I am trying to understand what's being said here in Z. Zhang - Emerging Topics in Computer Vision 2004

The author offers a geometric interpretation the constraints in finding intrinsic parameters using Homography between the model plane and its image

The author describes the model plane $(\Pi)$ by the following equation : $$ (\Pi) : \begin{bmatrix}\mathbf{r}_{3}^{\top}&\mathbf{r}_{3}^{\top}\mathbf{t}\end{bmatrix}\begin{bmatrix}\widehat{x}\\\widehat{y}\\\widehat{z}\\\epsilon\end{bmatrix}=0 $$ with $\mathbf{r}_{3}^{\top}$ being the third row of the rotation matrix $\mathcal{R}$ and $\mathbf{t}$ being the translation vector. This $1\times 4$ vector multiplied by the $4\times 1$ column variables provides the mathematical description to the model plane. Note that $\epsilon=0$ if the points are at infinity and $1$ if otherwise.

I have found from this website what appears to be some geometrical figure regarding this concept :

enter image description here

Z. Zhang makes the following statement that confused me :

This plane intersects the plane at infinity at a line, and we can easily see that $\begin{bmatrix}\mathbf{r}_{1}&0\end{bmatrix}^{\top}$ and $\begin{bmatrix}\mathbf{r}_{2}&0\end{bmatrix}^{\top}$ are two particular points on that line. Any point on it is a linear combination of these two points

From the figure above, I see two parallel lines on $w=1$ and if we can imagine two planes that intersect each one of them then they will intersect at infinity at a line.

My questions are :

  1. Is this what the author meant by his statement?
  2. Does this mean that if I have two parallel lines and that if I stand far enough that they will appear as if they are just one line (at infinity)?
  • $\begingroup$ I think Math.SE or even ComputerGraphics.SE may be more likely to provide an answer. But don't forget to post cross-references if you do cross-post your question. $\endgroup$
    – Ruslan
    Sep 22, 2021 at 20:04


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