Homogeneous coordinate representation of a vertical line Is there homogenous coordinate representation for a vertical line passing through an arbitrary point on the x axis (say C). Generally this is represented as: $x = C$ in euclidean geometry would it be $(-1/C, 0, 1)$ in homogenous coordinates (P2 space)?

Two ways to tackle this.

1. Using the equation. The equation of a line with vector $[a:b:c]$ is $ax+by+cz=0$ in homogeneous coordinates. If the plane is embedded at $z=1$ that's $ax+by+c=0$ in affine coordinates. Now you can write your $x=C$ as $1x+0y-C=0$ and obtain coordinates $[1:0:-C]$ for the line.

2. Joining points. The point on the $x$ axis is $[C:0:1]$ and the line at infinity for the vertical direction is $[0:1:0]$, infinitely far in $y$ direction. The line joining two points can be computed as the cross product, so $$\begin{bmatrix}C\\\0\\\1\end{bmatrix}\times\begin{bmatrix}0\\\1\\\0\end{bmatrix}=\begin{bmatrix}-1\\\0\\\C\end{bmatrix}$$ which is another representant of the same line.




Your $[-1/C:0:1]$ is yet a different representant of that same line, except if $C=0$. So you were right. In general with homogeneous coordinates you try to avoid divisions.