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| Line 1: |
Line 1: |
| [[File:FuncionLineal01.svg|290px|thumb|Three lines — the red and blue lines have the same slope, while the red and green ones have same y-intercept.]]
| | == maggio 2010danah Boyd Scarpe Mbt == |
| [[File:1D line.svg|310px|thumb|A representation of one [[line segment]]]]
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| {{General geometry}}
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| The notion of '''line''' or '''straight line''' was introduced by ancient mathematicians to represent [[Curvature|straight]] objects with negligible width and depth. Lines are an idealization of such objects. Until the seventeenth century, lines were defined like this: "The line is the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width. […] The straight line is that which is equally extended between its points"<ref>In (rather old) French: "La ligne est la première espece de quantité, laquelle a tant seulement une dimension à sçavoir longitude, sans aucune latitude ni profondité, & n'est autre chose que le flux ou coulement du poinct, lequel […] laissera de son mouvement imaginaire quelque vestige en long, exempt de toute latitude. […] La ligne droicte est celle qui est également estenduë entre ses poincts." Pages 7 and 8 of ''Les quinze livres des éléments géométriques d'Euclide Megarien, traduits de Grec en François, & augmentez de plusieurs figures & demonstrations, avec la corrections des erreurs commises és autres traductions'', by Pierre Mardele, Lyon, MDCXLV (1645).</ref>
| | Mi dà come un ronzio e il calcio inizia la mia nuova stagione di mentalità. Gran parte dei nostri centri di lavoro sulle tecnologie emergenti Web 2.0, tra cui Twitter, Facebook, YouTube, ecc MSR TR 2010 60 15 maggio 2010Sarita Yardi e Danah Boyd, Tweeting dalla Piazza della Città: Misurazione geografiche reti locali, in International Conference on Weblogs e Social Media, American Association for Artificial Intelligence, maggio 2010danah Boyd, Making Sense di privacy e pubblicità, no. <br><br>È più interessante di essere in una camera d'albergo e mangiare in un luogo diverso e esco. Le persone che si iscrivono in grado sicuramente hanno l'ambizione e vogliono progredire professionalmente. Registrato presso IATA come Approvato Consultant International Travel. <br><br>Forse alcuni dei giochi ottenere un po 'noioso nel tempo? Certo che hanno fatto, [http://www.rifugiamoci.it/ImgCatalogo/Animazione/menu.asp Scarpe Mbt] e questo è probabilmente il motivo ho smesso di giocare dopo aver completato il corso di base iniziale. Dopo alcuni falliti tentativi di modifica di questo, ecco un video clip di Newberg spiegare le sue opinioni durante la nostra intervista: Cosa ne pensi Pensi scansioni cerebrali e delle neuroscienze possono dirci nulla di significativo circa la religione Segui FaithWorld su Twitter RTRFaithWorldDoing meditazione regolare? aumenta la consapevolezza, riduce lo stress e [http://www.ilmercantedisogni.it/Slide/small/form.asp Occhiali Da Sole Gucci] aiuta l'intero sistema mente corpo per raggiungere lo stato di homeostasis.Mindfulness è la tecnica di meditazione che trovo più pratico, che porta la persona al 100% nel momento presente. <br><br>Sembra molto probabile però che sarebbe stato aggredito per il suo telefono cellulare e guardare anche se lui non era portando il gioco. So quanto la sua fede gli fortificato durante la sua malattia.. [10] Il 23 giugno un altro decreto firmato da Aguinaldo è stato rilasciato , sostituendo il governo dittatoriale con un governo rivoluzionario, con se stesso come presidente. <br><br>Egli doesn tirare pugni perché crede nel valore di sollevare altri up! Lo potete trovare in simbolista. Mia mamma lavora nell'ufficio del presidente di un collegio comunità statale con circa 6500 studenti nel sud rurale. Ben 1 su 3 madri che allattano al seno possono avere mastite. <br><br>In primo luogo, assicurarsi te stesso che il nome [http://www.ilmercantedisogni.it/Popups/fold.asp Pandora Roma] del bookmaker scelto per le scommesse [http://www.ilmercantedisogni.it/Slide/small/form.asp Gucci Occhiali] online è affidabile. Sono i camion militari che stanno facendo il problema e bloccando alcune delle strade principali.'. Non mi riferisco a molte delle loro esperienze, forse perché io lavoro in un campo dominato molto male, forse perché i miei hobby principali sono maschilista, forse perché mia madre non era particolarmente ricettivo ogni volta ho scelto di aprirsi a lei.<ul> |
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| [[Euclid]] described a line as "breadthless length", and introduced several [[postulate]]s as basic unprovable properties from which he constructed the geometry, which is now called [[Euclidean geometry]] to avoid confusion with other geometries which have been introduced since the end of nineteenth century (such as [[non-Euclidean geometry]], [[projective geometry]], and [[affine geometry]]).
