Studying about collineation

  1. Hello people!
    I'm studying about collineation. It seems to be simple, but I can't understand so much and I can't find so many things about this subject...
    Can someone explain to me what exactly is a Collineation? Is this the same thing as Affine Transformation?
    Is there examples that functions that are transformations but not collineation? I've found about F(x,y)=(x^3,y) but is there another one? Every book that I study just tells about this example, I wanted to see more...

    Thank you so much!!
     
  2. jcsd
  3. Simon Bridge

    Simon Bridge 14,649
    Science Advisor
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    Colinear points are points on the same line.
    Colineation is a projection that preserves the colinearity of the points.
    i.e. if you start out with a line, the projection should also be a line.

    i.e. a line in 3D space may be perpendicularly projected onto a plane and the projection is also a line.
    But a projection onto the surface of a sphere turns the line into a curve.

    You have found out about F(x,y)=(x^3,y) ... what about F(x,y)=(x,y^3)
    consider: F(x,y)=(x^n,y^m) ... what values do n and m have to take for the transformation to be a colineation?

    iirc: The spaces in the colineations do not need to be affine.
    An affine transformation would also be a colineation between affine spaces ... but not all colineations between affine spaces are affine transformations. Something like that.
     
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