Editing Parallel (geometry) (section)

== Euclidean parallelism==

===Two lines in a plane===

====Conditions for parallelism====

[[Image:Parallel transversal.svg|thumb|right|300px|As shown by the tick marks, lines ''a'' and ''b'' are parallel. This can be proved because the transversal ''t'' produces congruent corresponding angles <math>\theta</math>, shown here both to the right of the transversal, one above and adjacent to line ''a'' and the other above and adjacent to line ''b''.]]

Given parallel straight lines ''l'' and ''m'' in [[Euclidean space]], the following properties are equivalent:

#Every point on line ''m'' is located at exactly the same (minimum) distance from line ''l'' (''[[equidistant]] lines'').
#Line ''m'' is in the same plane as line ''l'' but does not intersect ''l'' (recall that lines extend to [[infinity]] in either direction).
#When lines ''m'' and ''l'' are both intersected by a third straight line (a [[Transversal (geometry)|transversal]]) in the same plane,  the [[corresponding angles]] of intersection with the transversal are [[Congruence (geometry)|congruent]].

Since these are equivalent properties, any one of them could be taken as the definition of parallel lines in Euclidean space, but the first and third properties involve measurement, and so, are "more complicated" than the second. Thus, the second property is the one usually chosen as the defining property of parallel lines in Euclidean geometry.<ref>{{harvnb|Wylie|1964|loc=pp. 92—94}}</ref> The other properties are then consequences of [[Euclid's Fifth Axiom|Euclid's Parallel Postulate]].

====History====
The definition of parallel lines as a pair of straight lines in a plane which do not meet appears as Definition 23 in Book I of [[Euclid's Elements]].<ref name=Euclid>{{harvnb|Heath|1956|loc=pp. 190–194}}</ref> Alternative definitions were discussed by other Greeks, often as part of an attempt to prove the [[parallel postulate]]. [[Proclus]] attributes a definition of parallel lines as equidistant lines to [[Posidonius]] and quotes [[Geminus]] in a similar vein. [[Simplicius of Cilicia|Simplicius]] also mentions Posidonius' definition as well as its modification by the philosopher Aganis.<ref name=Euclid />

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.<ref>{{harvnb|Richards|1988|loc=Chap. 4: Euclid and the English Schoolchild. pp. 161–200}}</ref> 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.<ref>{{citation|first=Lewis|last=Carroll|title=Euclid and His Modern Rivals|date=2009|orig-year=1879|publisher=Barnes & Noble|isbn=978-1-4351-2348-9}}</ref>

One of the early reform textbooks was James Maurice Wilson's ''Elementary Geometry'' of 1868.<ref>{{harvnb|Wilson|1868}}</ref> Wilson based his definition of parallel lines on the [[primitive notion]] of ''direction''. According to [[Wilhelm Killing]]<ref>''Einführung in die Grundlagen der Geometrie, I'', p. 5</ref> the idea may be traced back to [[Gottfried Wilhelm Leibniz|Leibniz]].<ref>{{harvnb|Heath|1956|loc= p. 194}}</ref> 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." {{harvtxt|Wilson|1868|loc=p. 2}} 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''." {{harvtxt|Wilson|1868|loc=p. 12}} [[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.<ref>{{harvnb|Richards|1988|loc=pp. 180–184}}</ref>

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.<ref>{{harvnb|Heath|1956|loc=p. 194}}</ref> 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.

==== Construction ====
The three properties above lead to three different methods of construction<ref>Only the third is a straightedge and compass construction, the first two are infinitary processes (they require an "infinite number of steps".)</ref> of parallel lines.
{{Clear|left}}
[[Image:Par-prob.png|thumb|left|250px|The problem: Draw a line through ''a'' parallel to ''l''.]]
{{Clear|left}}
<gallery widths="200px">
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°.
</gallery>

==== Distance between two parallel lines ====
{{Main|Distance between two parallel lines}}

Because parallel lines in a Euclidean plane are [[equidistant]] there is a unique distance between the two parallel lines. Given the equations of two non-vertical, non-horizontal parallel lines,
:<math>y = mx+b_1\,</math>
:<math>y = mx+b_2\,,</math>
the distance between the two lines can be found by locating two points (one on each line) that lie on a common perpendicular to the parallel lines and calculating the distance between them. Since the lines have slope ''m'', a common perpendicular would have slope −1/''m'' and we can take the line with equation ''y'' = −''x''/''m'' as a common perpendicular. Solve the linear systems
:<math>\begin{cases}
y = mx+b_1 \\
y = -x/m
\end{cases}</math>
and
:<math>\begin{cases}
y = mx+b_2 \\
y = -x/m
\end{cases}</math>
to get the coordinates of the points. The solutions to the linear systems are the points
:<math>\left( x_1,y_1 \right)\ = \left( \frac{-b_1m}{m^2+1},\frac{b_1}{m^2+1} \right)\,</math>
and
:<math>\left( x_2,y_2 \right)\ = \left( \frac{-b_2m}{m^2+1},\frac{b_2}{m^2+1} \right).</math>
These formulas still give the correct point coordinates even if the parallel lines are horizontal (i.e., ''m'' = 0). The distance between the points is
:<math>d = \sqrt{\left(x_2-x_1\right)^2 + \left(y_2-y_1\right)^2} = \sqrt{\left(\frac{b_1m-b_2m}{m^2+1}\right)^2 + \left(\frac{b_2-b_1}{m^2+1}\right)^2}\,,</math>
which reduces to
:<math>d = \frac{|b_2-b_1|}{\sqrt{m^2+1}}\,.</math>

When the lines are given by the general form of the equation of a line (horizontal and vertical lines are included):
:<math>ax+by+c_1=0\,</math>
:<math>ax+by+c_2=0,\,</math>
their distance can be expressed as
:<math>d = \frac{|c_2-c_1|}{\sqrt {a^2+b^2}}.</math>

===Two lines in three-dimensional space===
Two lines in the same [[three-dimensional space]] that do not intersect need not be parallel. Only if they are in a common plane are they called parallel; otherwise they are called [[skew lines]].

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.

===A line and a plane===
A line ''m'' and a plane ''q'' in three-dimensional space, the line not lying in that plane, are parallel if and only if they do not intersect.

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 planes===
Similar to the fact that parallel lines must be located in the same plane, parallel planes must be situated in the same three-dimensional space and contain no point in common.

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.