Correct notation for points and coordinates

Question

I am a little confused about the difference between $(X,Y,Z)$ and $(x,y,z)$

I saw a statement that says $(X,Y,Z)$ = $(x,y,1)$ with $x=\frac{X}{Z},y=\frac{Y}{Z}$ in the projective plane.

I feel like I am missing what the difference is (in this particular case) with capitalisation of $x$, $y$ and $z$.

Answer 1

There’s no standard. Different authors use different conventions, and you just have to figure out what capitalization and other text style variations convey on a case-by-case basis.

Answer 2

As amd mentioned, there is no universal standard for the meaning of capitalization in variables.

It is worth noting, however, that using an alternate capitalization of another variable is almost always done to emphasize the fact that two variables are related in a critical way. A common example is using the capitalized version of a function's name to represent its antiderivative or a particular transform of it.

In this case, the author likely chose to $X$, $Y$, and $Z$ in the way they did to emphasize that you're still looking at essentially the same $(x,y,z)$, which has been scaled down such that the third component is $1$. Here, $X$ simply represents altered version of the original $x$, and $Y$ represents an altered version of the original $y$.