普吕克坐标：修订间差异
删除的内容 添加的内容
Symplectopedia（留言 | 贡献） 无编辑摘要 |
過期十世的翻譯 |
||
| 第1行： | 第1行： | ||
{{Translating|time=2008-7-15}} |
|||
[[数学]]上，'''普吕克坐标'''是将[[射影空间|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。 |
[[数学]]上，'''普吕克坐标'''是将[[射影空间|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。 |
||
| 第21行： | 第20行： | ||
==到克莱因二次曲面的单射性和满射性== |
==到克莱因二次曲面的单射性和满射性== |
||
{{TransH}} |
|||
As not all Plücker coordinates are 0, suppose <math>p_{ij}\neq 0</math>. This means that the line L has no point in common with the line through the points represented by basisvectors <math>e_{i}</math> and <math>e_{j}</math>. The equations for the hyperplane through L and <math>e_{i}</math>, and the hyperplane through <math>e_{j}</math>, 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 <math>\alpha</math> is injective. |
|||
The image of <math>\alpha</math> is not the complete set of points in PG(5,K). One can check that the Plücker coordinates of a line L satisfy : |
|||
<math>p_{01}p_{23}+p_{02}p_{31}+p_{03}p_{12}=0</math> |
|||
One can show that <math>\alpha</math> is a surjection (and thus bijection) into the [[Klein quadric]]=<math>V(P_{01}P_{23}+P_{02}P_{31}+P_{03}P_{12})</math>. |
|||
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 <math>\mathbb{P}^5</math> as lines in <math>\mathbb{P}^3</math>. We can use this characterization to easily find certain classes of lines. |
|||
* If we want all the lines through the point <math>(x_0:x_1:x_2:x_3) \in \mathbb{P}^3</math>, a direct calculation yields that these are just the lines with Plucker co-ordinates <math>\,(p_{01}:p_{02}:p_{03}:p_{12}:p_{13}:p_{23})</math> where <math>\,x_0x_2p_{13} - x_0x_1p_{23} - x_1x_3p_{02} = 0</math> |
|||
* If we want all the lines in the plane <math>x_0 = a_1x_1 + a_2x_2 + a_3x_3</math>, another calculation shows that these are just the points with Plucker co-ordinates satisfying <math>a_1p_{01} + a_2p_{02} + a_3p_{03} = 0</math> |
|||
==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. |
|||
<!-- |
|||
===Other formulations of Plücker co-ordinates=== |
|||
Plucker co-ordinates can be formulated more generally using the ideas of [[multilinear algebra]] and the [[wedge product]]. This leads to a simple formulation of the generalisation of the Plücker mapping. |
|||
A Plücker map is a map <math>\pi</math> |
|||
:<math> |
|||
\begin{matrix} |
|||
\pi : Gr(n,N) &\rightarrow& \mathbb{P}(\wedge^n\mathbb{C}^N)\\ |
|||
span( v_1, \ldots, v_n ) &\mapsto& \mathbb{C}( v_1 \wedge \ldots \wedge v_n )\\ |
|||
\end{matrix} |
|||
</math> |
|||
where ''Gr''(''n'',''N'') is the [[Grassmannian]], i.e. the space of all ''n''-dimensional subspaces of an ''N''-dimensional [[vector space]]. The map described above is the particular case where ''n'' = 2, ''N'' = 4. |
|||
In the general context, the Grassmannian can be completely characterised as an intersection of quadrics, each coming from a relation on the Plücker co-ordinates that derives from linear algebra. |
|||
{{TransF}} |
|||
[[Category:射影几何|P]] |
[[Category:射影几何|P]] |
||
[[Category:多线性代数|P]] |
[[Category:多线性代数|P]] |
||
2008年7月16日 (三) 01:05的版本
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，穿过点和点。
定义为的行列式。
这蕴涵着和.
考虑六元组。不是所有6个都可以同时为0，因为如果是的话，所有的子矩阵都是零，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：
考虑和为L上不同点，其中而。 p'和q'不同的假设归结为。 可以检验： 这样，
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：从W到一个K上的5维摄影空间：
