Parallel (geometry)

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In geometry, the line parallel is the line in the unfilled field; that is, two lines in a plane that do not touch each other or touch at any point are said to be parallel. With extensions, lines and planes, or two planes, in a three-dimensional Euclidean space that does not share a point is said to be parallel. However, two lines in a non-compliant three-dimensional space must be in the same plane to be considered parallel; otherwise they are called slashes. The parallel plane is a plane in the same three-dimensional space that never met.

The parallel lines are the subject of Euclid's parallel postulates. Parallelism is primarily the property of affine geometry and Euclidean geometry is a particular example of this type of geometry. In some other geometries, such as hyperbolic geometry, the line can have analog properties called parallelism.

Video Parallel (geometry)

Simbol

Simbol paralelnya adalah ${\ displaystyle \ parallel}  ÂÂ$ . Misalnya, ${\ displaystyle AB \ parallel CD}  ÂÂ$ menunjukkan bahwa garis AB sejajar dengan line CD .

In the Unicode character set, the "parallel" and "not parallel" marks have codes U 2225 (?) And U 2226 (?), Respectively. Additionally, U 22D5 (?) Represents the relation "equals and parallel to".

Maps Parallel (geometry)

Euclidean Paralysis

Two lines in the

field

Conditions for parallelism

With the parallel line l and m in the Euclidean space, the following properties are equivalent:

1. Each point in the path is m located at the same distance (minimum) of the line l ( line equals ).
2. The m line is in the same field as the l but does not intersect l (remember that the line extends unlimited in both directions).
3. When the lines m and l both intersect with the third straight line (transversal) in the same plane, the intersection angle corresponding to transversal is congruent.

Since these are equivalent properties, one of them can be taken as a definition of parallel lines in Euclidean space, but the first and third properties involve measurements, etc., are "more complicated" than the second. Thus, the second property is usually chosen as the property defining the parallel lines in Euclidean geometry. Other properties are consequences of Euclid Parallel Postulates. Another property that also involves measurement is that parallel lines to each other have the same gradient (slope).

History

The definition of parallel lines as a pair of straight lines in an unfulfilled field appears as Definition 23 in Book I of the Euclid Element. The alternative definition is discussed by other Greeks, often as part of an attempt to prove a parallel postulate. Proclus associates the definition of parallel lines as the same line to Posidonius and quotes Geminus in the same vein. Simplicius also mentions Posidonius's definition and modification by the Aganist philosopher.

In the late nineteenth century, in England, Euclid's Elements was still a standard textbook in high school. The traditional treatment of geometry is suppressed to change by new developments in projective geometry and non-Euclidean geometry, so some new textbooks for the teaching of geometry are written at this time. The main difference between these reform texts, both between themselves and between them and Euclid, is the treatment of parallel lines. These reform texts are not without their criticism and one of them, Charles Dodgson (a.k.a. Lewis Carroll), wrote a drama, Euclid and His Modern Competition , in which these texts were reviled.

One of the early reform textbooks was James Geometry by James Maurice Wilson in 1868. Wilson bases his definition of parallel lines on primitive notions of direction. According to Wilhelm Killing the idea can be traced back to Leibniz. Wilson, without defining the direction because it is primitive, uses the term in another definition like the sixth definition, "Two straight lines that meet each other have different directions, and their direction difference is the angle between them. "Wilson (1868, p.2) In definition 15 he introduces parallel lines in this way; "Straight lines that have same direction , but are not part of the same straight line, are called parallel lines ." Wilson (1868, p.12) Augustus De Morgan reviews this text and states it as a failure, especially on the basis of this definition and the way Wilson uses it to prove things about parallel lines. Dodgson also devoted most of his drama (Act II, Scene VI §§1) to denounce Wilson's treatment of alignment. Wilson edited this concept from the third edition and higher than the text.

Another property, proposed by other reformers, used as a substitute for the definition of parallel lines, was not much better. The main difficulty, as Dodgson points out, is that to use it in this way requires additional axioms to be added to the system. The definition of the equidistant line of Posidonius, described by Francis Cuthbertson in his 1874 text Euclidean Geometry suffered the problem that the points found at a certain distance on one side of a straight line must be shown to form a straight line. This can not be proven and should be considered true. The corresponding angle formed by the transverse property, used by WD Cooley in its 1860 text, The geometry element, simplified and described requires evidence of the fact that if a transverse encounters a pair of lines in corresponding angular congruent then all transversal should do it. Again, a new axiom is needed to justify this claim.

Construction

The three properties above lead to three different parallel line construction methods.

The distance between two parallel lines

Karena garis-garis sejajar dalam bidang Euclidean berjarak sama ada jarak yang unik antara dua garis sejajar. Mengingat persamaan dua garis paralel non-vertikal, non-horizontal,

${\ displaystyle y = mx b_ {1} \,}  ÂÂ$

${\ displaystyle y = mx b_ {2} \ ,,}  ÂÂ$

the distance between two lines can be found by placing two points (one on each line) that lie on the common perpendicular to the parallel lines and calculating the distance between them. Since the line has a slope of m , a common perpendicular line will have a slope of -1/ m and we can take the line with the equation y = - x / m as a general perpendicular. Solve linear systems

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

dan

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

untuk mendapatkan koordinat poin. Solusi untuk sistem linear adalah poinnya

${\ displaystyle \ left (x_ {1}, y_ {1} \ right) \ = \ kiri ({\ frac {-b_ {1} m} {m ^ { 2} 1}}, {\ frac {b_ {1}} {m ^ {2} 1}} \ kanan) \,}  ÂÂ$

dan

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

Rumus-rumus ini masih memberikan titik koordinat yang benar bahkan jika garis-garis sejajar horisontal (yaitu, m = 0). Jarak antar titik adalah

