Plücker coordinates
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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 Julius Plücker (16 June 1801 – 22 May 1868) was a German mathematician and physicist. He made fundamental contributions to the field of analytical geometry and was a pioneer in the investigations of cathode rays that led eventually to the dis ...
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 geometry. T ...
to each line in projective 3-space, P3. 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 s ...
between the 4-dimensional space of lines in P3 and points on a
quadric In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is de ...
in P5 (projective 5-space). A predecessor and special case of
Grassmann coordinates Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mat ...
(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, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
), 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 ge ...
. 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 (physics), motion of points, Physical object, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause ...
used for
robot control Robotic control is the system that contributes to the movement of robots. This involves the mechanical aspects and programmable systems that makes it possible to control robots. Robotics could be controlled in various ways, which includes using ma ...
.


Geometric intuition

A line L 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, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
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 , 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 is ...
. For a fixed line, L, 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 , 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 algebra ...
. 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 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 geometry. T ...
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 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: Arts ...
, 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, pro ...
. : 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 ''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics. The term is used to indicate ...
, 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 In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved ...
). ''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 \epsilon := \angle BEC. Since \Delta BEC \cong \Delta BFG (by side-angle-side congruence), then \angle BFG = \epsilon. Since \angle BEC + \angle CED = 90^\circ, let \epsilon' := 90^\circ - \epsilon = \angle CED. By the
inscribed angle theorem In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle. Equivalently, an in ...
, \angle DEC = \angle DBC, so \angle DBC = \epsilon'. \angle HBF + \angle BFH + \angle FHB = 180^\circ; \epsilon' + \epsilon + \angle FHB = 180^\circ, \epsilon + \epsilon' = 90^\circ therefore \angle FHB = 90^\circ. 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 In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector- ...
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 P^3, let L be a line through distinct points x and y with
homogenous 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 geometry. ...
(x_0:x_1:x_2:x_3) and (y_0:y_1:y_2:y_3). The Plücker coordinates p_ are defined as follows: : (the skew symmetric matrix whose elements are ''p''''ij'' is also called the
Plücker matrix The Plücker matrix is a special skew-symmetric 4 × 4 matrix, which characterizes a straight line in projective space. The matrix is defined by 6 Plücker coordinates with 4 degrees of freedom. It is named after the German mat ...
)
This implies ''p''''ii'' = 0 and ''p''''ij'' = −''p''''ji'', reducing the possibilities to only six (4 choose 2) independent quantities. The sextuple : (p_:p_:p_:p_:p_:p_) 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. : M = \begin x_0 & y_0 \\ x_1 & y_1 \\ x_2 & y_2 \\ x_3 & y_3 \end 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 In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
; ''M'' has
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * H ...
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 multiplicat ...
Λ, : M' = M\Lambda . In particular, rows ''i'' and ''j'' of ''M′'' and ''M'' are related by : \begin x'_ & y'_\\x'_& y'_ \end = \begin x_ & y_\\x_& y_ \end \begin \lambda_ & \lambda_ \\ \lambda_ & \lambda_ \end . 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 P1) in P3 by ''G''1,3. We thus have a map: :\begin \alpha \colon \mathrm_ & \rightarrow \mathbf^5 \\ L & \mapsto L^, \end where : L^=(p_:p_:p_:p_:p_:p_) .


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: : (p_:p_:p_:p_:p_:p_)= (p^:p^:p^:p^:p^:p^) 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 \lambda. 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 ...
of (0,1,2,3); then : p_ = \lambda p^ .


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 : M = \begin 1 & 1 \\ x_1 & y_1 \\ x_2& y_2 \\ x_3 & y_3 \end , and setting x=(x_1,x_2,x_3) and y=(y_1,y_2,y_3), we have d=(p_,p_,p_)and m=(p_,p_,p_). Dually, we have d=(p^,p^,p^) and m=(p^,p^,p^).


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 : \begin x_0 & y_0 & z_0 \\ x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ x_3 & y_3 & z_3 \end are
linearly dependent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
, 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 : \begin 0 & = \begin x_0 & y_0 & z_0 \\ x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \end \\ pt& = \begin x_1 & y_1 \\ x_2 & y_2 \end z_0 - \begin x_0 & y_0 \\ x_2 & y_2 \end z_1 + \begin x_0 & y_0 \\ x_1 & y_1 \end z_2 \\ pt& = p_ z_0 - p_ z_1 + p_ z_2 . \\ pt& = p^ z_0 + p^ z_1 + p^ z_2 . \end The four possible planes obtained are as follows. : \begin 0 & = & + p_ z_0 & - p_ z_1 & + p_ z_2 & \\ 0 & = & - p_ z_0 & - p_ z_1 & & + p_ z_3 \\ 0 & = & +p_ z_0 & & - p_ z_2 & + p_ z_3 \\ 0 & = & & +p_ z_1 & + p_ z_2 & + p_ z_3 \end 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 : 0 = \sum_^3 p^ z_i , \qquad j = 0,\ldots,3 . 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 Injection or injected may refer to: Science and technology * Injective function, a mathematical function mapping distinct arguments to distinct values * Injection (medicine), insertion of liquid into the body with a syringe * Injection, in broadca ...
.


