Lines in Three-dimensional Space
For two given points in the plane, (x1, y1, z1) and (x2, y2, z2), the three determinants
determine the line containing them. Similarly, for two points in three-dimensional space (x1, y1, z1, w1) and (x2, y2, z2, w2), the line containing them is determined by the six determinants
This is the basis for a system of homogeneous line coordinates in three-dimensional space called Plücker coordinates. Six numbers in a set of coordinates only represent a line when they satisfy an additional equation. This system maps the space of lines in three-dimensional space to a projective space of dimension five, but with the additional requirement the space of lines is a manifold of dimension four.
More generally, the lines in n-dimensional projective space are determined by a system of n(n − 1)/2 homogeneous coordinates that satisfy a set of (n − 2)(n − 3)/2 conditions, resulting in a manifold of dimension 2(n − 1).
Read more about this topic: Line Coordinates
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