${\ displaystyle d = {\ sqrt {\ left ({\ frac {b_ {1} m-b_ {2} m} {m ^ {2} 1}} \ right) ^ {2} \ left ({\ frac {b_ {2} -b_ {1}} {m ^ {2} 1}} \ right) ^ {2}}} \ ,,}  ÂÂ$

yang mengurangi menjadi

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

Ketika garis diberikan oleh bentuk umum persamaan garis (garis horisontal dan vertikal disertakan):

${\ displaystyle ax c_ {1} = 0 \,}  ÂÂ$

${\ displaystyle ax c_ {2} = 0, \,}  ÂÂ$

jarak mereka dapat diekspresikan sebagai

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

Dua garis dalam ruang tiga dimensi

Two lines in the same three-dimensional space that do not intersect do not need to be parallel. Only if they are on a public plane, they are called parallel; otherwise they are called slashes.

Two different lines l and m in three-dimensional space are aligned if and only if the distance from the point P on line m to the nearest point on the l does not depend on the location P in the m path. This never applies to slashes.

Lines and fields

A line m and the field q in a three-dimensional space, a line that does not lie in that plane, is aligned if and only if they are not intersecting.

Equivalent, they are aligned if and only if the distance from the point P in the path m to the nearest point in the field q P in the m path.

Two planes

Similar to the fact that the parallel lines must lie in the same plane, the parallel plane must lie in the same three-dimensional space and contain no common ground.

Two different fields q and r are aligned if and only if the distance from the point P in the q field to the nearest point in the field r does not depend on the location P in the q field. This will never happen if both planes are not in the same three-dimensional space.

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Extensions for non-Euclidean geometry

In non-Euclidean geometry, it is more common to speak of geodesy than the line (straight). Geodesy is the shortest path between two points in a given geometry. In physics this can be interpreted as the path that the particle follows if no force is applied to it. In non-Euclidean geometry (elliptic or hyperbolic geometry) the three Euclidean properties mentioned above are unequal and only the second, (The line m is in the same plane as line l but does not inters l) since no measurement is useful in non geometry -Ellidean. In general the geometry of the three properties above gives three different types of curves, the same spacing curve , parallel geodesies and geodesics that share a common perpendicular, respectively.

Hyperbolic Geometry

While in Euclidean geometry, two geodeses can be intersect or parallel, in hyperbolic geometry, there are three possibilities. Two geodesics belonging to the same field can be:

1. cut , if they intersect at the same point on the plane,
2. parallel , if they do not intersect in the plane, but converge to a common boundary point in infinity (ideal point), or
3. very parallel , if they do not have a common boundary point to infinity.

In the literature ultra parallel geodesics is often called non-intersecting . Geodesy intersecting at infinity is called restrictive parallel .

As in the illustration through the point a not on the path l there are two restricted parallel lines, one for each direction of the ideal point of line l. They split lines that cut lines and very parallel lines with lines l .

The ultra-parallel line has one common perpendicular (ultraparallel theorem), and diverges on both sides of this general perpendicular.

Geometri bulat atau eliptik

In a rounded geometry, all geodesic is a large circle. The big circle splits the ball in two equal hemispheres and all the big circles intersect each other. Thus, there is no parallel geodesy to a particular geodesic, since all geodesics intersect. The same curve on the sphere is called parallel latitude analogous to the latitude in the globe. The parallel latitude can be generated by the intersection of the sphere with the plane parallel to the plane through the center of the ball.

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Reflexive variant

Jika l, m, n adalah tiga garis yang berbeda, maka ${\ displaystyle l \ parallel m \ \ land \ m \ parallel n \ \ menyiratkan \ l \ paralel n.}  ÂÂ$

In this case, parallelism is a transitive relationship. However, in the case of l = n , the superimposed line is not is considered to be parallel in Euclidean geometry. The binary relationship between parallel lines is clearly a symmetric relationship. According to Euclid's principle, parallelism is not reflexive relation and thus fail to be an equivalence relation. However, in the affine geometry the parallel pencil lines are taken as the equivalence class in the set of lines in which parallelism is the relation of equality.

For this purpose, Emil Artin (1957) adopted the definition of parallelism in which two lines are parallel if they have all or none of their points in common. Then the line is parallel to itself so that the reflexive and transitive nature belongs to this type of parallelism, creating the equivalent relation on the set of lines. In the study of the geometry of events, variants of this parallelism are used in the affine field.

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See also

• Clifford's parallel
• Restrict parallel
• Parallel curve
• The Ultraparallel Theorem

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Note

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References

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

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

Richards, Joan L. (1988), The Mathematical Vision: The Pursuit of Geometry in the Victorian England , Boston: Academic Press ISBN: 0-12-587445-6.genre = books & amp; rft.btitle = Mathematical Vision% 3A Pursuit Geometry in Victorian English & amp; rft.place = Boston & amp; rft.pub = Academic Press & amp; rft.date = 1988 & amp; rft. isbn = 0- 12-587445-6 & amp; rft.aulast = Richards & amp; rft.aufirst = Joan L. & amp; rfr_id = info% 3Asid% 2Fen.wikipedia.org% 3And% 28geometry% 29 "> < span>

Wilson, James Maurice (1868), Elementary Geometry (1st ed.), London: Macmillan and Co.