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!!
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.