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Homogeneous coordinate


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Projective Geometry

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Homogeneous coordinate

  1. 1. Homogeneous coordinate In Cartesian coordinate system, the coordinates of a point measures distance relatively, but homogeneous coordinate system serves for different purpose. The main propose of this system is calculation of infinitimum cum extension of affine coordinate. If we consider a plane and take a point at infinity, then coordinate (x, y) (in general both) are infinite. The calculations of such infinite quantities are interesting with homogeneous coordinate system. The point (x, y) of plane in homogeneous system is denoted by triple class of triples with which preserves proportionality If k is any real number except zero, the homogeneous coordinates and represents exactly the same point in the same way that we normally reduce our rational numbers to lowest terms. Therefore a simple representation for homogeneous coordinates is always preferred that we can. If is not equal to zero, we can multiply every coordinate of by 1/ to obtain equivalent point which is same as i.e. the Euclidean point (x,y) can be extended to homogeneous coordinate simply by adding „1” as third coordinate. For example, the points (2,1,1), (4,2,2) (200,100,100)… all corresponds to same Euclidean point (2,1). (200,100) (10,5) (4,2) Figure. 1.21 (2,1) Note Euclidean coordinate (x,y) corresponds to homogeneous coordinate (x,y,1) If we plot the points (2,1), (4,2) (6,3), (20,10), (200,100)… where the second coordinate is always double the first. Thus all the points along line x=2y can be written as (2y, ), where is a real number. But in homogeneous system homogeneous coordinates [ i.e. if , we can let be zero, and we obtain a “point at infinity” with ,0] , then is point at infinity but ratio of x and y is finite. Summary (1) Let be a point in , then homogeneous coordinate of is denoted by with , [The pair (2) Let Since The coordinate can be a point in R if and both are reals] be a line in R, then homogeneous coordinates of is equivalent to is same as is denoted by
  2. 2. The triple can be homogeneous coordinates of a line in R if not zero. [ can be a line in if and both are not zero] (3) The homogeneous coordinates of a point lies on the coordinate of line lies on the line i.e. or The point [Since lies on line and both are if and only if ] if and only if Note The incidence condition of a point on on the line implies same condition of the coordinate (4) If be coordinates of two lines point of intersection is given as If the lines given above are parallel, then intersection becomes as in with the possibility of R then their , then the point of Since the triple can be a homogeneous coordinates of in R if , can‟t be homogenous coordinate of any point. Thus we adopt the point of the form as an ideal point (point at infinity) in where two parallel lines meet. (5) Since the coordinate satisfy the incidence condition for all ideal points of the form we adopt as an ideal line (line at infinity) in . Things to Remember 1. Homogeneous coordinate of a line in Euclidean plane is 2. Homogeneous coordinate of a point in Euclidean plane is where x3 is not zero 3. If x3 is zero then represent a point at infinity 4. Homogeneous coordinate Preserves proportionality class 5. Homogeneous coordinate Is invented by Poncelete 6. Homogeneous coordinate represents line at infinity 7. Homogeneous coordinate represents point at infinity 8. By homogeneous coordinate calculation of infinitesimal is possible 9. Felix Klein provided an algebraic foundation for projective geometry in terms of "homogeneous coordinates," which had been discovered independently by K. W. Feuerbach and A. F. Mobius in 1827. 10. Euclidean coordinate (x,y) can be extended to homogeneous coordinate (x,y,1) 11. Homogeneous coordinate Preserves proportionality class Homogeneous coordinate Is invented by Poncelete