyperplane through $e_{j}$, can be expresed completely in function of the Plücker coordinates. As L is the unique common line on these two hyperplanes, the Plücker coordinates define the line and thus the map $\alpha$ is injective.

The image of $\alpha$ is not the complete set of points in PG(5,K). One can check that the Plücker coordinates of a line L satisfy : $p_{01}p_{23}+p_{02}p_{31}+p_{03}p_{12}=0$

One can show that $\alpha$ is a surjection (and thus bijection) into the Klein quadric=$V(P_{01}P_{23}+P_{02}P_{31}+P_{03}P_{12})$.

The lines in PG(3,K) thus correspond to the points on a quadric in PG(5,K).

Uses of the Plücker map

Using the Plücker map we can think of certain points in $\mathbb {P} ^{5}$ as lines in $\mathbb {P} ^{3}$. We can use this characterization to easily find certain classes of lines.

• If we want all the lines through the point $(x_{0}:x_{1}:x_{2}:x_{3})\in \mathbb {P} ^{3}$, a direct calculation yields that these are just the lines with Plucker co-ordinates $\,(p_{01}:p_{02}:p_{03}:p_{12}:p_{13}:p_{23})$ where $\,x_{0}x_{2}p_{13}-x_{0}x_{1}p_{23}-x_{1}x_{3}p_{02}=0$

• If we want all the lines in the plane $x_{0}=a_{1}x_{1}+a_{2}x_{2}+a_{3}x_{3}$, another calculation shows that these are just the points with Plucker co-ordinates satisfying $a_{1}p_{01}+a_{2}p_{02}+a_{3}p_{03}=0$

Line geometry

During the nineteenth century, line geometry was studied quite intensively. In terms of the formulation given above, this is a description of the intrinsic five-dimensional geometry on the quadric.