Download Natural Homogeneous Coordinates

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)
Non-ideal
(8,12,4)
(2,3)
Non-ideal
(2,7,0,2)
(1,3.5,0)
Ideal
(3,4,7,0)
no match