Parallellinien

Parallellinien are two distinct lines that lie in the same plane and never intersect, regardless of how far they are extended. In Euclidean geometry, this non-intersection implies that the lines have the same direction, commonly expressed by equal slopes in a Cartesian coordinate system.

Through any point not on a given line, there is exactly one line parallel to the given line.

Two parallel lines have a constant distance between them; the distance is measured along a line perpendicular to both.

Analytically, two lines ax+by=c and ax+by=d are parallel; the distance between them is |c-d|/sqrt(a^2+b^2).

In non-Euclidean geometries the meaning changes: in spherical geometry there are no parallel lines, since all great circles intersect; in hyperbolic geometry there are infinitely many parallel lines through a point.

In three dimensions, two lines are parallel if they have the same direction and do not intersect; lines can also be skew, meaning they do not lie in the same plane.

Parallellinien are used in tiling, architecture, and perspective drawing, where a pair of parallel guides helps maintain consistent spacing and alignment. See also perpendicular lines, skew lines, and the distance between parallel lines.