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 15,474
    Science Advisor
    Homework Helper
    Gold Member

    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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