| | <li>[http://morigele.com/bbs/read.php?tid=10012267&page=e#a] http://morigele.com/bbs/read.php?tid=10012267&page=e#a]]</li> |
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| In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in [[analytic geometry]], a line in the plane is often defined as the set of points whose coordinates satisfy a given [[linear equation]], but in a more abstract setting, such as [[incidence geometry]], a line may be an independent object, distinct from the set of points which lie on it.
| | <li>[http://bbs.ykdai.cn/showtopic-151322.aspx http://bbs.ykdai.cn/showtopic-151322.aspx]</li> |
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| When a geometry is described by a set of [[axiom]]s, the notion of a line is usually left undefined (a so-called [[primitive notion|primitive]] object). The properties of lines are then determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry. Thus in [[differential geometry]] a line may be interpreted as a [[geodesic]] (shortest path between points), while in some [[Projective geometry|projective geometries]] a line is a 2-dimensional vector space (all linear combinations of two independent vectors). This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a light ray as being a line.
| | <li>[http://bbs.thinkidea.net/forum.php?mod=viewthread&tid=693207&fromuid=192420 http://bbs.thinkidea.net/forum.php?mod=viewthread&tid=693207&fromuid=192420]</li> |
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| A [[line segment]] is a part of a line that is bounded by two distinct end points and contains every point on the line between its end points. Depending on how the line segment is defined, either of the two end points may or may not be part of the line segment. Two or more line segments may have some of the same relationships as lines, such as being parallel, intersecting, or skew.
| | <li>[http://clan.gamescraft.de/index.php?site=guestbook http://clan.gamescraft.de/index.php?site=guestbook]</li> |
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| ==Definitions versus descriptions==
| | </ul> |
| All definitions are ultimately circular in nature since they depend on concepts which must themselves have definitions, a dependence which can not be continued indefinitely without returning to the starting point. To avoid this vicious circle certain concepts must be taken as [[primitive notion|primitive]] concepts; terms which are given no definition.<ref>{{harvnb|Coxeter|1969|loc=pg. 4}}</ref> In geometry, it is frequently the case that the concept of line is taken as a primitive.<ref>{{harvnb|Faber|1983|loc=pg. 95}}</ref> In those situations where a line is a defined concept, as in [[coordinate geometry]], some other fundamental ideas are taken as primitives. When the line concept is a primitive, the behaviour and properties of lines are dictated by the [[axiom]]s which they must satisfy.
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| In a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too abstract to be dealt with. In this circumstance it is possible that a ''description'' or ''mental image'' of a primitive notion is provided to give a foundation to build the notion on which would formally be based on the (unstated) axioms. Descriptions of this type may be referred to, by some authors, as definitions in this informal style of presentation. These are not true definitions and could not be used in formal proofs of statements. The "definition" of line in [[Euclid's Elements]] falls into this category.<ref>{{harvnb|Faber|1983|loc=pg. 95}}</ref> Even in the case where a specific geometry is being considered (for example, [[Euclidean geometry]]), there is no generally accepted agreement among authors as to what an informal description of a line should be when the subject is not being treated formally.
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| == Ray ==<!-- This section is linked (older copy) from: [[Tangent]], [[Ray (mathematics)]], -->
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| Given a line and any point ''A'' on it, we may consider ''A'' as decomposing this line into two parts.
