Represenation of a Line and other linear geometrical elements

Line on a plane

On a plane line is a set of x=(x,y) points, where the following expression holds:

$
ax+by=c where $$ \sqrt{a^{2}+b^{2}}\neq 0$ .
Equivalently:
$$ \frac{a}{\sqrt{a^{2}+b^{2}}}x + \frac{b}{\sqrt{a^{2}+b^{2}}}y = \frac{c}{\sqrt{a^{2}+b^{2}}}$

Now the coefficients of x and of y squared add up to 1, we can define the
$$\boldsymbol{\omega}$ unit vector:
$$ \boldsymbol{\omega}= \left (\frac{a}{\sqrt{a^{2}+b^{2}}} , \frac{b}{\sqrt{a^{2}+b^{2}}} \right )$
and a t scalar:
$$ t= \frac{c}{\sqrt{a^{2}+b^{2}}}$
With these notations the expression of a line is:
$$ \boldsymbol{\omega}\boldsymbol{x}=t$ (1)
Thus, we are looking for the set of o points, where the projection of the location vector to a given $$ \boldsymbol{\omega}$ vector is constant.These points are located on a line perpendicular to vector $$ \boldsymbol{\omega}$ , and the distance of this line from the origin is t.

In order to parametrize the points of this line we look for the $$ \boldsymbol{\omega}_{\perp}$ unit vector perpendicular to vector $$ \boldsymbol{\omega}$ . For unique solution let us choose the sign of the determinant of these vectors, now we chose positive:
$$ \begin{vmatrix} \omega_{1} & \omega_{\perp1}\\ \omega_{2} & \omega_{\perp2} \end{vmatrix}=1$ (2)

Let us have the variable of integration the point s, with that we obtain the l points of an L line as follows:
$$ \mathbf{l}_{t,\boldsymbol{\omega}}\left ( s \right )=t\boldsymbol{\omega}+s\boldsymbol{\omega}_{\perp}$

This description still does not constitute a unique description, as when
$$ t\in \left \{ -\infty,\infty \right \}$ , then $$\mathbf{l}_{t,\boldsymbol{\omega}}=\mathbf{l}_{-t,\boldsymbol{-\omega}}$ . We should limit either t to positive numbers, or limit $$\boldsymbol{\omega}$ to one of the half-spaces. E.g., when $$\boldsymbol{\omega}= \left ( \cos \left ( \vartheta \right ),\sin \left ( \vartheta \right ) \right )$ , then either
$$ t\in \left \{ -\infty,\infty \right \}$ and $$ \vartheta\in \left \{ 0,\pi \right \}$
or
$$ t\in \left \{ 0,\infty \right \}$ and $$ \vartheta\in \left \{ 0,2\pi \right \}$

In the literature both conventions are present.

Linear elements in higher dimensions

The expression for the 2D line on $$\mathbf{x}\in \mathbb{R}^{2}$ determines sets of points such, that for a $$ t\in \mathbb{R}$ scalar and for a unit vector $$\boldsymbol{\omega}\in \mathbb{S}^{1}$ of a sphere of one degree of freedom ( $$\mathbb{S}$ ), the equation holds: $$ \boldsymbol{\omega}\boldsymbol{x}=t$ and with that $$ \left(t,\boldsymbol{\omega} \right ) \in \mathbb{R}\times \mathbb{S}^{1}$ equations determine a line with a direction. When we look at the parametrization of it in Eq. (2), s and t are interchangeable, since $$\boldsymbol{\omega}$ és $$\boldsymbol{\omega}_{\perp}$ determine each other apart from a sign. We could also say, that the parameter of our line is s and $$\boldsymbol{\omega_{\perp}}$ , the variable of integration is t, in the direction of $$\boldsymbol{\omega}$ .

In an n dimension space, Eq. (1) given that $$ \left(t,\boldsymbol{\omega} \right ) \in \mathbb{R}\times \mathbb{S}^{n-1}$ is an expression of a hyperplane perpendicular to the direction vector $$\boldsymbol{\omega}$ . Now to specify a single point on this plane, we need a set of direction vectors of a complete base of unit vectors $$\boldsymbol{\omega_{\perp,i}}$ , that we now with an off-hand notation order into matrix $$\boldsymbol{\Omega_{\perp} }$ , so now multiplied by a vector of $$\mathbf{s}\in \mathbb{R}^{n-1}$ we arrive into a point of the plane as follows:
$$ t\boldsymbol{\omega}+\mathbf{s}\boldsymbol{\Omega_{\perp} }$

If we choose, like we did before, for the parameters of the linear set $$ \left(t,\boldsymbol{\omega} \right )$ , then our expression describes points of the H hyperplane:
$$ \mathbf{H}_{\boldsymbol{\omega},t}\left ( \mathbf{s},\boldsymbol{\Omega_{\perp} } \right )=t\boldsymbol{\omega}+\mathbf{s}\boldsymbol{\Omega_{\perp} }=\mathbf{x}_{0}+\mathbf{s}\boldsymbol{\Omega_{\perp}}$

On the contrary, if we chose as the parameters the elements of the product $$ \mathbf{s}\boldsymbol{\Omega_{\perp} }$ , we obtain a line, with points along unit vector $$\boldsymbol{\omega}$ with variable of integration t:
$$\mathbf{L}_{\mathbf{s},\boldsymbol{\Omega_{\perp}}} \left ( t,\boldsymbol{\omega} \right )=t\boldsymbol{\omega}+\mathbf{s}\boldsymbol{\Omega_{\perp} }=t\boldsymbol{\omega}+\mathbf{x}_{0}$

Note, that the points of the H hyperplane is determined by n independent information contained in $$ \left(t,\boldsymbol{\omega} \right )$ together, while the L line is determined by the product $$ \mathbf{s}\boldsymbol{\Omega_{\perp} }$
with 2(n-1) independent elements, since additionally to the unit vector $$ \boldsymbol{\omega}$ we need the values of vector s as well. To reach a point in space we still need to define the vector base of $$ \boldsymbol{\Omega_{\perp}}$ , bearing no information on the object, it only constitutes the coordinate system choice.

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