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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)?

InKodeWeTrust
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1 Answers1

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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.

MvG
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  • The divide was done because I needed to do a perspective transformation (homography). And (I believe) it needs to be done with z =1 because you want the transformation occurring on the equivalent euclidean planes (i.e. image plane). – InKodeWeTrust Sep 27 '18 at 10:06
  • Homography using homogeneous coordinates without division should work perfectly fine. Dehomogenization with $z=1$ helps for points to imagene the point as a simple point in the Euclidean plane, with $x$ and $y$ coordinate. For lines this is less useful, and actively harmful for lines through the origin which have $c=0$. A slightly more useful normalization might be $b=-1$ because then $ax-y+c=0$ corresponds to $y=ax+c$, the common slope and intercept form of a line. But of course this fails for vertical lines as in your example. I can imagine no valid reason to dehomogenize a line here. – MvG Sep 27 '18 at 17:22