# 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:

where .
Equivalently:

Now the coefficients of x and of y squared add up to 1, we can define the
unit vector:

and a t scalar:

With these notations the expression of a line is:
(1)
Thus, we are looking for the set of o points, where the projection of the location vector to a given vector is constant.These points are located on a line perpendicular to vector , and the distance of this line from the origin is t.

In order to parametrize the points of this line we look for the unit vector perpendicular to vector . For unique solution let us choose the sign of the determinant of these vectors, now we chose positive:
(2)

Let us have the variable of integration the point s, with that we obtain the l points of an L line as follows:

This description still does not constitute a unique description, as when
, then . We should limit either t to positive numbers, or limit to one of the half-spaces. E.g., when , then either
and
or
and

In the literature both conventions are present.

### Linear elements in higher dimensions

The expression for the 2D line on determines sets of points such, that for a scalar and for a unit vector of a sphere of one degree of freedom ( ), the equation holds: and with that equations determine a line with a direction. When we look at the parametrization of it in Eq. (2), s and t are interchangeable, since és determine each other apart from a sign. We could also say, that the parameter of our line is s and , the variable of integration is t, in the direction of .

In an n dimension space, Eq. (1) given that is an expression of a hyperplane perpendicular to the direction vector . Now to specify a single point on this plane, we need a set of direction vectors of a complete base of unit vectors , that we now with an off-hand notation order into matrix , so now multiplied by a vector of we arrive into a point of the plane as follows:

If we choose, like we did before, for the parameters of the linear set , then our expression describes points of the H hyperplane:

On the contrary, if we chose as the parameters the elements of the product , we obtain a line, with points along unit vector with variable of integration t:

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