Editing Plücker coordinates (section)

==Uses==
Plücker coordinates allow concise solutions to problems of line geometry in 3-dimensional space, especially those involving [[incidence (geometry)|incidence]].

=== Line-line crossing ===
{{further|Line-line intersection}}

Two lines in {{tmath|\mathbb P^3}} are either [[skew lines|skew]] or [[coplanar]], and in the latter case they are either coincident or intersect in a unique point. If {{mvar|p<sub>ij</sub>}} and {{mvar|p&prime;<sub>ij</sub>}} are the Plücker coordinates of two lines, then they are coplanar precisely when

:<math>\mathbf d \cdot \mathbf m' + \mathbf m \cdot \mathbf d' = 0,</math>

as shown by

: <math>
\begin{align}
0 & = p_{01}p'_{23} + p_{02}p'_{31} + p_{03}p'_{12} + p_{23}p'_{01} + p_{31}p'_{02} + p_{12}p'_{03} \\[5pt]
& = \begin{vmatrix}x_0&y_0&x'_0&y'_0\\
x_1&y_1&x'_1&y'_1\\
x_2&y_2&x'_2&y'_2\\
x_3&y_3&x'_3&y'_3\end{vmatrix}.
\end{align}
</math>

When the lines are skew, the sign of the result indicates the sense of crossing: positive if a right-handed screw takes {{mvar|L}} into {{mvar|L&prime;}}, else negative.

The quadratic Plücker relation essentially states that a line is coplanar with itself.

=== Line-line join ===
In the event that two lines are coplanar but not parallel, their common plane has equation

:<math>0 = (\mathbf m \cdot \mathbf d')x_0 + (\mathbf d \times \mathbf d')\cdot \mathbf x,</math>

where <math>x=(x_1,x_2,x_3).</math>

The slightest perturbation will destroy the existence of a common plane, and near-parallelism of the lines will cause numeric difficulties in finding such a plane even if it does exist.

=== Line-line meet ===
Dually, two coplanar lines, neither of which contains the origin, have common point

: <math>(x_0:\mathbf x) = (\mathbf d \cdot \mathbf m': \mathbf m \times \mathbf m').</math>

To handle lines not meeting this restriction, see the references.

=== Plane-line meet ===
{{further|Plane-line intersection}}

Given a plane with equation

: <math> 0 =  a^0x_0 + a^1x_1 + a^2x_2 + a^3x_3 , </math>

or more concisely,
:<math>0 = a^0x_0 + \mathbf a \cdot \mathbf x;</math>
and given a line not in it with Plücker coordinates {{math|('''d''' : '''m''')}}, then their point of intersection is

: <math>(x_0 : \mathbf x) = (\mathbf a \cdot \mathbf d : \mathbf a \times \mathbf m - a_0\mathbf d) .</math>

The point coordinates, {{math|(''x''{{sub|0}} : ''x''{{sub|1}} : ''x''{{sub|2}} : ''x''{{sub|3}})}}, can also be expressed in terms of Plücker coordinates as

: <math> x_i = \sum_{j \ne i} a^j p_{ij} , \qquad i = 0 \ldots 3 . </math>

=== Point-line join ===
Dually, given a point {{math|(''y''<sub>0</sub> : '''y''')}} and a line not containing it, their common plane has equation

: <math>0 = (\mathbf y \cdot \mathbf m) x_0 + (\mathbf y \times \mathbf d - y_0 \mathbf m)\cdot \mathbf x.</math>

The plane coordinates, {{math|(''a''{{sup|0}} : ''a''{{sup|1}} : ''a''{{sup|2}} : ''a''{{sup|3}})}}, can also be expressed in terms of dual Plücker coordinates as

: <math> a^i = \sum_{j \ne i} y_j p^{ij} , \qquad i = 0 \ldots 3 . </math>

=== Line families ===
Because the [[Klein quadric]] is in {{tmath|\mathbb P^5}}, it contains linear subspaces of dimensions one and two (but no higher). These correspond to one- and two-parameter families of lines in {{tmath|\mathbb P^3}}.

For example, suppose {{mvar|L, L&prime;}} are distinct lines in {{tmath|\mathbb P^3}} determined by points {{math|'''x''', '''y'''}} and {{math|'''x'''&prime;, '''y'''&prime;}}, respectively. Linear combinations of their determining points give linear combinations of their Plücker coordinates, generating a one-parameter family of lines containing {{mvar|L}} and {{math|''L''&prime;}}. This corresponds to a one-dimensional linear subspace belonging to the Klein quadric.

==== Lines in plane ====
If three distinct and non-parallel lines are coplanar; their linear combinations generate a two-parameter family of lines, all the lines in the plane. This corresponds to a two-dimensional linear subspace belonging to the Klein quadric.

==== Lines through point ====
If three distinct and non-coplanar lines intersect in a point, their linear combinations generate a two-parameter family of lines, all the lines through the point. This also corresponds to a two-dimensional linear subspace belonging to the Klein quadric.

==== Ruled surface ====
A [[ruled surface]] is a family of lines that is not necessarily linear. It corresponds to a curve on the Klein quadric. For example, a [[hyperboloid of one sheet]] is a quadric surface in {{tmath|\mathbb P^3}} ruled by two different families of lines, one line of each passing through each point of the surface; each family corresponds under the Plücker map to a [[conic section]] within the Klein quadric in {{tmath|\mathbb P^5}}.

=== Line geometry ===
During the nineteenth century, ''line geometry'' was studied intensively. In terms of the bijection given above, this is a description of the intrinsic geometry of the Klein quadric.

==== Ray tracing ====
Line geometry is extensively used in [[Ray tracing (graphics)|ray tracing]] application where the geometry and intersections of rays need to be calculated in 3D. An implementation is described in
[http://www.flipcode.com/archives/Introduction_To_Plcker_Coordinates.shtml Introduction to Plücker Coordinates] written for the Ray Tracing forum by Thouis Jones.