Principle of Duality

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

$$\mathbf{l}^{\top} \mathbf{x} = \mathbf{x}^{\top} \mathbf{l} = 0$$ (1.40)

It is therefore possible to obtain dual formations when the parameters of a line $\mathbf{l}$ are replaced with those of a point $\mathbf {x}$.

From this consideration arises the principle of duality (Duality Principle) which guarantees that the solution of the dual problem, where the meanings of line and point are swapped, is also a solution to the original problem.

For example, as seen in the previous 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, while 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.