Line

There are several formulations to express the concept of a line.

In the most general case, the multidimensional case, a line, the locus of points $\mathbf{x} \in \mathbb{R}^n$ of dimension 1, takes the form

$$\mathbf{x} = \mathbf{p} + t \mathbf{v}$$ (1.22)

where $\mathbf{p} \in \mathbb{R}^n$ is a generic point of origin, $\mathbf{v} \in \mathbb{R}^n$ is the direction vector, and $t \in \mathbb{R}$ is a scalar. In this case, it is referred to as a parametric ray.

In most applications, the line is a typical concept in two-dimensional space. In this space, ignoring the equation of the line written in explicit form $y=mx + q$ as it presents singularities, we focus on the line written in implicit form. The equation of the line written in implicit form is:

$$a x + b y + c = 0$$ (1.23)

This representation is very useful because it allows considering both horizontal and vertical lines without any singularities. The parameter $c$ equals zero when the line passes through the origin and, of course, the line passes through a point $(x',y')$ when $c=-ax'-by'$.

In the two-dimensional case, the equation of the parametric ray (1.22) reduces to the equation of the implicit line with parameters

$$(a,b) \cdot \mathbf{v} = 0 \quad c = - (a,b) \cdot \mathbf{p}$$ (1.24)

From the first of the equations (1.24), it is seen that the vector formed by the parameters $(a,b)$ and the direction vector are orthogonal to each other. The generator vector of the line is indeed proportional, for example, to $\mathbf{v} \propto (-b, a)$ or $\mathbf{v} \propto (\frac{1}{a},-\frac{1}{b})$. The vector $\mathbf{v}'$ orthogonal to the given line is simply $\mathbf{v}' \propto (a,b)$ and the line orthogonal to the given one has an implicit equation written in the form

$$bx - ay + c' = 0$$ (1.25)

where $c'$ is obtained by selecting the point on the original line through which the perpendicular must pass.

The parameters of the line written in implicit form are homogeneous (the equation (1.23) is indeed called the homogeneous equation of the line), meaning they represent a vector subspace of $\mathbb{R}^3$: any multiple of those parameters represents the same line. Thus, these parameters are defined up to a multiplicative factor. This consideration suggests another way to represent a line and a generic hyperplane.

Lines, written in homogeneous implicit form, must satisfy the equation (dot product):

$$\mathbf{l}^{\top} \mathbf{x} = 0$$ (1.26)

with $\mathbf{x} \in \mathbb{R}^{3}$ being the point in homogeneous coordinates and $\mathbf{l}=(a,b,c)^{\top}$ the parameters of the line. For homogeneous coordinates see the previous section 1.4 while for the implications of this notation, on the point-line duality, see subsection 1.5.7.

Since the implicit line is known up to a multiplication factor, there are infinitely many ways to express the same line. One possible normalization of the line is obtained by dividing the parameters by the length $\sqrt{a^2 +b^2}$. In this case, a particular solution of the line is obtained as the parameters correspond to those of a line written in polar coordinates in the same form as equation (1.46), and consequently, with this normalization, the parameter $c$ represents the minimum distance between the line and the origin of the axes.

Finally, since the line is a hyperplane in 2 dimensions, its equation can be written as shown in equation (1.49).