Plücker coordinates

In geometry, Plücker coordinates, introduced by Julius Plücker in the 19th century, are a way to assign six homogenous coordinates to each line in projective 3-space, "P"³. Because they satisfy a quadratic constraint, they establish a one-to-one correspondence between the 4-dimensional space of lines in "P"³ and points on a quadric in "P"⁵ (projective 5-space). 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.

Geometric intuition

A line "L" in 3-dimensional Euclidean space is determined by two distinct points that it contains, or by two distinct planes that contain it. Consider the first case, with points "x" = ("x"₁,"x"₂,"x"₃) and "y" = ("y"₁,"y"₂,"y"₃). The vector displacement from "x" to "y" is nonzero because the points are distinct, and represents the "direction" of the line. That is, every displacement between points on "L" is a scalar multiple of "d" = "y"−"x". If a physical particle of unit mass were to move from "x" to "y", it would have a moment about the origin. The geometric equivalent is a vector whose direction is perpendicular to the plane containing "L" and the origin, and whose length equals the area of the triangle formed by the displacement and the origin. Treating the points as displacements from the origin, the moment is "m" = "x"×"y", where "×" denotes the vector cross product. The area of the triangle is proportional to the length of the segment between "x" and "y", considered as the base of the triangle; it is not changed by sliding the base along the line, parallel to itself. By definition the moment vector is perpendicular to every displacement along the line, so "d"•"m" = 0, where "•" denotes the vector dot product.

Although neither "d" nor "m" alone is sufficient to determine "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

: ("d":"m") = ("d"₁:"d"₂:"d"₃:"m"₁:"m"₂:"m"₃)

may be considered homogeneous coordinates for "L", in the sense that all pairs (λ"d":λ"m"), 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.

: Example. Let "x" = (2,3,7) and "y" = (2,1,0). Then ("d":"m") = (0:−2:−7:−7:14:−4).

Alternatively, let the equations for points "x" of two distinct planes containing "L" be

: 0 = "a" + "a"•"x": 0 = "b" + "b"•"x" .

Then their respective planes are perpendicular to vectors "a" and "b", and the direction of "L" must be perpendicular to both. Hence we may set "d" = "a"×"b", which is nonzero because "a" and "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

:

Dual coordinates are convenient in some computations, and we can show that they are equivalent to primary coordinates. Specifically, let ("i","j","k","l") be an even permutation of (0,1,2,3); then

: $p\_\{ij\}\; =\; p^\{kl\}\; .\; ,!$

Geometry

To relate back to the geometric intuition, take "x"₀ = 0 as the plane at infinity; thus the coordinates of points "not" at infinity can normalized so that "x"₀ = 1. Then "M" becomes

:

and setting "x" = ("x"₁,"x"₂,"x"₃) and "y" = ("y"₁,"y"₂,"y"₃), we have "d" = ("p"₀₁,"p"₀₂,"p"₀₃) and "m" = ("p"₂₃,"p"₃₁,"p"₁₂).

Dually, we have "d" = ("p"²³,"p"³¹,"p"¹²) and "m" = ("p"⁰¹,"p"⁰²,"p"⁰³).

Bijection between lines and Klein quadric

Plane equations

If the point z = ("z"₀:"z"₁:"z"₂:"z"₃) lies on "L", then the columns of

:

are linearly dependent, so that the rank of this larger matrix is still 2. This implies that all 3×3 submatrices have determinant zero, generating four (4 choose 3) plane equations, such as

:

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

Point equations

Letting ("x"₀:"x"₁:"x"₂:"x"₃) be the point coordinates, four possible points on a line each have coordinates "x"_"i" = "p"_"ij", for "j" = 0…3. Some of these possible points may be inadmissible because all coordinates are zero, but since at least one Plücker coordinate is nonzero, at least two distinct points are guaranteed.

Bijectivity

If ("q"₀₁:"q"₀₂:"q"₀₃:"q"₂₃:"q"₃₁:"q"₁₂) are the homogeneous coordinates of a point in "P"⁵, without loss of generality assume that "q"₀₁ is nonzero. Then the matrix

:

has rank 2, and so its columns are distinct points defining a line "L". When the "P"⁵ coordinates, "q"_"ij", satisfy the quadratic Plücker relation, they are the Plücker coordinates of "L". To see this, first normalize "q"₀₁ to 1. Then we immediately have that for the Plücker coordinates computed from "M", "p"_"ij" = "q"_"ij", except for

: $p\_\{23\}\; =\; -\; q\_\{03\}\; q\_\{12\}\; -\; q\_\{02\}\; q\_\{31\}\; .\; ,!$

But if the "q"_"ij" satisfy the Plücker relation "q"₂₃+"q"₀₂"q"₃₁+"q"₀₃"q"₁₂ = 0, then "p"_"23" = "q"_"23", completing the set of identities.

Consequently, α is a surjection onto the algebraic variety consisting of the set of zeros of the quadratic polynomial

: $p\_\{01\}p\_\{23\}+p\_\{02\}p\_\{31\}+p\_\{03\}p\_\{12\}\; .\; ,!$

And since α is also an injection, the lines in "P"³ are thus in bijective correspondence with the points of this quadric in "P"⁵, called the Plücker quadric or Klein quadric.

Uses

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

Line-line crossing

Two lines in "P"³ are either skew or coplanar, and in the latter case they are either coincident or intersect in a unique point. If "p"_"ij" and "p"′_"ij" are the Plücker coordinates of two lines, then they are coplanar precisely when "d"⋅"m"′+"m"⋅"d"′ = 0, 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.

Line-line join

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

: 0 = ("m"•"d"′)"x"₀ + ("d"×"d"′)•"x" ,

where "x" = ("x"₁,"x"₂,"x"₃).

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

: ("x"₀ : "x") = (d•m′:m×m′) .

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

Plane-line meet

Given a plane with equation

: $0\; =\; a^0x\_0\; +\; a^1x\_1\; +\; a^2x\_2\; +\; a^3x\_3\; ,\; ,!$

or more concisely 0 = "a"⁰"x"₀+"a"•"x"; and given a line not in it with Plücker coordinates ("d":"m"), then their point of intersection is

: ("x"₀ : "x") = ("a"•"d" : "a"×"m" − "a"₀"d") .

The point coordinates, ("x"₀:"x"₁:"x"₂:"x"₃), can also be expressed in terms of Plücker coordinates as

: $x\_i\; =\; sum\_\{j\; e\; i\}\; a^j\; p\_\{ij\}\; ,\; qquad\; i\; =\; 0\; ldots\; 3\; .\; ,!$

Point-line join

Dually, given a point ("y"₀:"y") and a line not containing it, their common plane has equation

: 0 = ("y"•"m") "x"₀ + ("y"×"d"−"y"₀"m")•"x" .

The plane coordinates, ("a"⁰:"a"¹:"a"²:"a"³), can also be expressed in terms of dual Plücker coordinates as

: $a^i\; =\; sum\_\{j\; e\; i\}\; y\_j\; p^\{ij\}\; ,\; qquad\; i\; =\; 0\; ldots\; 3\; .\; ,!$

Line families

Because the Klein quadric is in "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 "P"³.

For example, suppose "L" and "L"′ are distinct lines in "P"³ 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 "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 "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.

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