In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, Plücker coordinates, introduced by
Julius Plücker in the 19th century, are a way to assign six
homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometr ...
to each
line
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Art ...
in
projective 3-space, P
3. Because they satisfy a quadratic constraint, they establish a
one-to-one correspondence
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
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
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
), Plücker coordinates arise naturally in
geometric algebra
In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the ...
. They have proved useful for
computer graphics
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
, and also can be extended to coordinates for the
screws and wrenches in the theory of
kinematics
Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fiel ...
used for
robot control.
Geometric intuition
A line
in 3-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
is determined by two distinct points that it contains, or by two distinct planes that contain it. Consider the first case, with points
and
. The vector displacement from
to
is nonzero because the points are distinct, and represents the ''direction'' of the line. That is, every displacement between points on
is a scalar multiple of
. If a physical particle of unit mass were to move from
to
, it would have a
moment
Moment or Moments may refer to:
* Present time
Music
* The Moments, American R&B vocal group Albums
* ''Moment'' (Dark Tranquillity album), 2020
* ''Moment'' (Speed album), 1998
* ''Moments'' (Darude album)
* ''Moments'' (Christine Guldbrand ...
about the origin. The geometric equivalent is a vector whose direction is perpendicular to the plane containing
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 , where "×" denotes the vector
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
. For a fixed line,
, the area of the triangle is proportional to the length of the segment between
and
, 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 , where "⋅" denotes the vector
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
.
Although neither
nor
alone is sufficient to determine
, together the pair does so uniquely, up to a common (nonzero) scalar multiple which depends on the distance between
and
. That is, the coordinates
: (''d'':''m'') = (''d''
1:''d''
2:''d''
3:''m''
1:''m''
2:''m''
3)
may be considered
homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometr ...
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
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Points ...
,
lines, and a
plane "at infinity", in the sense of
projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, ...
.
: 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
:
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''.
''Proof 1'': Need to show that ''m'' = ''a'' ''b'' − ''b'' ''a'' = ''r'' × ''d'' = ''r'' × (''a'' × ''b'').
what is "r"?
Without loss of generality, let ''a'' ⋅ ''a'' = ''b'' ⋅ ''b'' = 1.
Point ''B'' is the origin. Line ''L'' passes through point ''D'' and is orthogonal to the plane of the picture. The two planes pass through ''CD'' and ''DE'' and are both orthogonal to the plane of the picture. Points ''C'' and ''E'' are the closest points on those planes to the origin ''B'', therefore angles ''BCD'' and ''BED'' are right angles and so the points ''B'', ''C'', ''D'', ''E'' lie on a circle (due to a corollary of
Thales's theorem). ''BD'' is the diameter of that circle.
: ''a'' := BE/ , , BE, , , ''b'' := BC/ , , BC, , ,''r'' := BD, −''a'' = , , BE, , = , , BF, , ,−''b'' = , , BC, , = , , BG, , , ''m'' = ''ab'' − ''ba'' = FG, , , ''d'', , = , , ''a'' × ''b'', , = sin(FBG)
Angle ''BHF'' is a right angle due to the following argument. Let
. Since
(by side-angle-side congruence), then
. Since
, let
. By the
inscribed angle theorem,
, so
.
;
,
therefore
. Then ''DHF'' must be a right angle as well.
Angles ''DCF'' and ''DHF'' are right angles, so the four points C, D, H, F lie on a circle, and (by the
intersecting secants theorem
The intersecting secant theorem or just secant theorem describes the relation of line segments created by two intersecting secants and the associated circle.
For two lines ''AD'' and ''BC'' that intersect each other in ''P'' and some circle in '' ...
)
, , BF, , , , BC, , = , , BH, , , , BD, , , that is ''ab'' sin(FBG) = , , BH, , , , ''r'', , sin(FBG), 2(area of triangle BFG) = ''ab'' sin(FBG) = , , BH, , , , FG, , = , , BH, , , , ''r'', , sin(FBG),
, , ''m'', , = , , FG, , = , , ''r'', , sin(FBG) = , , ''r'', , , , ''d'', , , check direction and ''m'' = ''r'' × ''d''. ∎
''Proof 2'':
Let ''a'' ⋅ ''a'' = ''b'' ⋅ ''b'' = 1. This implies that
: ''a'' = −, , BE, , , ''b'' = −, , BC, , .
