Plücker coordinates

Because they satisfy a quadratic constraint, they establish a one-to-one correspondence between the 4-dimensional space of lines in ⁠

A predecessor and special case of Grassmann coordinates (which describe k-dimensional linear subspaces, or flats, in an n-dimensional Euclidean space), Plücker coordinates arise naturally in geometric algebra.

They have proved useful for computer graphics, and also can be extended to coordinates for the screws and wrenches in the theory of kinematics used for robot control.

The vector displacement from x to y is nonzero because the points are distinct, and represents the direction of the line.

If a physical particle of unit mass were to move from x to y, it would have a moment about the origin of the coordinate system.

Treating the points as displacements from the origin, the moment is m = x × y, where "×" denotes the vector cross product.

Although neither direction d nor moment m alone is sufficient to determine the line L, together the pair does so uniquely, up to a common (nonzero) scalar multiple which depends on the distance between x and y.

That is, the coordinates may be considered homogeneous coordinates for L, in the sense that all pairs (λd : λm), for λ ≠ 0, can be produced by points on L and only L, and any such pair determines a unique line so long as d is not zero and d ⋅ m = 0.

Furthermore, this approach extends to include points, lines, and a plane "at infinity", in the sense of projective geometry.

Hence we may set d = a × b, which is nonzero because a, b are neither zero nor parallel (the planes being distinct and intersecting).

If point x satisfies both plane equations, then it also satisfies the linear combination That is, is a vector perpendicular to displacements to points on L from the origin; it is, in fact, a moment consistent with the d previously defined from a and b.

Proof 1: Need to show that Without loss of generality, let Point B is the origin.

Although the usual algebraic definition tends to obscure the relationship, (d : m) are the Plücker coordinates of L. In a 3-dimensional projective space ⁠

The sextuple is uniquely determined by L up to a common nonzero scale factor.

The Plücker coordinate pij is the determinant of rows i and j of M. Because x and y are distinct points, the columns of M are linearly independent; M has rank 2.

Let M′ be a second matrix, with columns x′, y′ a different pair of distinct points on L. Then the columns of M′ are linear combinations of the columns of M; so for some 2×2 nonsingular matrix Λ, In particular, rows i and j of M′ and M are related by 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, det Λ.

Let L be a line contained in distinct planes a and b with homogeneous coefficients (a0 : a1 : a2 : a3) and (b0 : b1 : b2 : b3), respectively.

lies on L, then the columns of are linearly dependent, so that the rank of this larger matrix is still 2.

Since both 3×3 determinants have duplicate columns, the right hand side is identically zero.

Then the matrix has rank 2, and so its columns are distinct points defining a line L. When the ⁠

Plücker coordinates allow concise solutions to problems of line geometry in 3-dimensional space, especially those involving incidence.

⁠ are either skew or coplanar, and in the latter case they are either coincident or intersect in a unique point.

If pij and p′ij are the Plücker coordinates of two lines, then they are coplanar precisely when as shown by When the lines are skew, the sign of the result indicates the sense of crossing: positive if a right-handed screw takes L into L′, else negative.

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

In the event that two lines are coplanar but not parallel, their common plane has equation where

This corresponds to a one-dimensional linear subspace belonging to the Klein quadric.

This corresponds to a two-dimensional linear subspace belonging to the Klein quadric.

This also corresponds to a two-dimensional linear subspace belonging to the Klein quadric.

In terms of the bijection given above, this is a description of the intrinsic geometry of the Klein quadric.

An implementation is described in Introduction to Plücker Coordinates written for the Ray Tracing forum by Thouis Jones.