Паралелност (геометрија)

Линијско цртање паралелних линија и кривих.

Паралелност у геометрији представља однос између два геометријска објекта. Примери дефинисања паралелности у еуклидској геометрији су:

• Две праве су паралелне уколико се налазе у једној равни и не секу се.
• Права је паралелна са равни, уколико са њом нема пресечних тачака. (еуклидски простор)
• Две различите равни су паралелне уколико се не секу. (еуклидски простор)

У тродимензионом простору треба разликовати појам паралелности и мимоилажења. Уколико две праве не леже у истој равни и не секу се, онда су исте мимоилазне а не паралелне. По сличном концепту постоји и појам мимоилазних равни у еуклидским просторима већих димензија. У еуклидском простору Rⁿ, за два афина потпростора a₁ + V₁ и a₂ + V₂ каже се да су паралелни ако је један од одговарајућих векторских потпростора V₁ и V₂ потпростор другог. У геометријском простору тачака у којем је уведен појам бесконачно далеких тачака, два геометријска објекта су паралелна уколико се секу у бесконачно далекој тачки.

Parallel lines are the subject of Euclid's parallel postulate. Parallelism is primarily a property of affine geometries and Euclidean geometry is a special instance of this type of geometry. In some other geometries, such as hyperbolic geometry, lines can have analogous properties that are referred to as parallelism.

Симбол

The parallel symbol is $\parallel$. For example, $AB\parallel CD$ indicates that line AB is parallel to line CD.

In the Unicode character set, the "parallel" and "not parallel" signs have codepoints U+2225 (∥) and U+2226 (∦), respectively. In addition, U+22D5 (⋕) represents the relation "equal and parallel to".

The same symbol is used for a binary function in electrical engineering (the parallel operator). It is distinct from the double-vertical-line brackets that indicate a norm, as well as from the logical or operator (||) in several programming languages.

Еуклидски паралелизам

Two lines in a plane

Conditions for parallelism

As shown by the tick marks, lines a and b are parallel. This can be proved because the transversal t produces congruent corresponding angles $\theta$, 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:

1. Every point on line m is located at exactly the same (minimum) distance from line l (equidistant lines).
2. Line m is in the same plane as line l but does not intersect l (recall that lines extend to infinity in either direction).
3. When lines m and l are both intersected by a third straight line (a transversal) in the same plane, the corresponding angles of intersection with the transversal are 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. The other properties are then consequences of Euclid's Parallel Postulate. Another property that also involves measurement is that lines parallel to each other have the same gradient (slope).

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. 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 also mentions Posidonius' definition as well as its modification by the philosopher Aganis.

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 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." Wilson (1868, 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." Wilson (1868, 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.

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.

Construction

The three properties above lead to three different methods of construction of parallel lines.

The problem: Draw a line through a parallel to l.

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

Distance between two parallel lines

Главни чланак: Distance between two 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,

$y=mx+b_{1}\,$

$y=mx+b_{2}\,,$

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

${\begin{cases}y=mx+b_{1}\\y=-x/m\end{cases}}$

and

${\begin{cases}y=mx+b_{2}\\y=-x/m\end{cases}}$

to get the coordinates of the points. The solutions to the linear systems are the points

$\left(x_{1},y_{1}\right)\ =\left({\frac {-b_{1}m}{m^{2}+1}},{\frac {b_{1}}{m^{2}+1}}\right)\,$

and

$\left(x_{2},y_{2}\right)\ =\left({\frac {-b_{2}m}{m^{2}+1}},{\frac {b_{2}}{m^{2}+1}}\right).$

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

$d={\sqrt {\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}}={\sqrt {\left({\frac {b_{1}m-b_{2}m}{m^{2}+1}}\right)^{2}+\left({\frac {b_{2}-b_{1}}{m^{2}+1}}\right)^{2}}}\,,$

which reduces to

$d={\frac {|b_{2}-b_{1}|}{\sqrt {m^{2}+1}}}\,.$

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

$ax+by+c_{1}=0\,$

$ax+by+c_{2}=0,\,$

their distance can be expressed as

$d={\frac {|c_{2}-c_{1}|}{\sqrt {a^{2}+b^{2}}}}.$

Литература

• Heath, Thomas L. (1956), The Thirteen Books of Euclid's Elements (2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925] изд.), New York: Dover Publications 

(3 vols.): ISBN 0-486-60088-2 (vol. 1), ISBN 0-486-60089-0 (vol. 2), ISBN 0-486-60090-4 (vol. 3). Heath's authoritative translation plus extensive historical research and detailed commentary throughout the text.

