# Visualizing projective plane intersection at infinity

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 :

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)?
• 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. Sep 22, 2021 at 20:04