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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
- <math> (0:y:z:0) = \mu (1 - \lambda) (0:1:0:0) + \mu \lambda (0:0:1:0) <math>
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.
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
- <math> \mathbf{A} : (X_A:Y_A:Z_A:W_A), <math>
- <math> \mathbf{B} : (X_B:Y_B:Z_B:W_B). <math>
It is desired to find their linear combination <math> a \mathbf{A} + b \mathbf{B} <math> 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 <math> W_A \ne 0 <math> and <math> W_B \ne 0 <math>. Their linear combination is
- <math> a (X_A:Y_A:Z_A:W_A) + b(X_B:Y_B:Z_B:W_B) \ <math>
- <math> = (a X_A:a Y_A:a Z_A:W_A) + (b X_B:b Y_B:b Z_B:W_B) \ <math>
- <math> = \left( a {X_A \over W_A} : a {Y_A \over W_A} : a {Z_A \over W_A} : 1 \right) + \left( b {X_B \over W_B} : b {Y_B \over W_B} : b {Z_B \over W_B} : 1 \right) <math>
- <math> = \left( a {X_A \over W_A} + b {X_B \over W_B} : a {Y_A \over W_A} + b {Y_B \over W_B} : a {Z_A \over W_A} + b {Z_B \over W_B} : 1 \right) . <math>
Both Points Are At Infinity
If both points are on the plane at infinity, then <math> W_A = 0 <math> and <math> W_B = 0 <math>. Their linear combination is
- <math> 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) <math>
- <math> = (a X_A + b X_B : a Y_A + b Y_B : a Z_A + b Z_B : 0). <math>
One Point Is Affine And The Other At Infinity
Let the first point be affine, so that <math> W_A \ne 0 <math>. Then
- <math> a (X_A:Y_A:Z_A:W_A) + b(X_B:Y_B:Z_B:0) <math>
- <math> = a (0:0:0:0) + b (X_B:Y_B:Z_B:0), <math>
- <math> = (b X_B:b Y_B:b Z_B:0), <math>
which means that the point at infinity is "dominant".
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