Point at infinity

As distance elements are the elements ( points, lines, and so on ) that are added to a - dimensional affine space to expand this projective to a room, the projective completion of the affine space, conversely produced by slitting one -dimensional projective space always one - dimensional affine space.

A remote point (also: point at infinity or improper point) is introduced as the " intersection " of a family of parallel straight lines. A far point is thus the mathematical specification of the speech that " parallel lines meet at infinity ". The image of a remote point in a perspective representation is called vanishing point.

All remote points of a plane form the line at infinity ( at infinity, improper line).

In the spatial ( three-dimensional ) geometry there is one line at infinity to each family of parallel planes. The line at infinity together form the remote plane ( plane at infinity, improper level).

More remote levels and correspondingly higher dimensional remote elements are available in areas of higher dimension:

At the conclusion of a projective -dimensional affine space the space is added to a remote hyperplane, ie a remote -dimensional space. Conversely, one -dimensional sub-space, that is, a hyperplane of the projective space to the remote hyperplane at the " slits " of -dimensional projective space. All points of these selected hyperplane to be remote points, their subspaces to remote lines, etc., and all other points of the projective space, the actual points, then we construct the affine space.

The slots of a projective plane by selecting a projective line as the line at infinity is a way to introduce in any geometrically characterized levels projective coordinates using affine coordinates in synthetic geometry. These coordinates then form a Ternärkörper.