Natural Homogeneous Coordinates In projective geometry parallel lines intersect at a point. The point at infinity is called an ideal point.   There is an ideal point for every slope. The collection of ideal points is called an ideal line. We might think of the line as a circle.

Representing Points in the Projective Plane A coordinate pair (x, y) is not sufficient to represent both ordinary points and ideal points. We use triples, (x,y,z), to represent points in the projective plane

Representing Ideal Points Using Triples: (x,y,z)  Consider two distinct parallel lines: ax + by + cz = 0 ax + by + c'z = 0 (c not equal to c') (c - c') * z = 0 hence z = 0. We use z=0 to represent ideal points. 

Projective Coordinate Triples And Cartesian Coordinate Pairs Let z = 1. Then a projective coordinate line given by ax + by +cz=0 becomes ax + by + c = 0.  The second equation corresponds to the equation of a line in Euclidean coordinates. We can make the correspondence between points (x,y,1) in parallel coordinates and points (x,y) in Euclidean coordinates

Projective Points And Euclidean Points If a point ( x, y, 1) is on the line ax + by + cz=0, so is point (px, py, p) for any p. Note: apx + bpy + cpz = p * (ax + by + cz)=0 Multiple projective coordinate points correspond to the same Euclidean coordinate. To obtain Euclidean coordinates from non-ideal points represented as projective coordinates, divide by the last coordinate so it becomes 1.

Examples Type Projective Euclidean Non-ideal (4,6,2) (2,3) (8,12,4) (2,7,0,2) (1,3.5,0) Ideal (3,4,7,0) no match