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Represenation of a Line and other linear geometrical elements

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.
 \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:
\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 \}
$ 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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