Line passing through two points

For two points $(x_0,y_0)$ and $(x_1, y_1)$ in Cartesian space $\mathbb{R}^2$, there exists an implicit line with the equation

$$(y_1 - y_0) x - (x_1 - x_0) y - y_1 x_0 + x_1 y_0 = 0$$ (1.27)

where it is clearly visible that there are no singularities and all values are admissible.

Indicating with $(d_x, d_y)$ the difference between the two points, the line passing through a point $(x_0,y_0)$ and directed along the vector $(d_x, d_y)$ has the equation

$$d_y x - d_x y + y_0 d_x - x_0 d_y = 0$$ (1.28)

Generalizing to the n-dimensional case, the equation of the line in $\mathbb{R}^{n}$, passing through two points $\mathbf {p}$ and $\mathbf{q}$ written in homogeneous form, is the locus of points $\mathbf{x} \in \mathbb{R}^n$ such that

$$\mathbf{x} = (1-t) \mathbf{p} + t \mathbf{q} = \mathbf{p} + t ( \mathbf{q} - \mathbf{p} )$$ (1.29)

the equation of the parametric ray with $t \in \mathbb{R}$ scalar value. The values of $\mathbf {x}$ associated with values $t \in [0,1]$ are internal points to the segment $(\mathbf{p},\mathbf{q})$.

Using instead the homogeneous coordinates, limited to the two-dimensional Cartesian case, the following remarkable result is obtained: the line with parameters $\mathbf{l}=(a,b,c)^{\top}$, passing through the points $\mathbf{x}_1$ and $\mathbf{x}_2$, is obtained as

$$\mathbf{l}=\mathbf{x}_1 \times \mathbf{x}_2$$ (1.30)

since any point $\mathbf {x}$, to belong to the line, must satisfy the equation (1.26).