Parallel (geometry)

In Euclidean geometry is defined: Two lines are parallel if they lie in a plane and do not intersect. In addition, you shall determine that each should be a straight line parallel to itself.

It is often said of parallel straight lines, which do not coincide, intersect with each other " at infinity ". This statement gets a precise meaning when the Euclidean space is extended to a projective space.

In three-dimensional Euclidean space is valid:

• Two straight lines that do not lie in a plane are called skew. ( Also, they do not intersect but are not parallel. )
• A straight line is parallel to a plane if it is entirely in this plane or does not intersect.
• Two planes are parallel if they coincide or do not intersect. One speaks of parallel planes.

Analog ways of speaking are valid for Euclidean and affine geometries in arbitrary dimension and for analytical geometry ( the geometry in Euclidean vector spaces ). In particular, two lines are parallel in a vector space if their direction vectors are linearly dependent (or proportional) are.

Properties

In the plane Euclidean and affine geometry applies:

• For every line and every point that does not lie on the straight line, there is exactly one line that is parallel to the given line and passes through the given point ( the Parallel by this point ).

This statement is called, because it is needed in an axiomatic structure of Euclidean geometry as an axiom the axiom of parallels. In analytic geometry ( geometry in Euclidean vector spaces ), however, it is provable (ie a set ). In affine spaces of arbitrary dimension applies:

• The relation "parallel" between straight lines forms an equivalence relation, the lines can thus be divided into equivalence classes of parallel straight lines. Such an equivalence class is called a parallel class and forms a special bunch.
• If you add an affine space for each parallel class " distant infinite" one (also called " inauthentic " ) point ( farthest point ) which describes the then ever intersect two lines of the band, one obtains a projective space as a projective completion of the affine space.

In Euclidean geometry also applies to arbitrary dimension of space:

• In parallel straight lines g and h is the distance of all points of G to the straight line h is constant ( and vice versa), the straight lines are always the same distance from each other. The same applies to parallel planes.

In the non-Euclidean geometry applies: If you replace the parallel axiom by requiring for each line and each point which does not lie on the straight line, there are at least two lines through the point which does not intersect the given line, we obtain a non-Euclidean geometry, namely the hyperbolic.

Generalization of affine spaces

In one - dimensional affine space above a body affinity partial spaces can be described as cosets of linear subspaces of the space to be associated with coordinate vector. Then and. We define now:

• The rooms and are parallel when or applies.

Parallelism can be defined as equivalent Alone with geometric terms:

• And the spaces are connected in parallel, if there is a parallel displacement of the affinity area so as to apply or.

Written vectorially equivalent to a displacement vector (it can be selected for example from the first representation ) and the statement is then

• And the spaces are connected in parallel, if there is a shift, so that, or applies.

Most of this very general definition is limited to affine subspaces that are at least one-dimensional, because otherwise for the purposes of definition, the empty set and one-point quantities at any subspace were parallel.

Properties

• The generalized parallelism is on the set of -dimensional subspaces of a -dimensional affine space ( for fixed ) is an equivalence relation. An equivalence class is called a parallel class of planes, especially for a parallel class of hyperplanes.
• In the language of projective geometry is such a parallel group of dimensional levels from all levels which intersect at one -dimensional ( projective ) subspace of the remote hyperplane. Therefore, one also speaks of a sheaf. ( The terms bundle and tufts in projective geometry Projective see room # Projective subspace. )
• On the set of all affine subspaces ( of arbitrary dimension ) is the parallelism symmetrically and reflexive, but not transitive, so in general no equivalence relation.

Related terms

The idea of the parallel curve is also used in other situations in which most characterize transmitted by the constant distance.

• In a parallel displacement of each point is shifted by a " constant amount in the same direction "

• A parallel curve to a flat curve is obtained by plotting each point on the curve by a constant amount in the direction of the normal at that point.

• A parallel body to a ( closed ) convex body obtained if one's body " to r increases ," ie, adds all points whose distance is less than or equal to r, by forming the union of all balls of radius r, centered in the body is.

• Two vectors which show exactly in opposite direction are anti-parallel.

Generalizations for finite geometries

In the finite geometry, the concept of parallelism is defined ( as an equivalence relation) in a more general form for block plans. Finite affine and projective geometries can be considered as a special block plans. The classification of " straight lines", which are referred to in the finite geometry as " blocks ", in "Parallel droves " is generalized in the theory of block designs with the concept of resolution of a block diagram. A further generalization of the resolution is the concept of tactical decomposition.