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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.

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Natural Homogeneous Coordinates

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  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 z=0 to represent ideal points. 

  4. 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

  5. Projective Points AndEuclidean 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.

  6. Examples

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