I need to comprehend projective plane as a prerequisite for some other topic. But I can't understand what it really looks like. How can I make this plane natural to me?
Moreover I want to understand why homogenous equations like $X^2+Y^2=Z^2$ boil down to like $x^2+y^2=1$ in this plane.
I know the definition of the plane as the set of all lines passing through the origin and also have some experience working with projective transformations in Olympiad geometry problems.
From the definition as lines you have that a point $(X,Y,Z)\neq(0,0,0)$ determines a line through the origin.
If the point happens to have $Z\neq0$ then the point $(X/Z,Y/Z,1)$ determines the same line through the origin.
This means that $(X,Y,Z)$ and $(x,y,1)$, with $x=X/Z, y=Y/Z$, are representing the same point of the projective plane.
If we take $X^2+Y^2=Z^2$ and look only in the part of the projective plane where $Z\neq0$ then we can divide both side by $Z$ and get $(X/Z)^2+(Y/Z)^2=1$, which is the equation $x^2+y^2=1$ in the coordinates $(x,y)$.
Of course, the points such that $x^2+y^2=1$ are only those points for which $Z\neq0$ (the complements of a line in the projective plane).
To have the complete picture we also need to study it at points of the projective plane where $Z=0$. So the solutions of $x^2+y^2=1$ are not the whole picture.
If $Z=0$,then either $X\neq0$ or $Y\neq$. Since the equation is symmetric with respect to $X,Y$, it is enough to look at the case $Y\neq0$. In this case we can proceed similarly. We divide by $Y^2$ and get $(X/Y)^2+1=(Z/Y)^2$. Calling $\tilde{x}:=X/Y$ and $\tilde{z}:=Z/Y$ we get $\tilde{x}^2+1=\tilde{z}^2$.