equation 0 = (m•d′)x0 + (d×d′)•x , where 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 (x0 : x) = (d•m′:m×m′) . To handle lines not meeting this restriction, see the references. Plane-line meet Given a plane with equation or more concisely 0 = a0x0+a•x; and given a line not in it with Plücker coordinates (d:m), then their point of intersection is (x0 : x) = (a•d : a×m − a0d) . The point coordinates, (x0:x1:x2:x3), can also be expressed in terms of Plücker coordinates as Point-line join Dually, given a point (y0:y) and a line not containing it, their common plane has equation 0 = (y•m) x0 + (y×d−y0m)•x . The plane coordinates, (a0:a1:a2:a3), can also be expressed in terms of dual Plücker coordinates as Line families Because the Klein quadric is in P5, it contains linear subspaces of dimensions one and two (but no higher). These correspond to one- and two-parameter families of lines in P3. For example, suppose L and L′ are distinct lines in P3 determined by points x, y and x′, y′, respectively. Linear combinations of their determining points give linear combinations of their Plücker coordinates, generating a one-parameter family of lines containing L and L′. 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 P3 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