Parallel (geometry)

In geometry, parallel lines are lines in a plane which do not meet. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. However, two lines in three-dimensional space which do not meet must be in a common plane to be considered parallel. Parallel planes are planes in the same three-dimensional space. Parallel lines are the subject of Euclid's parallel postulate. Parallelism is 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; the parallel symbol is ∥. For example, A B ∥ C D 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". Given parallel straight lines l and m in Euclidean space, the following properties are equivalent: Every point on line m is located at the same distance from line l.

Line m is in the same plane as line l but does not intersect l. When lines m and l are both intersected by a third straight line 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, so, are "more complicated" than the second. Thus, the second property is the one chosen as the defining property of parallel lines in Euclidean geometry; the other properties are consequences of Euclid's Parallel Postulate. Another property that involves measurement is that lines parallel to each other have the same gradient; 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 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 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, wrote a play 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, the difference of their directions is the angle between them."

Wilson In definition 15 he introduces parallel lines in this way. Wilson Augustus De Morgan reviewed this text and declared it a failure on the basis of this definition and the way Wilson used it to prove things about parallel lines. Dodgson devotes a large section of his play 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 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 and explained requires a proof of the fact that if one transversal meets a pair of lines in congruent corresponding angles all transversals must do so.

Again, a new axiom is needed to justify this statement. The three properties above lead to three different methods of construction of 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, y = m x + b 1 y = m x + b 2

Angle of parallelism

In hyperbolic geometry, the angle of parallelism Π, is the angle at one vertex of a right hyperbolic triangle that has two asymptotic parallel sides. The angle depends on the segment length a between the right angle and the vertex of the angle of parallelism. Given a point off of a line, if we drop a perpendicular to the line from the point a is the distance along this perpendicular segment, φ or Π is the least angle such that the line drawn through the point at that angle does not intersect the given line. Since two sides are asymptotic parallel, lim a → 0 Π = 1 2 π and lim a → ∞ Π = 0. There are five equivalent expressions that relate Π and a: sin Π = sech a = 1 cosh a = 2 e a + e − a, cos Π = tanh a = e a − e − a e a + e − a, tan Π = csch a = 1 sinh a = 2 e a − e − a, tan = e − a, Π = 1 2 π − gd , where sinh, tanh and csch are hyperbolic functions and gd is the Gudermannian function. János Bolyai discovered a construction which gives the asymptotic parallel s to a line r passing through a point A not on r.

Drop a perpendicular from A onto B on r. Choose any point C on r different from B. Erect a perpendicular t to r at C. Drop a perpendicular from A onto D on t. Length DA is longer than CB, but shorter than CA. Draw a circle around C with radius equal to DA, it will intersect the segment AB at a point E. The angle BEC is independent of the length BC, depending only on AB. Construct s through A at angle BEC from AB. Sin B E C = sinh B C sinh C E = sinh B C sinh D A = sinh B C sin A C D sinh C A = sinh B C cos A C B sinh C A = sinh B C tanh C A tanh C B sinh C A = cosh B C cosh C A = cosh B C cosh C B cosh A B = 1 cosh A B. See Trigonometry of right triangles for the formulas used here; the angle of parallelism was developed in 1840 in the German publication "Geometrische Untersuchungen zur Theory der Parallellinien" by Nikolai Lobachevsky. This publication became known in English after the Texas professor G. B. Halsted produced a transla