Quadratic relation

The image of α is not the complete set of points in P5; the Plücker coordinates of a line ''L'' satisfy the quadratic Plücker relation : \begin 0 & = p_p^+p_p^+p_p^ \\ & = p_p_+p_p_+p_p_. \end For proof, write this homogeneous polynomial as determinants and use
Laplace expansion In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression of the determinant of an matrix as a weighted sum of minors, which are the determinants of some submatrices of . Spec ...
(in reverse). : \begin 0 & = \beginx_0&y_0\\x_1&y_1\end\beginx_2&y_2\\x_3&y_3\end+ \beginx_0&y_0\\x_2&y_2\end\beginx_3&y_3\\x_1&y_1\end+ \beginx_0&y_0\\x_3&y_3\end\beginx_1&y_1\\x_2&y_2\end \\ pt& = (x_0 y_1-y_0 x_1)\beginx_2&y_2\\x_3&y_3\end- (x_0 y_2-y_0 x_2)\beginx_1&y_1\\x_3&y_3\end+ (x_0 y_3-y_0 x_3)\beginx_1&y_1\\x_2&y_2\end \\ pt& = x_0 \left(y_1\beginx_2&y_2\\x_3&y_3\end- y_2\beginx_1&y_1\\x_3&y_3\end+ y_3\beginx_1&y_1\\x_2&y_2\end\right) -y_0 \left(x_1\beginx_2&y_2\\x_3&y_3\end- x_2\beginx_1&y_1\\x_3&y_3\end+ x_3\beginx_1&y_1\\x_2&y_2\end\right) \\ pt& = x_0 \beginx_1&y_1&y_1\\x_2&y_2&y_2\\x_3&y_3&y_3\end -y_0 \beginx_1&x_1&y_1\\x_2&x_2&y_2\\x_3&x_3&y_3\end \end Since both 3×3 determinants have duplicate columns, the right hand side is identically zero. Another proof may be done like this: Since vector : d = \left( p_, p_, p_ \right) is perpendicular to vector : m = \left( p_, p_, p_ \right) (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 P5, without loss of generality assume that ''q''01 is nonzero. Then the matrix : M = \begin q_ & 0 \\ 0 & q_ \\ -q_ & q_ \\ q_ & q_ \end has rank 2, and so its columns are distinct points defining a line ''L''. When the P5 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_ = - q_ q_ - q_ q_ . 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 In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
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. Mo ...
consisting of the set of zeros of the quadratic polynomial : p_p_+p_p_+p_p_ . And since α is also an injection, the lines in P3 are thus in
bijective 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 s ...
correspondence with the points of this
quadric In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is de ...
in P5, 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 q ...
.


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 P3 are either
skew Skew may refer to: In mathematics * Skew lines, neither parallel nor intersecting. * Skew normal distribution, a probability distribution * Skew field or division ring * Skew-Hermitian matrix * Skew lattice * Skew polygon, whose vertices do not ...
or
coplanar In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. Howe ...
, 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 : \begin 0 & = p_p'_ + p_p'_ + p_p'_ + p_p'_ + p_p'_ + p_p'_ \\ pt& = \beginx_0&y_0&x'_0&y'_0\\ x_1&y_1&x'_1&y'_1\\ x_2&y_2&x'_2&y'_2\\ x_3&y_3&x'_3&y'_3\end. \end 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_ a^j p_ , \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_ y_j p^ , \qquad i = 0 \ldots 3 .


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 q ...
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 In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, the ...
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 In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by defo ...
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 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 specia ...
within the Klein quadric in P5.


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 Coordinates
written for the Ray Tracing forum by Thouis Jones.


See also

* Flat projective plane *
Plücker matrix The Plücker matrix is a special skew-symmetric 4 × 4 matrix, which characterizes a straight line in projective space. The matrix is defined by 6 Plücker coordinates with 4 degrees of freedom. It is named after the German mat ...


References

* *
From the German: ''Grundzüge der Mathematik, Band II: Geometrie''. Vandenhoeck & Ruprecht. * * * * * * * * {{DEFAULTSORT:Plucker Coordinates Projective geometry Multilinear algebra Geometric algebra Coordinate systems