# Q: Points in general position

Question. The above picture only show an affine part of the respective projective spaces. What happens if some of the points lie at infinity?

## 4 thoughts on “Q: Points in general position”

1. $\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} -1 \\ 1 \\ 1 \end{bmatrix}$ are not in general position in $P(\mathbb{R}^3) = \mathbb{R}^2 \cup P(\mathbb{R}^2)$. Furthermore $\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} -1 \\ 0 \end{bmatrix} \left(= \begin{bmatrix} 1 \\ 0 \end{bmatrix}\right), \begin{bmatrix} 0 \\ 1 \end{bmatrix}$ are not in general position in $P(\mathbb{R}^2) = \mathbb{R} \cup P(\mathbb{R})$.

Question. Let $U = \text{span}\left(\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}\right)$. Then $P(U)$ is a line. Is there a way to draw this line? Post your pictures 😀

• Nice figures and good follow-up question! Last weeks tutorial should help 😉

• Well, I think I have to rephrase my question (cos it doesnt express what it was intended to do)…
When I imagine a line, I think of it as affine combinations of two points, which leads to my…
Question. Take some arbitrary point and a point at infinity. Is there a line in the projective space s.t. both points lie on it? Can I visualize this line?

• Talking about “points at infinity” implies using a particular model of a projective space.

In the hemisphere model, you may see the points at the boundary as points at infinity. Clearly, there is a line through such a point on the boundary and a second point in the interior of the hemisphere.

In the affine model, you may understand the points at infinity as directions. To require that a line passes through a certain point at infinity means prescribing a direction for the line. Again, a finite point and a direction determine a unique line.

Question: What happens if you choose both points at infinity?