**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.