# Representation of a line as the span of two vectors

According to Harley, Zisserman (Multiple View Geometry) a line can be represented via the span of a matrix $W^T$ defined as:

$$\begin{bmatrix} A^T\\ B^T \end{bmatrix}$$ where $A$ and $B$ are two non-coincident points.

Hartley and Zisserman stated, that the span of $W^T$ is the pencil of points on the line.

When I read this definition I think of $A$ and $B$ as two vectors. When these are linear dependent the span is a line and the statement is true. But what if the two vectors are linear independent? Then the span is plane and I have no idea how this works with the definition above.

But I'm sure I miss something important here.

## 1 Answer

Remember that you’re working in $$\mathbb P^3$$, not $$\mathbb R^3$$. Even if $$A$$ and $$B$$ are linearly dependent in $$\mathbb R^3$$, they won’t be when represented in homogeneous coordinates, since those coordinates will correspond to distinct lines through the origin in $$\mathbb R^4$$.

You left out an important part of that sentence that I think might help you understand what’s going on. It reads, in full, “The span of $$W^T$$ is the pencil of points $$\lambda A+\mu B$$ on the line.” I.e., the line consists of the points $$\begin{bmatrix}A&B\end{bmatrix}\begin{bmatrix}\lambda\\\mu\end{bmatrix}$$

For example, take the line $$x-y=0$$, $$z=0$$. We can set $$W=\begin{bmatrix}1&1&0&1\\0&0&0&1\end{bmatrix},$$ which yields the points $$(\lambda, \lambda, 0, \lambda+\mu)^T$$, or equivalently, $$\left(\frac\lambda{\lambda+\mu}, \frac\lambda{\lambda+\mu},0,1\right)^T$$, which is clearly the right set of points. The two points used to form $$W$$ are certainly linearly dependent in $$\mathbb R^3$$, since one of them is the origin, but they’re not linearly dependent when represented in homogeneous coordinates.

Similarly, if we take the line through $$(1,0,0)$$ and $$(2,0,0)$$, we get the matrix $$W=\begin{bmatrix}1&0&0&1\\2&0&0&1\end{bmatrix},$$ which yields the points $$\left({\lambda+2\mu\over\lambda+\mu},0,0,1\right)^T$$. Play around with some other examples yourself to see how this works. (Note that you can’t rescale like I did when $$\lambda=-\mu$$, but that case corresponds to the point at infinity, which is on every line.)

The next paragraph in the book explains why this works, and the example on the following page is actually rather interesting because one of the points used to represent the $$x$$-axis is the point at infinity.