Editing Plücker coordinates (section)

== Algebraic definition ==

=== Primal coordinates ===
In a 3-dimensional projective space {{tmath|\mathbb P^3}}, let {{mvar|L}} be a line through distinct points {{mvar|x}} and {{mvar|y}} with [[homogeneous coordinates]] {{math|(''x''{{sub|0}} : ''x''{{sub|1}} : ''x''{{sub|2}} : ''x''{{sub|3}})}} and {{math|(''y''{{sub|0}} : ''y''{{sub|1}} : ''y''{{sub|2}} : ''y''{{sub|3}})}}.

The Plücker coordinates {{mvar|p{{sub|ij}}}} are defined as follows:
:<math>p_{ij} = \begin{vmatrix} x_{i} & y_{i} \\ x_{j} & y_{j}\end{vmatrix} = x_{i}y_{j}-x_{j}y_{i} . </math>

(the skew symmetric matrix whose elements are {{mvar|p<sub>ij</sub>}} is also called the [[Plücker matrix]] )<br>
This implies {{math|1=''p<sub>ii</sub>'' = 0}} and {{math|1=''p<sub>ij</sub>'' = −''p<sub>ji</sub>''}}, reducing the possibilities to only six (4 [[binomial coefficient|choose]] 2) independent quantities. The sextuple

: <math>(p_{01}:p_{02}:p_{03}:p_{23}:p_{31}:p_{12}) </math>

is uniquely determined by {{mvar|L}} up to a common nonzero scale factor. Furthermore, not all six components can be zero.
Thus the Plücker coordinates of {{mvar|L}} may be considered as homogeneous coordinates of a point in a 5-dimensional projective space, as suggested by the colon notation.

To see these facts, let {{mvar|M}} be the 4×2 matrix with the point coordinates as columns.

: <math> M = \begin{bmatrix} x_0 &  y_0 \\ x_1 & y_1 \\ x_2 & y_2 \\ x_3 & y_3 \end{bmatrix}</math>

The Plücker coordinate {{mvar|p<sub>ij</sub>}} is the determinant of rows {{mvar|i}} and {{mvar|j}} of {{mvar|M}}.
Because {{math|'''x'''}} and {{math|'''y'''}} are distinct points, the columns of {{mvar|M}} are [[linear independence|linearly independent]]; {{mvar|M}} has [[rank (linear algebra)|rank]] 2. Let {{mvar|M&prime;}} be a second matrix, with columns {{math|'''x&prime;''', '''y&prime;'''}} a different pair of distinct points on {{mvar|L}}. Then the columns of {{mvar|M&prime;}} are [[linear combination]]s of the columns of {{mvar|M}}; so for some 2×2 [[nonsingular matrix]] {{math|Λ}},

: <math> M' = M\Lambda . </math>

In particular, rows {{mvar|i}} and {{mvar|j}} of {{mvar|M&prime;}} and {{mvar|M}} are related by

: <math> \begin{bmatrix} x'_{i} & y'_{i}\\x'_{j}& y'_{j} \end{bmatrix} = \begin{bmatrix} x_{i} & y_{i}\\x_{j}& y_{j} \end{bmatrix} \begin{bmatrix} \lambda_{00} & \lambda_{01} \\ \lambda_{10} & \lambda_{11} \end{bmatrix} . </math>

Therefore, the determinant of the left side 2×2 matrix equals the product of the determinants of the right side 2×2 matrices, the latter of which is a fixed scalar, {{math|det Λ}}. Furthermore, all six 2×2 subdeterminants in {{mvar|M}} cannot be zero because the rank of {{mvar|M}} is 2.

=== Plücker map ===
Denote the set of all lines (linear images of {{tmath|\mathbb P^1}}) in {{tmath|\mathbb P^3}} by {{math|''G''<sub>1,3</sub>}}.  We thus have a map:
:<math>\begin{align}
\alpha \colon \mathrm{G}_{1,3} & \rightarrow \mathbb P^5 \\
L & \mapsto L^{\alpha},
\end{align}</math>
where
:<math> L^{\alpha}=(p_{01}:p_{02}:p_{03}:p_{23}:p_{31}:p_{12}) . </math>

=== Dual coordinates ===
Alternatively, a line can be described as the intersection of two planes. Let {{mvar|L}} 
be a line contained in distinct planes {{math|'''a'''}} and {{math|'''b'''}} with homogeneous coefficients {{math|(''a''{{sup|0}} : ''a''{{sup|1}} : ''a''{{sup|2}} : ''a''{{sup|3}})}} and {{math|(''b''{{sup|0}} : ''b''{{sup|1}} : ''b''{{sup|2}} : ''b''{{sup|3}})}}, respectively. (The first plane equation is <math display=inline>\sum_k a^k x_k =0,</math> for example.) The dual Plücker coordinate {{mvar|p<sup>ij</sup>}} is

:<math>p^{ij} = \begin{vmatrix} a^{i} & a^{j} \\ b^{i} & b^{j}\end{vmatrix}  = a^{i}b^{j}-a^{j}b^{i} . </math>

Dual coordinates are convenient in some computations, and they are equivalent to primary coordinates:
: <math>
(p_{01}:p_{02}:p_{03}:p_{23}:p_{31}:p_{12})=
(p^{23}:p^{31}:p^{12}:p^{01}:p^{02}:p^{03})
</math>
Here, equality between the two vectors in homogeneous coordinates means that the numbers on the right side are equal to the numbers on the left side up to some common scaling factor {{math|λ}}. Specifically, let {{math|(''i'', ''j'', ''k'', ''ℓ'')}} be an [[even permutation]] of {{math|(0, 1, 2, 3)}}; then

: <math>p_{ij} = \lambda p^{k\ell} .  </math>

=== Geometry ===
To relate back to the geometric intuition, take {{math|1=''x''<sub>0</sub> = 0}} as the plane at infinity; thus the coordinates of points ''not'' at infinity can be normalized so that {{math|1=''x''<sub>0</sub> = 1}}. Then {{mvar|M}} becomes

: <math> M = \begin{bmatrix} 1 &  1 \\ x_1 & y_1 \\ x_2& y_2 \\ x_3 & y_3 \end{bmatrix} , </math>

and setting <math>x=(x_1,x_2,x_3)</math> and <math>y=(y_1,y_2,y_3)</math>, we have <math>d=(p_{01},p_{02},p_{03})</math>and <math>m=(p_{23},p_{31},p_{12})</math>.

Dually, we have <math>d=(p^{23},p^{31},p^{12})</math> and <math>m=(p^{01},p^{02},p^{03}).</math>