# What transformations can be set by projecting a straight line onto a straight line

What transformations can be set by projecting a straight line onto a straight line (without adding an infinitely distant point)? I said that the homothety with coefficient $$k \neq 1$$ and the reflection. But I was told that this is not true. Where am i wrong?

• Please, complete the background. Transformations of plane? What do you mean with "projecting"? – Martín Vacas Vignolo Dec 31 '18 at 10:40
• @MartínVacasVignolo I think that it was about homography, something like you put one point on the plane and make straight lines across this point , and they should intersect two another straight lines. I don't really understand how correctly formulate this. This is the function between points of two lines on the real plane $\mathbb{R}^2$ but in the context of $\mathbb{RP}^1$ – Just do it Dec 31 '18 at 10:48
• @MartínVacasVignolo I just started to study projective geometry and it is still difficult for me to formulate my thoughts – Just do it Dec 31 '18 at 10:49

The set of transformations of the (affine) plane that preserve lines is the set of affine transformations.

This is equivalent to the set of projective transformations of the protectively completed affine plane.

The category of transformations you indicated was much narrower: in fact they were all isomorphisms of the plane.

Even if you meant that the transformations have to be nonsingular, there are still more homographies than just the homothetic transformations.

• And identity projection and parallel transfer, yes? – Just do it Dec 31 '18 at 11:15
• @Arsenii those are affine isomorphisms, yes. – rschwieb Dec 31 '18 at 18:22
• but how we can construct it, if we have not without infinity distant point? how this "identity" projection should be construct? – Just do it Dec 31 '18 at 23:14
• @Arsenii I don't understand, what do the ideal points have to do with constructing affine transformations? Also, I am not sure what you mean by "projection." It has many meanings. What exactly are you using it for? – rschwieb Jan 1 at 0:06
• I do not know the exact mathematical description of this process. I can say this: Imagine that you have a plane, and there are two lines on it and some point not lying on one of the two lines. So we can start to draw through a point and a set of points of one of the lines - a new set of lines. They will intersect the second line at some points, which we will call the images of the points of the first line in this map. One can visualize that if a point lies, for example, between two parallel lines, then we will get a reflection about a certain point – Just do it Jan 1 at 0:25