Duality Principle

A concept that will be useful below is the point-line duality principle. This principle is based on the commutative property of the dot product applied to the equation of a line written in implicit form, where the points on the line are expressed as homogeneous coordinates:

$$\mathbf{l}^{\top} \mathbf{x} = \mathbf{x}^{\top} \mathbf{l} = 0$$ (1.76)

It is therefore possible to obtain dual formulations by replacing the parameters of a line $\mathbf{l}$ with those of one of its points $\mathbf {x}$.

This observation gives rise to the duality principle (Duality Principle), which guarantees that the solution to the dual problem, in which the meanings of line and point are exchanged, is also a solution to the original problem.

For example, as seen in the preceding sections, given two points $\mathbf {p}$ and $\mathbf{q}$, it is possible to define a line $\mathbf{l} = \mathbf{p} \times \mathbf{q}$ passing through them, whereas given two lines $\mathbf{l}$ and $\mathbf{m}$, it is possible to define a point $\mathbf{x} = \mathbf{l} \times \mathbf{m}$ as their intersection.