Homogeneous Coordinates - Introduction

Introduction

The projective plane can be thought of as the Euclidean plane with additional points, so called points at infinity, added. There is a point at infinity for each direction, informally defined as the limit of a point that moves in that direction away from a fixed point. Parallel lines in the Euclidean plane are said to intersect at a point at infinity corresponding to their common direction. A given point (x, y) on the Euclidean plane is identified with two ratios (X/Z, Y/Z), so the point (x, y) corresponds to the triple (X, Y, Z) = (xZ, yZ, Z) where Z ≠ 0. Such a triple is a set of homogeneous coordinates for the point (x, y). Note that, since ratios are used, multiplying the three homogeneous coordinates by a common, non-zero factor does not change the point represented – unlike Cartesian coordinates, a single point can be represented by infinitely many homogeneous coordinates.

The equation of a line through the point (a, b) may be written l(x − a) + m(y − b) = 0 where l and m are not both 0. In parametric form this can be written x = a + mt, y = b − lt. Let Z=1/t, so the coordinates of a point on the line may be written (a + m/Z, b − l/Z)=((aZ + m)/Z, (bZ − l)/Z). In homogeneous coordinates this becomes (aZ + m, bZ − l, Z). In the limit as t approaches infinity, in other words as the point moves away from (a, b), Z becomes 0 and the homogeneous coordinates of the point become (m, −l, 0). So (m, −l, 0) are defined as homogeneous coordinates of the point at infinity corresponding to the direction of the line l(x − a) + m(y − b) = 0.

To summarize:

• Any point in the projective plane is represented by a triple (X, Y, Z), called the homogeneous coordinates or projective coordinates of the point, where X, Y and Z are not all 0.
• The point represented by a given set of homogeneous coordinates is unchanged if the coordinates are multiplied by a common factor.
• Conversely, two sets of homogeneous coordinates represent the same point if and only if one is obtained from the other by multiplying by a common factor.
• When Z is not 0 the point represented is the point (X/Z, Y/Z) in the Euclidean plane.
• When Z is 0 the point represented is a point at infinity.

Note that the triple (0, 0, 0) is omitted and does not represent any point. The origin is represented by (0, 0, 1).