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| Each such part is called a ''ray'' (or ''half-line'') and the point ''A'' is called its ''initial point''. The point A is considered to be a member of the ray.<ref>On occasion we may consider a ray without its initial point. Such rays are called ''open'' rays, in contrast to the typical ray which would be said to be ''closed''.</ref> Intuitively, a ray consists of those points on a line passing through ''A'' and proceeding indefinitely, starting at ''A'', in one direction only along the line. However, in order to use this concept of a ray in proofs a more precise definition is required.
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| Given distinct points ''A'' and ''B'', they determine a unique '''ray''' with initial point ''A''. As two points define a unique line, this ray consists of all the points between ''A'' and ''B'' (including ''A'' and ''B'') and all the points ''C'' on the line through ''A'' and ''B'' such that ''B'' is between ''A'' and ''C''.<ref>{{harvnb|Wylie, Jr.|1964|loc=pg. 59, Definition 3}}</ref> This is, at times, also expressed as the set of all points ''C'' such that ''A'' is not between ''B'' and ''C''.<ref>{{harvnb|Pedoe|1988|loc=pg. 2}}</ref> A point ''D'', on the line determined by ''A'' and ''B'' but not in the ray with initial point ''A'' determined by ''B'', will determine another ray with initial point ''A''. With respect to the ''AB'' ray, the ''AD'' ray is called the ''opposite ray''.
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| [[File:Ray (A, B, C).svg|500px|center|Ray]]
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| Thus, we would say that two different points, ''A'' and ''B'', define a line and a decomposition of this line into the [[disjoint union]] of an open segment {{open-open|''A'', ''B''}} and two rays, ''BC'' and ''AD'' (the point ''D'' is not drawn in the diagram, but is to the left of ''A'' on the line ''AB''). These are not opposite rays since they have different initial points.
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| The definition of a ray depends upon the notion of betweenness for points on a line. It follows that rays exist only for geometries for which this notion exists, typically [[Euclidean geometry]] or [[affine geometry]] over an [[ordered field]]. On the other hand, rays do not exist in [[projective geometry]] nor in a geometry over a non-ordered field, like the [[complex number]]s or any [[finite field]].
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| In [[topology]], a ray in a space ''X'' is a continuous embedding '''R'''<sup>+</sup> → ''X''. It is used to define the important concept of [[end (topology)|end]] of the space.
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| ==Euclidean geometry==
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| {{see also|Euclidean geometry}}
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| When geometry was first formalised by [[Euclid]] in the ''[[Euclid's Elements|Elements]]'', he defined a line to be "breadthless length" with a straight line being a line "which lies evenly with the points on itself".<ref>Faber, Appendix A, p. 291.</ref> These definitions serve little purpose since they use terms which are not, themselves, defined. In fact, Euclid did not use these definitions in this work and probably included them just to make it clear to the reader what was being discussed. In modern geometry, a line is simply taken as an undefined object with properties given by [[axiom]]s,<ref>Faber, Part III, p. 95.</ref> but is sometimes defined as a set of points obeying a linear relationship when some other fundamental concept is left undefined.
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| In an [[axiom]]atic formulation of Euclidean geometry, such as that of [[Hilbert's axioms|Hilbert]] (Euclid's original axioms contained various flaws which have been corrected by modern mathematicians),<ref>Faber, Part III, p. 108.</ref> a line is stated to have certain properties which relate it to other lines and [[point (geometry)|points]]. For example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect in at most one point.<ref>Faber, Appendix B, p. 300.</ref> In two [[dimension]]s, i.e., the Euclidean [[plane (mathematics)|plane]], two lines which do not intersect are called [[Parallel (geometry)|parallel]]. In higher dimensions, two lines that do not intersect may be parallel if they are contained in a [[Plane (geometry)|plane]], or [[Skew lines|skew]] if they are not.
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| Any collection of finitely many lines partitions the plane into [[convex polygon]]s (possibly unbounded); this partition is known as an [[arrangement of lines]].
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| ===Cartesian plane===
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| {{main|Linear equation}}
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| Lines in a [[Cartesian plane]] or, more generally, in [[affine coordinates]], can be described algebraically by ''linear'' equations. In two dimensions, the equation for non-vertical lines is often given in the ''[[slope-intercept form]]'':
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| :<math> y = mx + c \,</math>
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| where:
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| : ''m'' is the [[slope]] or [[slope|gradient]] of the line.
