Parallelism (geometry)

In Euclidean geometry one defines: Two straight lines are parallel if they lie in one plane and do not intersect. In addition, one stipulates that every straight line should be parallel to itself. Two straight lines are called true parallel if they are parallel but not identical.

It is often said of genuinely parallel straight lines that they intersect "at infinity" . This statement gets a precise sense when the Euclidean space is expanded to a projective space .

In three-dimensional Euclidean space the following also applies:

• Two straight lines that do not lie in one plane are called skewed . (They also have no intersection, but are not parallel.)
• A straight line is parallel to a plane if it lies entirely in this plane or does not intersect it.
• Two planes are parallel if they coincide or do not intersect. One speaks of parallel levels .

Analogous ways of speaking apply to Euclidean and affine geometries in any dimension and to analytic geometry (the geometry in Euclidean vector spaces ). In particular, two straight lines in a vector space are parallel if their direction vectors are linearly dependent (or proportional).

properties

In plane Euclidean and affine geometry:

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

This statement is called the axiom of parallels , since it is required as an axiom for an axiomatic structure of Euclidean geometry . In analytic geometry (geometry in Euclidean vector spaces), however, it is provable (i.e. a proposition ). In affine spaces of any dimension the following applies:

• The “parallel” relationship between straight lines forms an equivalence relationship , so the straight lines can be divided into equivalence classes of straight lines parallel to one another. Such an equivalence class is known as a family of parallels and forms a special tuft .
• If one adds an "infinitely distant" (also "improper" ) point ( far point ) to an affine space for each family of parallels , at which two straight lines of the family intersect, one obtains a projective space as the projective closure of the affine space.

In Euclidean geometry the following applies for any dimension of space:

• In the case of parallel straight lines  and , the distance between all points from the straight line is  constant (and vice versa), so the straight lines are always the same distance from each other. The same applies to parallel planes. ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle g}$ ${\displaystyle h}$

Outside of Euclidean geometry applies: If you replace the parallel axiom by the requirement to every line and every point that is not on the line, there are at least two lines through the point, which is precisely not to cut the given , the result is a non-Euclidean geometry , namely the hyperbolic .

Generalization for affine spaces

In a -dimensional affine space over a body , affine subspaces can be described as secondary classes of linear subspaces of the associated coordinate vector space. Then is and . One now defines: ${\displaystyle n}$ ${\displaystyle A}$ ${\displaystyle K}$ ${\displaystyle A_{1},A_{2}}$ ${\displaystyle U_{1},U_{2}<K^{n}}$ ${\displaystyle A}$ ${\displaystyle A_{1}=P_{1}+U_{1}}$ ${\displaystyle A_{2}=P_{2}+U_{2}}$

• The spaces and are parallel if or applies. ${\displaystyle A_{1}}$ ${\displaystyle A_{2}}$ ${\displaystyle U_{1}\subseteq U_{2}}$ ${\displaystyle U_{2}\subseteq U_{1}}$

Using geometric terms alone, parallelism can be defined as:

• The spaces and are parallel if there is a parallel shift of the affine space such that or holds. ${\displaystyle A_{1}}$ ${\displaystyle A_{2}}$ ${\displaystyle \tau }$ ${\displaystyle A}$ ${\displaystyle \tau (A_{1})\subseteq A_{2}}$ ${\displaystyle A_{2}\subseteq \tau (A_{1})}$

Written vectorially corresponds to a displacement vector (it can be chosen from the first representation , for example ) and the statement then reads ${\displaystyle \tau }$ ${\displaystyle {\vec {v}}\in K^{n}}$ ${\displaystyle {\vec {v}}={\overrightarrow {P_{1}P_{2}}}}$

• The spaces and are parallel if there is a displacement such that or holds. ${\displaystyle A_{1}}$ ${\displaystyle A_{2}}$ ${\displaystyle {\vec {v}}\in K^{n}}$ ${\displaystyle A_{1}+{\vec {v}}\subseteq A_{2}}$ ${\displaystyle A_{2}\subseteq A_{1}+{\vec {v}}}$

