Encyclopedia > Homogeneous co-ordinates

  Article Content

Homogeneous co-ordinates

The introduction for projective space of homogeneous co-ordinates makes calculations possible, as Cartesian co-ordinates do in Euclidean space. The homogeneous co-ordinates of a point of projective space of dimension n are usually written as (x:y:z: ... :w), a row vector of length n+1, other than (0:0:0: ... :0). Two sets of co-ordinates that are proportional denote the same point of projective space: for any non-zero scalar c from the underlying field K, (cx:cy:cz: ... :cw) denotes the same point. Therefore this system of co-ordinates can be explained as follows: if the projective space is constructed from a vector space V of dimension n+1, introduce co-ordinates in V by choosing a basis, and use these in P(V), the equivalence classes of proportional non-zero vectors in V.

Taking the example of projective space of dimension three, there will be homogeneous co-ordinates (x:y:z:w). The plane at infinity is usually identified with the set of points with w = 0. Away from this plane we can use (x/w, y/w, z/w) as an ordinary Cartesian system; therefore the affine space complementary to the plane at infinity is co-ordinatised in a familiar way, with a basis corresponding to (1:0:0:1), (0:1:0:1), (0:0:1:1).

If we try to intersect the two planes defined by equations x = w and x = 2w then we clearly will derive first w = 0 and then x = 0. That tells us that the intersection is contained in the plane at infinity, and consists of all points with co-ordinates (0:y:z;0). It is a line, and in fact the line joining (0:1:0:0) and (0:0:1:0). It cannot be given by a single equation in the co-ordinates. In fact a line in three-dimensional projective space corresponds to a two-dimensional subspace of the underlying four-dimensional vector space, therefore given by two linear conditions.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
East Farmingdale, New York

... Latino of any race. There are 1,693 households out of which 37.7% have children under the age of 18 living with them, 58.8% are married couples living together, 12.8% ...

 
 
 
This page was created in 52.8 ms