Remote element

Visualization of the far point
Comparable to the far point, straight lines that are parallel in reality meet in the perspective representation at a point, the vanishing point . In contrast to the vanishing point, however , the far point is not a point on the plane of the drawing (i.e. not - like the vanishing point here - identical to a point on the drawn door), but is located outside the set of "real" points.

As remote elements are the elements (points, lines, and so on), leading to a - dimensional affine space be added to this to a projective space , the projective completion of the affine space to expand, conversely, caused by slots one -dimensional projective space always a -dimensional affine space. ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$

A far point (also: infinitely distant point or improper point ) is introduced as the “intersection” of a family of parallel straight lines. A distant point is the mathematical specification of the way of speaking that "parallels intersect at infinity". The image of a far point in a perspective representation is called a vanishing point .

All far points of a plane form its far line ( infinitely distant straight line , improper straight line ).

In spatial (three-dimensional) geometry there is a long-distance line for every family of parallel planes. The long line together form the distance plane ( infinitely distant plane , improper plane ).

There are further remote levels and correspondingly higher-dimensional remote elements in rooms of higher dimensions:

With the projective closure of a -dimensional affine space, a far hyperplane , i.e. a -dimensional far space , is added to the space. Conversely, when “slitting” a -dimensional projective space, a -dimensional subspace, that is, a hyperplane of the projective space becomes a remote hyperplane. All points of this selected hyperplane become distant points, their subspaces become distant lines, etc., all other points of the projective space, the actual points, then form the affine space. ${\displaystyle n}$ ${\displaystyle n-1}$ ${\displaystyle n}$ ${\displaystyle n-1}$

The slotting of a projective plane by selecting a projective straight line as a distance line is a possibility in synthetic geometry to introduce projective coordinates into any geometrically characterized planes with the aid of affine coordinates . These coordinates then form a ternary body .

literature

• Hanfried Lenz : Lectures on projective geometry , Leipzig 1965
• Günter Pickert : Ebene incidence geometry , 2nd edition, Frankfurt am Main 1968
• Hermann Schaal: Linear Algebra and Analytical Geometry, Volume II , Vieweg 1980, ISBN 3-528-13057-1