Usually this very general definition is restricted to affine subspaces that are at least one-dimensional, since otherwise the empty set and single-point sets would be parallel to any subspace in the sense of the definition.

properties

• The generalized parallelism is an equivalence relation on the set of -dimensional subspaces of a -dimensional affine space (for fixed ). An equivalence class is called a family of parallels of planes , especially as a family of parallels of hyperplanes . ${\displaystyle k}$ ${\displaystyle n}$ ${\displaystyle 1\leq k<n}$ ${\displaystyle k=n-1}$
• In the language of projective geometry, such a family of parallels of -dimensional planes consists of all planes that intersect in a -dimensional (projective) subspace of the remote hyperplane. Therefore one speaks of a plane cluster . (For the terms bundle and tuft in projective geometry, see Projective Space # Projective Subspace .) ${\displaystyle k}$ ${\displaystyle k-1}$
• On the set of all affine subspaces (of any dimension ), the parallelism is symmetrical and reflexive, but not transitive, i.e. generally no equivalence relation. ${\displaystyle 1\leq k<n}$ ${\displaystyle n>2}$

Related terms

Parallel straight lines and curves

The idea of ​​the parallel course is also used in other situations, whereby the characterization is usually transferred through the constant distance.

• With a parallel shift, each point is shifted by a “constant amount in the same direction”

(in vector spaces:) . ${\displaystyle x\mapsto x+a}$

Thus, lines and half-straight lines can also run parallel to one another, although these special cases are not covered by the Euclidean definition.

• A curve parallel to a plane curve is obtained by applying a constant amount in the direction of the normal to this point at each point of the curve.

(for a curve these are the curves if the normalized normal vector is closed). ${\displaystyle \gamma (s)\in \mathbb {R} ^{2}}$ ${\displaystyle \gamma (s)\pm an(s)}$ ${\displaystyle n(s)}$ ${\displaystyle \gamma (s)}$

(Example: concentric circles)

• A body parallel to a (closed) convex body is obtained if the body is " enlarged by r ", i.e. That is, add all points whose distance is less than or equal to r by taking the union of all spheres of radius r whose center lies in the body.

(In vector spaces:, where the sphere of radius r is around the origin.) ${\displaystyle K+B_{r}=\{x+y\mid x\in K,y\in B_{r}\}}$ ${\displaystyle B_{r}=\{y\mid \left\|y\right\|\leq r\}}$

• Two vectors that point exactly in opposite directions are anti-parallel .

Generalizations for finite geometries

In finite geometry , the concept of parallelism (as an equivalence relation) is also defined in a more general form for block plans . Finite affine and projective geometries can be understood as special block plans. The division of the “straight lines”, also called “blocks” in finite geometry, into “sets of parallels” is generalized in the theory of block plans to the concept of dissolving a block plan . Another generalization of resolution is the concept of tactical decomposition .

See also

Other positional relationships of straight lines are:

• Copunctality
• Orthogonality
• Complanarity

Individual evidence

1. ↑ Manfred Andrie, Paul Meier: Linear Algebra and Geometry for Engineers: An application-oriented introduction with exercises . Springer, 3rd edition, 2013, ISBN 9783642957987 , p. 202
2. ↑ plane parallelism. Retrieved November 17, 2012 . 
3. ^ Parallelism of vectors. Retrieved November 17, 2012 . 
4. ^ Affine Geometry - Lecture at the University of Jena. (PDF; 94 kB) Retrieved November 17, 2012 . 
5. ^ Translation. WolframMathWorld, accessed November 17, 2012 . 
6. ↑ Four types of symmetry. Retrieved November 17, 2012 . 
7. ↑ parallel curves . Retrieved November 17, 2012 . 
8. ↑ Parallel bodies and convex bodies. Retrieved November 17, 2012 . 
9. ↑ Antiparallel. WolframMathWorld, accessed November 17, 2012 .