Parallel (geometry)

 ( Elementary Geometry ) 

At the end of the nineteenth century, in England, Euclid's Elements was still the standard textbook in secondary schools. The traditional treatment of geometry was being pressured to change by the new developments in projective geometry and non-Euclidean geometry, so several new textbooks for the teaching of geometry were written at this time. A major difference between these reform texts, both between themselves and between them and Euclid, is the treatment of parallel lines. These reform texts were not without their critics and one of them, Charles Dodgson (a.k.a. Lewis Carroll), wrote a play, Euclid and His Modern Rivals, in which these texts are lambasted.

One of the early reform textbooks was James Maurice Wilson's Elementary Geometry of 1868. Wilson based his definition of parallel lines on the primitive notion of direction. According to Wilhelm Killing Einführung in die Grundlagen der Geometrie, I, p. 5 the idea may be traced back to Leibniz. Wilson, without defining direction since it is a primitive, uses the term in other definitions such as his sixth definition, "Two straight lines that meet one another have different directions, and the difference of their directions is the angle between them." In definition 15 he introduces parallel lines in this way; "Straight lines which have the same direction, but are not parts of the same straight line, are called parallel lines." Augustus De Morgan reviewed this text and declared it a failure, primarily on the basis of this definition and the way Wilson used it to prove things about parallel lines. Dodgson also devotes a large section of his play (Act II, Scene VI § 1) to denouncing Wilson's treatment of parallels. Wilson edited this concept out of the third and higher editions of his text.

Other properties, proposed by other reformers, used as replacements for the definition of parallel lines, did not fare much better. The main difficulty, as pointed out by Dodgson, was that to use them in this way required additional axioms to be added to the system. The equidistant line definition of Posidonius, expounded by Francis Cuthbertson in his 1874 text Euclidean Geometry suffers from the problem that the points that are found at a fixed given distance on one side of a straight line must be shown to form a straight line. This can not be proved and must be assumed to be true. The corresponding angles formed by a transversal property, used by W. D. Cooley in his 1860 text, The Elements of Geometry, simplified and explained requires a proof of the fact that if one transversal meets a pair of lines in congruent corresponding angles then all transversals must do so. Again, a new axiom is needed to justify this statement.

image:Par-equi.png|Property 1: Line m has everywhere the same distance to line l. image:Par-para.png|Property 2: Take a random line through a that intersects l in x. Move point x to infinity. image:Par-perp.png|Property 3: Both l and m share a transversal line through a that intersect them at 90°.

$y\; =\; mx+b\_1\backslash ,$

$y\; =\; mx+b\_2\backslash ,,$

$\backslash begin\{cases\}$

$\backslash begin\{cases\}$

$\backslash left(\; x\_1,y\_1\; \backslash right)\backslash \; =\; \backslash left(\; \backslash frac\{-b\_1m\}\{m^2+1\},\backslash frac\{b\_1\}\{m^2+1\}\; \backslash right)\backslash ,$

$\backslash left(\; x\_2,y\_2\; \backslash right)\backslash \; =\; \backslash left(\; \backslash frac\{-b\_2m\}\{m^2+1\},\backslash frac\{b\_2\}\{m^2+1\}\; \backslash right).$

$d\; =\; \backslash sqrt\{\backslash left(x\_2-x\_1\backslash right)^2\; +\; \backslash left(y\_2-y\_1\backslash right)^2\}\; =\; \backslash sqrt\{\backslash left(\backslash frac\{b\_1m-b\_2m\}\{m^2+1\}\backslash right)^2\; +\; \backslash left(\backslash frac\{b\_2-b\_1\}\{m^2+1\}\backslash right)^2\}\backslash ,,$

$d\; =\; \backslash frac${\sqrt{m^2+1}}\,.

When the lines are given by the general form of the equation of a line (horizontal and vertical lines are included):

$ax+by+c\_1=0\backslash ,$

$ax+by+c\_2=0,\backslash ,$

$d\; =\; \backslash frac${\sqrt {a^2+b^2}}.

Two distinct lines l and m in three-dimensional space are parallel if and only if the distance from a point P on line m to the nearest point on line l is independent of the location of P on line m. This never holds for skew lines.

Equivalently, they are parallel if and only if the distance from a point P on line m to the nearest point in plane q is independent of the location of P on line m.

Two distinct planes q and r are parallel if and only if the distance from a point P in plane q to the nearest point in plane r is independent of the location of P in plane q. This will never hold if the two planes are not in the same three-dimensional space.

In non-Euclidean geometry (elliptic or hyperbolic geometry) the three Euclidean properties mentioned above are not equivalent and only the second one (Line m is in the same plane as line l but does not intersect l) is useful in non-Euclidean geometries, since it involves no measurements. In general geometry the three properties above give three different types of curves, equidistant curves, parallel geodesics and geodesics sharing a common perpendicular, respectively.

1. intersecting, if they intersect in a common point in the plane,
2. parallel, if they do not intersect in the plane, but converge to a common limit point at infinity (ideal point), or
3. ultra parallel, if they do not have a common limit point at infinity.

In the literature ultra parallel geodesics are often called non-intersecting. Geodesics intersecting at infinity are called limiting parallel.

As in the illustration through a point a not on line l there are two limiting parallel lines, one for each direction ideal point of line l. They separate the lines intersecting line l and those that are ultra parallel to line l.

Ultra parallel lines have single common perpendicular (ultraparallel theorem), and diverge on both sides of this common perpendicular.

In this case, parallelism is a transitive relation. However, in case l = n, the superimposed lines are not considered parallel in Euclidean geometry. The binary relation between parallel lines is evidently a symmetric relation. According to Euclid's tenets, parallelism is not a reflexive relation and thus fails to be an equivalence relation. Nevertheless, in affine geometry a pencil of parallel lines is taken as an equivalence class in the set of lines where parallelism is an equivalence relation.H. S. M. Coxeter (1961) Introduction to Geometry, p 192, John Wiley & SonsWanda Szmielew (1983) From Affine to Euclidean Geometry, p 17, D. Reidel Andy Liu (2011) "Is parallelism an equivalence relation?", The College Mathematics Journal 42(5):372

To this end, Emil Artin (1957) adopted a definition of parallelism where two lines are parallel if they have all or none of their points in common.Emil Artin (1957) Geometric Algebra, page 52 via Internet Archive Then a line is parallel to itself so that the reflexive and transitive properties belong to this type of parallelism, creating an equivalence relation on the set of lines. In the study of incidence geometry, this variant of parallelism is used in the affine plane.

• Clifford parallel
• Collinearity
• Concurrent lines
• Limiting parallel
• Parallel curve
• Ultraparallel theorem

(3 vols.): (vol. 1), (vol. 2), (vol. 3). Heath's authoritative translation plus extensive historical research and detailed commentary throughout the text.