Plücker Coordinates for Lines
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A line l in 3-space spanned by vectors
has Plücker coordinates pi,j:=
xi yj -
xj yi.
These identify the line l with a point
pl in 5-space
satisfying the quadratic Plücker equation
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= 0.
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Conversely, every point in 5-space satisfying this
Plücker equation
is the Plücker coordinate of a line in 3-space.