Natural Homogeneous Coordinates

1
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.

2
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

3
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 z0 to represent ideal points. 

4
Projective Coordinate Triples And Cartesian
Coordinate Pairs

• Let z 1.
• Then a projective coordinate line given by
• ax by cz0 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

5
Projective Points AndEuclidean Points

• If a point ( x, y, 1) is on the line ax by
  cz0, 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.

6
Examples