I am following an introductory course in geometry, and I'm working on some problems involving projective geometry. In these problems (and in some similar ones in syllabus), we have to determine the projective transformations which map certain conic sections onto other ones.
A hint I am given for this question reads: use the fact that $(0:0:1)$ is the intersection of the $X$ and $Y$ axes, and in an example problem in the syllabus, they use the fact that $(1:0:0)$ is the intersection point of the $X$ axis and the $Z$ axis. This confuses me.
In my mind, the axes obviously intersect at the point $(0, 0, 0)$ (in $\mathbb{R}^{3}$ at least). Looking at it now I would think that the point $(0:0:0)$ is not really defined in projective space, but the other so called 'points of intersection' seem plainly wrong to me.
The point $(1:0:0)$ for example. Is that not represented in Euclidian space by the line which runs along the $x$-axis? I find it difficult to see how this would be a point of intersection, even looking at it projectively.
My question is: how does one define the point of intersection of axes in projective space?
Many thanks, Hugo.