Homogeneous co-ordinates

In mathematics, homogeneous co-ordinates, introduced by August Ferdinand Möbius, make calculations possible in projective space just 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). The line is given by the equation

(0: y : z :0) = μ(1 - λ)(0:1:0:0) + μλ(0:0:1:0)

where μ is a scaling factor. The scaling factor can be adjusted to normalize the co-ordinates (0:y:z:0), thereby eliminating one of the two degrees of freedom. The result is a set of points with only one degree of freedom, as is expected for a line.

Table of contents

1 Linear Combinations of Points Described with Homogeneous Co-ordinates

1.1 Both Points Are Affine
1.2 Both Points Are At Infinity
1.3 One Point Is Affine And The Other At Infinity

Linear Combinations of Points Described with Homogeneous Co-ordinates

Let there be a pair of points A and B in projective 3-space, whose homogeneous co-ordinates are

It is desired to find their linear combination where a and b are coefficients which can be adjusted at will. There are three cases to consider:

both points belong to affine 3-space,
both points belong to the plane at infinity,
one point is affine and the other one is at infinity.

The X, Y, and Z co-ordinates can be considered as numerators, whereas the W coordinate can be considered as a denominator. To add homogeneous coordinates it is necessary that the denominator be common. Otherwise it is necessary to rescale the co-ordinates until all the denominators are common. Homogeneous co-ordinates are equivalent up to any uniform rescaling.

Both Points Are Affine

If both points are in affine 3-space, then and . Their linear combination is

Both Points Are At Infinity

If both points are on the plane at infinity, then $W A = 0$ and $W B = 0$ . Their linear combination is

a (X A : Y A : Z A : W A) + b (X B : Y B : Z B : W B) = (a X A : a Y A : a Z A :0) + (b X B : b Y B : b Z B :0)

= (a X A + b X B : a Y A + b Y B : a Z A + b Z B :0).

One Point Is Affine And The Other At Infinity

Let the first point be affine, so that . Then

a (X A : Y A : Z A : W A) + b (X B : Y B : Z B :0)

= a (0:0:0:0) + b (X B : Y B : Z B :0),

= (b X B : b Y B : b Z B :0),

which means that the point at infinity is "dominant".