Magnetic reconnection

29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church.

In geometry, Plücker coordinates, introduced by Julius Plücker in the 19th century, are a way to assign six homogeneous coordinates to each line in projective 3-space, P³. Because they satisfy a quadratic constraint, they establish a one-to-one correspondence between the 4-dimensional space of lines in P³ and points on a quadric in P⁵ (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

Displacement and moment of two points on line

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 = (x₁,x₂,x₃) and y = (y₁,y₂,y₃). 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) = (d₁:d₂:d₃:m₁:m₂:m₃)

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 a₀ = 2, a = (−1,0,0) and b₀ = −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 3-dimensional projective space, P³, let L be a line containing distinct points x and y with homogeneous coordinates (x₀:x₁:x₂:x₃) and (y₀:y₁:y₂:y₃), respectively. Let M be the 4×2 matrix with these coordinates as columns.

$M=\left[\begin{array}{cc}{x}_0& y_0\\ x_1& y_1\\ x_2& y_2\\ x_3& y_3\end{array}\right]$

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,

$M^{\prime }=M\Lambda .$

In particular, rows i and j of M′ and M are related by

$\left[\begin{array}{cc}x{\text{'}}_i& y{\text{'}}_i\\ x{\text{'}}_j& y{\text{'}}_j\end{array}\right]=\left[\begin{array}{cc}{x}_i& y_i\\ x_j& y_j\end{array}\right]\left[\begin{array}{cc}{\lambda }_{00}& {\lambda }_{01}\\ {\lambda }_{10}& {\lambda }_{11}\end{array}\right].$

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,

$p_{ij}$	$=\left|\begin{array}{cc}{x}_i& y_i\\ x_j& y_j\end{array}\right|$
	$=x_i{y}_j-x_j{y}_i.$

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

$(p_{01}:p_{02}:p_{03}:p_{23}:p_{31}:p_{12})$

is uniquely determined by L, up to a common nonzero scale factor. Furthermore, all six components cannot be zero, because if they were, all 2×2 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 5-dimensional projective space.

Plücker map

Denote the set of all lines (linear images of P¹) in P³ by G_1,3. We thus have a map:

$\begin{array}{cc}\alpha :{\mathrm{G}}_{1,3}& \to {𝐏}^5\\ L& \mapsto L^{\alpha },\end{array}$

where

$L^{\alpha }=(p_{01}:p_{02}:p_{03}:p_{23}:p_{31}:p_{12}).$

Dual coordinates

Alternatively, 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. (The first plane equation is 0 = ∑_k a^kx_k, for example.) Let N be the 2×4 matrix with these coordinates as rows.

$N=\left[\begin{array}{cccc}{a}^0& a^1& a^2& a^3\\ b^0& b^1& b^2& b^3\end{array}\right]$

We define dual Plücker coordinate p^ij as the determinant of columns i and j of N,

$p^{ij}$	$=\left|\begin{array}{cc}{a}^i& a^j\\ b^i& b^j\end{array}\right|$
	$=a^i{b}^j-a^j{b}^i.$

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 x₀ = 0 as the plane at infinity; thus the coordinates of points not at infinity can be normalized so that x₀ = 1. Then M becomes

$M=\left[\begin{array}{cc}1& 1\\ x_1& y_1\\ x_2& y_2\\ x_3& y_3\end{array}\right],$

and setting x = (x₁,x₂,x₃) and y = (y₁,y₂,y₃), we have d = (p₀₁,p₀₂,p₀₃) and m = (p₂₃,p₃₁,p₁₂).

Dually, we have d = (p²³,p³¹,p¹²) and m = (p⁰¹,p⁰²,p⁰³).

Bijection between lines and Klein quadric

Plane equations

If the point z = (z₀:z₁:z₂:z₃) lies on L, then the columns of

$\left[\begin{array}{ccc}{x}_0& y_0& z_0\\ x_1& y_1& z_1\\ x_2& y_2& z_2\\ x_3& y_3& z_3\end{array}\right]$

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

$0$	$=\left|\begin{array}{ccc}{x}_0& y_0& z_0\\ x_1& y_1& z_1\\ x_2& y_2& z_2\end{array}\right|$
	$=\left|\begin{array}{cc}{x}_1& y_1\\ x_2& y_2\end{array}\right|z_0-\left|\begin{array}{cc}{x}_0& y_0\\ x_2& y_2\end{array}\right|z_1+\left|\begin{array}{cc}{x}_0& y_0\\ x_1& y_1\end{array}\right|z_2$
	$=p_{12}{z}_0-p_{02}{z}_1+p_{01}{z}_2.$
	$=p^{03}{z}_0+p^{13}{z}_1+p^{23}{z}_2.$

The four possible planes obtained are as follows.

$\begin{array}{cccccc}0& =& +p_{12}{z}_0& -p_{02}{z}_1& +p_{01}{z}_2& \\ 0& =& -p_{31}{z}_0& -p_{03}{z}_1& & +p_{01}{z}_3\\ 0& =& +p_{23}{z}_0& & -p_{03}{z}_2& +p_{02}{z}_3\\ 0& =& & +p_{23}{z}_1& +p_{31}{z}_2& +p_{12}{z}_3\end{array}$

Using dual coordinates, and letting (a⁰:a¹:a²:a³) be the line coefficients, each of these is simply aⁱ = p^ij, or

$0=\sum _{i=0}^3{p}^{ij}{z}_i,j=0,\dots ,3.$

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.

