Editing Plücker coordinates (section)

=== 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}}.