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| : ''c'' is the [[y-intercept]] of the line.
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| : ''x'' is the [[independent variable]] of the function ''y'' = ''f''(''x'').
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| The slope of the line through points ''A''(''x''<sub>a</sub>, ''y''<sub>a</sub>) and ''B''(''x''<sub>b</sub>, ''y''<sub>b</sub>), when ''x''<sub>a</sub> ≠ ''x''<sub>b</sub>, is given by ''m'' = (''y''<sub>b</sub> − ''y''<sub>a</sub>)/(''x''<sub>b</sub> − ''x''<sub>a</sub>)
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| and the equation of this line can be written y = ''m''(''x'' − ''x''<sub>a</sub>) + ''y''<sub>a</sub>.
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| In '''R'''<sup>2</sup>, every line ''L'' (including vertical lines) is described by a linear equation of the form
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| :<math>L=\{(x,y)\mid ax+by=c\} \,</math>
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| with fixed real [[coefficient]]s ''a'', ''b'' and ''c'' such that ''a'' and ''b'' are not both zero. Using this form, vertical lines correspond to the equations with ''b'' = 0.
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| There are many variant ways to write the equation of a line which can all be converted from one to another by algebraic manipulation. These forms (see [[Linear equation]] for other forms) are generally named by the type of information (data) about the line that is needed to write down the form. Some of the important data of a line is its slope, [[root of a function|x-intercept]], known points on the line and y-intercept.
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| The equation of the line passing through two different points <math>P_0 = ( x_0, y_0 )</math> and <math>P_1 = (x_1, y_1)</math> may be written as
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| :<math>(y - y_0)(x_1 - x_0) = (y_1 - y_0)(x - x_0)</math>.
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| If ''x<sub>0</sub>'' ≠ ''x<sub>1</sub>'', this equation may be rewritten as
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| :<math>y=(x-x_0)\,\frac{y_1-y_0}{x_1-x_0}+y_0</math>
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| or
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| :<math>y=x\,\frac{y_1-y_0}{x_1-x_0}+\frac{x_1y_0-x_0y_1}{x_1-x_0}\,.</math>
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| In three dimensions, lines can ''not'' be described by a single linear equation, so they are frequently described by [[parametric equations]]:
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| :<math> x = x_0 + at \,</math>
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| :<math> y = y_0 + bt \,</math>
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| :<math> z = z_0 + ct \,</math>
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| where:
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| : ''x'', ''y'', and ''z'' are all functions of the independent variable ''t'' which ranges over the real numbers.
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| : (''x''<sub>0</sub>, ''y''<sub>0</sub>, ''z''<sub>0</sub>) is any point on the line.
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| : ''a'', ''b'', and ''c'' are related to the slope of the line, such that the [[vector (geometric)|vector]] (''a'', ''b'', ''c'') is parallel to the line.
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| They may also be described as the simultaneous solutions of two [[linear equation]]s
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| :<math> a_1x+b_1y+c_1z-d_1=0 \,</math>
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| :<math> a_2x+b_2y+c_2z-d_2=0 \,</math> | |
| such that <math> (a_1,b_1,c_1)</math> and <math> (a_2,b_2,c_2)</math> are not proportional (the relations <math> a_1=ta_2,b_1=tb_2,c_1=tc_2 </math> imply {{math|1=''t'' = 0}}). This follows since in three dimensions a single linear equation typically describes a [[plane (geometry)|plane]] and a line is what is common to two distinct intersecting planes.
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| ====Normal form==== | |
| The ''[[normal (geometry)|normal]] segment'' for a given line is defined to be the line segment drawn from the [[origin (mathematics)|origin]] perpendicular to the line. This segment joins the origin with the closest point on the line to the origin. The ''normal form'' of the equation of a straight line on the plane is given by:
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| :: <math> y \sin \theta + x \cos \theta - p = 0,\,</math>
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| where ''θ'' is the angle of inclination of the normal segment, and ''p'' is the (signed) length of the normal segment. The normal form can be derived from the general form by dividing all of the coefficients by
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| ::<math>\frac{|c|}{-c}\sqrt{a^2 + b^2}.</math>
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| This form is also called the Hesse standard form,{{citation needed|date=October 2012}} after the German mathematician [[Otto Hesse|Ludwig Otto Hesse]].
