My textbook, Multiple View Geometry in Computer Vision by Hartley and Zisserman says the following:
Ideal points and the line at infinity. Homogeneous vectors $\mathbf{x} = (x_1, x_2, x_3)^T$ such that $x_3 \not= 0$ correspond to finite points in $\mathbb{R}^2$. One may augment $\mathbb{R}^2$ by adding points with last coordinate $x_3 = 0$. The resulting space is the set of all homogeneous $3$-vectors, namely the projective space $\mathbb{P}^2$. The points with last coordinate $x_3 = 0$ are known as ideal points, or points at infinity. The set of all ideal points may be written $(x_1, x_2, 0)^T$, with a particular point specified by the ratio $x_1 : x_2$. Note that this set lies on a single line, the line at infinity, denoted by the vector $\mathbf{l}_{\infty} = (0, 0, 1)^T$. Indeed, one verifies that $(0, 0, 1)(x_1, x_2, 0)^T = 0$.
Using result 2.2 one finds that a line $\mathbf{l} = (a, b, c)^T$ intersects $\mathbf{l}_{\infty}$ in the ideal point $(b, -a, 0)^T$ (since $(b, -a, 0)\mathbf{l} = 0$). A line $\mathbf{l}' = (a, b, c')^T$ parallel to $\mathbf{l}$ intersects $\mathbf{l}_{\infty}$ in the same ideal point $(b, -a, 0)^T$ irrespective of the value of $c'$. In inhomogeneous notation, $(b, -a)^T$ is a vector tangent to the line, and orthogonal to the line normal $(a, b)$, and so represents the line's direction. As the line's direction varies the ideal point $(b, -a, 0)^T$ varies over $\mathbf{l}_\infty$. For these reasons the line at infinity can be thought of as the set of directions of lines in the plane.
My confusion is with regards to this part:
In inhomogeneous notation, $(b, -a)^T$ is a vector tangent to the line, and orthogonal to the line normal $(a, b)$, and so represents the line's direction. As the line's direction varies the ideal point $(b, -a, 0)^T$ varies over $\mathbf{l}_\infty$. For these reasons the line at infinity can be thought of as the set of directions of lines in the plane.
Specifically, it is not clear to me as to precisely which line the author is referring to:
In inhomogeneous notation, $(b, -a)^T$ is a vector tangent to the line, and orthogonal to the line normal $(a, b)$, and so represents the line's direction.
If my understanding is correct, the inhomogeneous vector $(b, -a)^T$ is normal to the line $\mathbf{l}$ (and $\mathbf{l}'$) described by the vector $(a, b, c)$ (or $(a, b, c')$ in the case of $\mathbf{l}'$), and the inhomogeneous vector $(a, b)^T$ is the tangent to the same lines (and so represents the direction of the lines). However, this is phrased differently from what the author has written, and, given the vagueness of this part, it is unclear what line the author is precisely referring to.
I would greatly appreciate it if people could please take the time to clarify this.
Best Answer
No, the inhomogeneous vector $(a,b)$ is normal to the line $\mathbf l = (a,b,c)^T$. Go back to how this homogeneous vector representation of a line is defined: The line is the set of all points $\mathbf x$ such that $\mathbf l^T\mathbf x=0$. Expanded, this equation is $ax+by+c=0$, which is just the (I hope) familiar point-normal equation of the line.
The point of this passage is that each ideal point corresponds to a unique direction in the plane; all lines go in this direction intersect at that ideal point.