A question about transformations of the euclidean plane

Question

A few days ago a began taking a course on geometry. Immediately I started reading up on affine transformations, transformations of the euclidean plane, isometrices etc. The auther defines a relation between ordered tripples of $\mathbb{R}^3$ by $a=(a_1,a_2,a_3)$ is related to $b=(b_1,b_2,b_3)$ if there is a nonzero integer $k$ s.t. $a=kb$. For some reason the auther associates each point $(c_1,c_2)$ of $\mathbb{R}^2$ with an equivalence class $C$ of the aforementioned relation where if $(x_1,x_2,x_3)\in C$ then $x_3\neq0$. So there is a unique class which $(c_1,c_2,1)$ belongs to. So typically when the auther speaks of $(c_1,c_2)$ she speaks of $(c_1,c_2,1)$. When the auther talks about tranformations from the plane to the plane she is instead considering $\mathbb{R}^3$ using the above tripples. I'm very confused by this. What is the point of it? If we want to study transformations from the plane to the plane why do we have to involve these equivalence classes and all this work with $3\times3$ matrices instead of just $2\times2$ matrices?

Answer 1

The reasons for all of this should become clearer as you get further into the material. The short answer, however, is that you can only represent linear transformations of the plane with $2\times2$ matrices. This means that, for instance, you can’t even represent a simple translation in this way. You could, of course, include translations by including terms of the form $\mathbf x-\mathbf p$ in the calculations, but that makes composing translations with other transformations inconvenient. By using homogeneous coordinates (which is what these equivalence classes are), you can represent translations and other important nonlinear transformations as $3\times3$ matrices, so that composition of transformations corresponds to simple matrix multiplication.

A handy mental model to use for this is to embed the Euclidean plane into $\mathbb R^3\setminus\{0\}$ by identifying it with the plane $z=1$. Each point on this plane defines a unique line through the origin—these lines are your equivalence classes.

There’s a lot more that can be said about this, but I’ll leave that to your instructor and other sources.

Answer 2

As a note the projective plane consists of the following points,

$$(a,b,1),\quad a,b\in{\Bbb R},$$ $$(a,1,0),\quad a\in{\Bbb R},$$ and $$(1,0,0).$$