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:
“Was seiz’d by the spirit that trails in the lines underfoot,
The rim, the sediment that stands for all the water and all the land
of the globe.
Fascinated, my eyes reverting from the south, dropt, to follow those slender windrows,
Chaff, straw, splinters of wood, weeds, and the sea-gluten,
Scum, scales from shining rocks, leaves of salt-lettuce, left by the tide,”
—Walt Whitman (1819–1892)
“I take SPACE to be the central fact to man born in America.... I spell it large because it comes large here. Large and without mercy.”
—Charles Olson (1910–1970)