Intersection of Two Lines

Let $\ell_1$ and $\ell_2$ be two lines with parameters $\mathbf{l}_1$ and $\mathbf{l}_2$ intersecting at the point $\mathbf {x}$ expressed in homogeneous coordinates. To obtain the point of intersection, it is necessary to solve a homogeneous system in the form

$$\left\{ \begin{array}{rl} \mathbf{l}_1^{\top} \mathbf{x} &... ... \mathbf{l}_2^{\top} \mathbf{x} & = 0 \\ \end{array} \right.$$ (1.39)

The system, of type $\mathbf{A}\mathbf{x}=0$, can also be extended to the case of n intersecting lines, with $n > 2$, resulting in an overdetermined system that can be solved using the SVD or QR decomposition technique. The solution to the overdetermined problem, affected by noise, represents the point that minimizes the algebraic residue of equation (1.39).

In the case of only two lines, the system (1.39) directly provides the solution.

The intersection between two lines $\mathbf{l}_1$ and $\mathbf{l}_2$, written in implicit form (1.23), is the point $\mathbf{x} = \mathbf{l}_1 \times \mathbf{l}_2$ expressed in homogeneous coordinates, where $\times$ is the cross product.

It is noteworthy that, since homogeneous coordinates can represent points at infinity, this particular formalism also accommodates the case where the two lines are parallel.