According to the
vector triple product formula,
: ''r'' × (''a'' × ''b'') = (''r'' ⋅ ''b'') ''a'' − (''r'' ⋅ ''a'') ''b''
Then
When , , ''r'', , = 0, the line ''L'' passes the origin with direction ''d''. If , , ''r'', , > 0, the line has direction ''d''; the plane that includes the origin and the line ''L'' has normal vector ''m''; the line is tangent to a circle on that plane (normal to ''m'' and perpendicular to the plane of the picture) centered at the origin and with radius , , ''r'', , .
: 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
Primal coordinates
In a 3-dimensional projective space
, let
be a line through distinct points
and
with
homogenous coordinates and
.
The Plücker coordinates
are defined as follows:
:
(the skew symmetric matrix whose elements are ''p''
''ij'' is also called the
Plücker matrix )
This implies ''p''
''ii'' = 0 and ''p''
''ij'' = −''p''
''ji'', reducing the possibilities to only six (4
choose 2) independent quantities. The sextuple
:
is uniquely determined by ''L'' up to a common nonzero scale factor. Furthermore, not all six components can be zero.
Thus the Plücker coordinates of ''L'' may be considered as homogeneous coordinates of a point in a 5-dimensional projective space, as suggested by the colon notation.
To see these facts, let ''M'' be the 4×2 matrix with the point coordinates as columns.
:
The Plücker coordinate ''p''
''ij'' 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′ 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
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplic ...
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. Furthermore, all six 2×2 subdeterminants in ''M'' cannot be zero because the rank of ''M'' is 2.
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:
:
where
:
Dual coordinates
Alternatively, a line can be described as the intersection of two planes. 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 Σ
''k'' ''a''
''k''''x''
''k''=0, for example.) The dual Plücker coordinate ''p''
''ij'' is
:
Dual coordinates are convenient in some computations, and they are equivalent to primary coordinates:
:
Here, equality between the two vectors in homogeneous coordinates means that the numbers on the right side are equal to the numbers on the left side up to some common scaling factor
. Specifically, let (''i'',''j'',''k'',''ℓ'') be an
even permutation
In mathematics, when ''X'' is a finite set with at least two elements, the permutations of ''X'' (i.e. the bijective functions from ''X'' to ''X'') fall into two classes of equal size: the even permutations and the odd permutations. If any total o ...
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 be normalized so that ''x''
0 = 1. Then ''M'' becomes
:
and setting
and
, we have
and
.
Dually, we have
and
.
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 non-vacuous 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
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
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
In mathematics, the lines of a 3-dimensional projective space, ''S'', can be viewed as points of a 5-dimensional projective space, ''T''. In that 5-space, the points that represent each line in ''S'' lie on a quadric, ''Q'' known as the Klein qu ...
.
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
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
:
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
:
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
:
Line families
Because the
Klein quadric
In mathematics, the lines of a 3-dimensional projective space, ''S'', can be viewed as points of a 5-dimensional projective space, ''T''. In that 5-space, the points that represent each line in ''S'' lie on a quadric, ''Q'' known as the Klein qu ...
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
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...
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.
Ray tracing
Line geometry is extensively used in
ray tracing application where the geometry and intersections of rays need to be calculated in 3D. An implementation is described i
Introduction to Plücker Coordinateswritten for the Ray Tracing forum by Thouis Jones.
See also
*
Flat projective plane
*
Plücker matrix
References
*
*
From the German: ''Grundzüge der Mathematik, Band II: Geometrie''. Vandenhoeck & Ruprecht.
*
*
*
*
*
*
*
*
{{DEFAULTSORT:Plucker Coordinates
Projective geometry
Multilinear algebra
Geometric algebra
Coordinate systems