From Wikipedia, the free encyclopedia
This article is about the classic case of
lines in projective 3space. For general Plücker coordinates, see
Plücker
embedding.
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 3space,
P^{3}. Because they satisfy a quadratic
constraint, they establish a onetoone
correspondence between the 4dimensional space of lines in
P^{3} and points on a quadric in
P^{5} (projective 5space). A predecessor and
special case of Grassmann
coordinates (which describe kdimensional linear
subspaces, or flats, in an ndimensional 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
Displacement and moment of two points on line
A line L in 3dimensional 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
twice 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
(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.
 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

0 
= a (b +
b•x) −
b (a +
a•x) 

= (a b − b
a)•x . 
That is, m =
a b − b
a 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.
 Example. Let a_{0} = 2,
a = (−1,0,0) and
b_{0} = −7,
b = (0,7,−2). Then
(d:m) =
(0:−2:−7:−7:14:−4).
Although the usual algebraic definition tends to obscure the
relationship,
(d:m) are the
Plücker coordinates of L.
Algebraic
definition
In a 3dimensional projective space, P^{3}, let
L be a line containing distinct points x
and y with homogeneous coordinates
(x_{0}:x_{1}:x_{2}:
x_{3}) and
(y_{0}:y_{1}:y_{2}:
y_{3}), respectively. Let M be the 4×2 matrix
with these coordinates as columns.
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′ and
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.
Primary
coordinates
With this motivation, we define Plücker coordinate
p_{ij} as the determinant of rows
i and j of M,

This implies p_{ii} = 0 and
p_{ij} =
−p_{ji}, reducing the possibilities to
only six (4 choose 2) independent quantities.
As we have seen, the sixtuple
is uniquely determined by L, up to a common nonzero
scale factor. Furthermore, all six components cannot be zero,
because if they were, all
subdeterminants in M would be zero and the rank of
M at most one, contradicting the assumption that
x and y are distinct. Thus the
Plücker coordinates of L, as suggested by the colons, may
be considered homogeneous coordinates of a point in a 5dimensional
projective space.
Plücker
map
Denote the set of all lines (linear images of
P^{1}) in P^{3} by
G_{1,3}. We thus have a map α from G_{1,3} to a
5dimensional projective space, given by
where
Dual
coordinates
Alternatively, let L be a line contained in distinct
planes a and b with homogeneous
coefficients
(a^{0}:a^{1}:a^{2}:
a^{3}) and
(b^{0}:b^{1}:b^{2}:
b^{3}), respectively. (The first plane equation is
0 =
∑_{k} a^{k}x_{
k}, for example.) Let N be the 2×4 matrix
with these coordinates as rows.
We define dual Plücker coordinate
p^{ij} as the determinant of columns
i and j of N,

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
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

The four possible planes obtained are as follows.
Using dual coordinates, and letting
(a^{0}:a^{1}:a^{2}:
a^{3}) be the line coefficients, each of these is
simply a^{i} =
p^{ij}, or
Each Plücker coordinate appears in two of the four equations,
each time multiplying a different variable; and as at least one of
the coordinates is nonzero, we are guaranteed nonvacuous equations
for two distinct planes intersecting in L. Thus the
Plücker coordinates of a line determine that line uniquely, and the
map α is an injection.
Quadratic
relation
The image of α is not the complete set of points in
P^{5}; the Plücker coordinates of a line
L satisfy the quadratic Plücker relation

For proof, write this homogeneous polynomial as determinants and
use Laplace
expansion (in reverse).

Since both 3×3 determinants have duplicate columns, the right
hand side is identically zero.
Another proof may be done like this: Since vector

is perpendicular to vector

(see above), the scalar product of d and m must be zero!
q.e.d.
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
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
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 3dimensional space, especially those involving incidence.
Lineline
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 righthanded screw takes
L into L′, else negative.
The quadratic Plücker relation essentially states that a line is
coplanar with itself.
Lineline
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 nearparallelism of the lines will cause numeric
difficulties in finding such a plane even if it does exist.
Lineline
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.
Planeline
meet
Given a plane with equation
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
Pointline
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
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 twoparameter 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 oneparameter family of lines containing
L and L′. This corresponds to a onedimensional
linear subspace belonging to the Klein quadric.
Lines in
plane
If three distinct and nonparallel lines are coplanar; their
linear combinations generate a twoparameter family of lines, all
the lines in the plane. This corresponds to a twodimensional
linear subspace belonging to the Klein quadric.
Lines
through point
If three distinct and noncoplanar lines intersect in a point,
their linear combinations generate a twoparameter family of lines,
all the lines through the point. This also corresponds to a
twodimensional 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.
References
 Hodge, W. V. D.; D. Pedoe (1994).
Methods of Algebraic Geometry, Volume I
(Book II). Cambridge University Press.
ISBN
9780521469005.
 Behnke, H.; F. Bachmann, K. Fladt,
H. Kunle (eds.) (1984). Fundamentals of Mathematics, Volume II:
Geometry. trans. S. H. Gould. MIT Press. ISBN
9780262520942.
From the German: Grundzüge der Mathematik, Band II:
Geometrie. Vandenhoeck & Ruprecht.
 Kuptsov, L.P.
(2001), "Plücker coordinates", in
Hazewinkel, Michiel, Encyclopaedia of
Mathematics, Kluwer Academic Publishers, ISBN
9781556080104, http://eom.springer.de/P/p072890.htm
 Stolfi, Jorge (1991). Oriented
Projective Geometry. Academic Press. ISBN
9780126720259.
From original Stanford Ph.D. dissertation, Primitives
for Computational Geometry, available as DEC SRC Research Report
36.
 Shoemake, Ken (1998). "Plücker Coordinate
Tutorial". Ray Tracing News 11
(1). http://www.acm.org/tog/resources/RTNews/html/rtnv11n1.html#art3.
 Mason, Matthew T.; J. Kenneth
Salisbury (1985). Robot Hands and the Mechanics of
Manipulation. MIT
Press. ISBN
9780262132053.
 Hohmeyer, M.; S. Teller (1999).
"Determining the Lines
Through Four Lines" (PDF). Journal of Graphics
Tools (A K Peters) 4 (3): 11–22.
ISSN 10867651. http://people.csail.mit.edu/seth/pubs/TellerHohmeyerJGT2000.pdf.