• Richards, Joan L. (1988), Mathematical Visions: The Pursuit of Geometry in Victorian England, Boston: Academic Press, ISBN 0-12-587445-6 
• Wilson, James Maurice (1868), Elementary Geometry (1st изд.), London: Macmillan and Co. 
• Wylie Jr., C. R. (1964), Foundations of Geometry, McGraw–Hill 
• Papadopoulos, Athanase; Théret, Guillaume (2014), La théorie des parallèles de Johann Heinrich Lambert : Présentation, traduction et commentaires, Paris: Collection Sciences dans l'histoire, Librairie Albert Blanchard, ISBN 978-2-85367-266-5 
• Nicholas M. Patrikalakis and Takashi Maekawa, Shape Interrogation for Computer Aided Design and Manufacturing, Springer, 2002, ISBN 3540424547, 9783540424543, pp. 408. [2]
• A'Campo, Norbert and Papadopoulos, Athanase, (2012) Notes on hyperbolic geometry, in: Strasbourg Master class on Geometry, pp. 1–182, IRMA Lectures in Mathematics and Theoretical Physics, Vol. 18, Zürich: European Mathematical Society (EMS), 461 pages, SBN ISBN 978-3-03719-105-7, DOI 10.4171–105.
• Coxeter, H. S. M., (1942) Non-Euclidean geometry, University of Toronto Press, Toronto
• Fenchel, Werner (1989). Elementary geometry in hyperbolic space. De Gruyter Studies in mathematics. 11. Berlin-New York: Walter de Gruyter & Co. 
• Fenchel, Werner; Nielsen, Jakob (2003). Asmus L. Schmidt, ур. Discontinuous groups of isometries in the hyperbolic plane. De Gruyter Studies in mathematics. 29. Berlin: Walter de Gruyter & Co. 
• Lobachevsky, Nikolai I., (2010) Pangeometry, Edited and translated by Athanase Papadopoulos, Heritage of European Mathematics, Vol. 4. Zürich: European Mathematical Society (EMS). xii, 310~p, ISBN 978-3-03719-087-6/hbk
• Milnor, John W., (1982) Hyperbolic geometry: The first 150 years, Bull. Amer. Math. Soc. (N.S.) Volume 6, Number 1, pp. 9–24.
• Reynolds, William F., (1993) Hyperbolic Geometry on a Hyperboloid, American Mathematical Monthly 100:442–455.
• Stillwell, John (1996). Sources of hyperbolic geometry. History of Mathematics. 10. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-0529-9. MR 1402697. 
• Samuels, David, (March 2006) Knit Theory Discover Magazine, volume 27, Number 3.
• James W. Anderson, Hyperbolic Geometry, Springer 2005, ISBN 1-85233-934-9
• James W. Cannon, William J. Floyd, Richard Kenyon, and Walter R. Parry (1997) Hyperbolic Geometry, MSRI Publications, volume 31.
• Meserve, Bruce E. (1983) [1959], Fundamental Concepts of Geometry, Dover, ISBN 0-486-63415-9 
• Papadopoulos, Athanase (2015), Euler, la géométrie sphérique et le calcul des variations. In: Leonhard Euler : Mathématicien, physicien et théoricien de la musique (dir. X. Hascher et A. Papadopoulos), CNRS Editions, Paris, ISBN 978-2-271-08331-9 
• Van Brummelen, Glen (2013). Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry. Princeton University Press. ISBN 9780691148922. Приступљено 31. 12. 2014. CS1 одржавање: Формат датума (веза)
• Roshdi Rashed and Athanase Papadopoulos (2017) Menelaus' Spherics: Early Translation and al-Mahani'/alHarawi's version. Critical edition of Menelaus' Spherics from the Arabic manuscripts, with historical and mathematical commentaries, De Gruyter Series: Scientia Graeco-Arabica 21 ISBN 978-3-11-057142-4

Спољашње везе

• Constructing a parallel line through a given point with compass and straightedge