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Studying about collineation

by cristina89
Tags: collineation, studying
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Mar27-14, 06:28 AM
P: 29
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!!
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Simon Bridge
Mar27-14, 08:43 PM
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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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