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

$0$	$=p_{01}{p}^{01}+p_{02}{p}^{02}+p_{03}{p}^{03}$
	$=p_{01}{p}_{23}+p_{02}{p}_{31}+p_{03}{p}_{12}.$

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

$0$	$=\left|\begin{array}{cc}{x}_0& y_0\\ x_1& y_1\end{array}\right|\left|\begin{array}{cc}{x}_2& y_2\\ x_3& y_3\end{array}\right|+\left|\begin{array}{cc}{x}_0& y_0\\ x_2& y_2\end{array}\right|\left|\begin{array}{cc}{x}_3& y_3\\ x_1& y_1\end{array}\right|+\left|\begin{array}{cc}{x}_0& y_0\\ x_3& y_3\end{array}\right|\left|\begin{array}{cc}{x}_1& y_1\\ x_2& y_2\end{array}\right|$
	$=(x_0{y}_1-y_0{x}_1)\left|\begin{array}{cc}{x}_2& y_2\\ x_3& y_3\end{array}\right|-(x_0{y}_2-y_0{x}_2)\left|\begin{array}{cc}{x}_1& y_1\\ x_3& y_3\end{array}\right|+(x_0{y}_3-y_0{x}_3)\left|\begin{array}{cc}{x}_1& y_1\\ x_2& y_2\end{array}\right|$
	$=x_0(y_1\left|\begin{array}{cc}{x}_2& y_2\\ x_3& y_3\end{array}\right|-y_2\left|\begin{array}{cc}{x}_1& y_1\\ x_3& y_3\end{array}\right|+y_3\left|\begin{array}{cc}{x}_1& y_1\\ x_2& y_2\end{array}\right|)-y_0(x_1\left|\begin{array}{cc}{x}_2& y_2\\ x_3& y_3\end{array}\right|-x_2\left|\begin{array}{cc}{x}_1& y_1\\ x_3& y_3\end{array}\right|+x_3\left|\begin{array}{cc}{x}_1& y_1\\ x_2& y_2\end{array}\right|)$
	$=x_0\left|\begin{array}{ccc}{x}_1& y_1& y_1\\ x_2& y_2& y_2\\ x_3& y_3& y_3\end{array}\right|-y_0\left|\begin{array}{ccc}{x}_1& x_1& y_1\\ x_2& x_2& y_2\\ x_3& x_3& y_3\end{array}\right|$

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

Another proof may be done like this: Since vector

$d=(p_{01},p_{02},p_{03})$

is perpendicular to vector

$m=(p_{23},p_{31},p_{12})$

(see above), the scalar product of d and m must be zero! q.e.d.

Point equations

Letting (x₀:x₁:x₂:x₃) 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 (q₀₁:q₀₂:q₀₃:q₂₃:q₃₁:q₁₂) are the homogeneous coordinates of a point in P⁵, without loss of generality assume that q₀₁ is nonzero. Then the matrix

$M=\left[\begin{array}{cc}{q}_{01}& 0\\ 0& q_{01}\\ -q_{12}& q_{02}\\ q_{31}& q_{03}\end{array}\right]$

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 q₀₁ 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 q₂₃+q₀₂q₃₁+q₀₃q₁₂ = 0, then p₂₃ = q₂₃, 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 P³ are thus in bijective correspondence with the points of this quadric in P⁵, 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 P³ 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

$0$	$=p_{01}p{\text{'}}_{23}+p_{02}p{\text{'}}_{31}+p_{03}p{\text{'}}_{12}+p_{23}p{\text{'}}_{01}+p_{31}p{\text{'}}_{02}+p_{12}p{\text{'}}_{03}$
	$=\left|\begin{array}{cccc}{x}_0& y_0& x{\text{'}}_0& y{\text{'}}_0\\ x_1& y_1& x{\text{'}}_1& y{\text{'}}_1\\ x_2& y_2& x{\text{'}}_2& y{\text{'}}_2\\ x_3& y_3& x{\text{'}}_3& y{\text{'}}_3\end{array}\right|.$

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′)x₀ + (d×d′)•x ,

where x = (x₁,x₂,x₃).

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

(x₀ : x) = (d•m′:m×m′) .

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

Plane-line meet

Given a plane with equation

$0=a^0{x}_0+a^1{x}_1+a^2{x}_2+a^3{x}_3,$

or more concisely 0 = a⁰x₀+a•x; and given a line not in it with Plücker coordinates (d:m), then their point of intersection is

(x₀ : x) = (a•d : a×m − a₀d) .

The point coordinates, (x₀:x₁:x₂:x₃), can also be expressed in terms of Plücker coordinates as

$x_i=\sum _{j\ne i}{a}^j{p}_{ij},i=0\dots 3.$

Point-line join

Dually, given a point (y₀:y) and a line not containing it, their common plane has equation

0 = (y•m) x₀ + (y×d−y₀m)•x .

The plane coordinates, (a⁰:a¹:a²:a³), can also be expressed in terms of dual Plücker coordinates as

$a^i=\sum _{j\ne i}{y}_j{p}^{ij},i=0\dots 3.$

Line families

Because the Klein quadric 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 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 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 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 in Introduction to Pluecker Coordinates written for the Ray Tracing forum by Thouis Jones.

See also

• Flat projective plane

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  From the German: Grundzüge der Mathematik, Band II: Geometrie. Vandenhoeck & Ruprecht.

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534