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
Famous quotes containing the words lines and/or space:
“It is the Late city that first defies the land, contradicts Nature in the lines of its silhouette, denies all Nature. It wants to be something different from and higher than Nature. These high-pitched gables, these Baroque cupolas, spires, and pinnacles, neither are, nor desire to be, related with anything in Nature. And then begins the gigantic megalopolis, the city-as-world, which suffers nothing beside itself and sets about annihilating the country picture.”
—Oswald Spengler (1880–1936)
“This moment exhibits infinite space, but there is a space also wherein all moments are infinitely exhibited, and the everlasting duration of infinite space is another region and room of joys.”
—Thomas Traherne (1636–1674)