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| Unlike the slope-intercept and intercept forms, this form can represent any line but also requires only two finite parameters, ''θ'' and ''p'', to be specified. Note that if the line is through the origin (''c'' = 0, ''p'' = 0), one drops the |''c''|/(−''c'') term to compute sin''θ'' and cos''θ''.
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| ===Polar coordinates=== | |
| In [[polar coordinates]] on the Euclidean plane a line is expressed as
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| ::<math>r=\frac{mr\cos\theta+b}{\sin\theta},</math>
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| :where ''m'' is the slope of the line and b is the ''y''-intercept. When ''θ'' = 0 the graph will be undefined. The equation can be rewritten to eliminate discontinuities:
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| ::<math>r\sin\theta=mr\cos\theta+b.\,</math>
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| ===Vector equation===
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| The vector equation of the line through points A and B is given by '''r''' = '''OA''' + λ'''AB''' (where λ is a [[scalar (mathematics)|scalar]]).
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| If '''a''' is vector '''OA''' and '''b''' is vector '''OB''', then the equation of the line can be written: '''r''' = '''a''' + λ('''b''' − '''a''').
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| A ray starting at point ''A'' is described by limiting λ. One ray is obtained if λ ≥ 0, and the opposite ray comes from λ ≤ 0.
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| ===Euclidean space===
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| In [[Euclidean space]], '''R'''<sup>''n''</sup> (and analogously in every other [[affine space]]), the line ''L'' passing through two different points ''a'' and ''b'' (considered as vectors) is the subset
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| :<math>L = \{(1-t)\,a+t\,b\mid t\in\mathbb{R}\}</math>
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| The direction of the line is from ''a'' (''t'' = 0) to ''b'' (''t'' = 1), or in other words, in the direction of the vector ''b'' − ''a''. Different choices of ''a'' and ''b'' can yield the same line.
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| ====Collinear points====
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| {{Main|Collinearity}}
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| Three points are said to be ''collinear'' if they lie on the same line. Three points ''[[general position|usually]]'' determine a [[plane (geometry)|plane]], but in the case of three collinear points this does ''not'' happen.
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| In [[affine coordinates]], in ''n''-dimensional space the points ''X''=(''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>n</sub>), ''Y''=(''y''<sub>1</sub>, ''y''<sub>2</sub>, ..., ''y''<sub>n</sub>), and ''Z''=(''z''<sub>1</sub>, ''z''<sub>2</sub>, ..., ''z''<sub>n</sub>) are collinear if the [[matrix (mathematics)|matrix]]
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| :<math>\begin{bmatrix}
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| 1 & x_1 & x_2 & \dots & x_n \\
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| 1 & y_1 & y_2 & \dots & y_n \\
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| 1 & z_1 & z_2 & \dots & z_n
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| \end{bmatrix}
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| </math> | |
| has a [[rank (linear algebra)|rank]] less than 3. In particular, for three points in the plane (''n'' = 2), above matrix is square and the points are collinear if and only if its [[determinant]] is zero.
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| When the distance ''d''(''a'',''b''){{which|date=June 2013}} between two points ''a'' and ''b'' is defined, the collinearity between three points may be expressed by:<ref>[[Alessandro Padoa]], Un nouveau système de définitions pour la géométrie euclidienne, [[International Congress of Mathematicians]], 1900</ref><ref>[[Bertrand Russell]], [[The Principles of Mathematics]], p.410</ref>
| |
| :The points ''a'', ''b'' and ''c'' are collinear if and only if ''d''(''x'',''a'') = ''d''(''c'',''a'') and ''d''(''x'',''b'') = ''d''(''c'',''b'') implies ''x''=''c''.
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| In [[Euclidean geometry]] this property is true, since if ''c'' is not on the line determined by ''a'' and ''b'' there will be another point (not equal to ''c'') which is just as far from ''a'' and ''b'' as the point ''c'' is (visualize the point on the other side of the line which is the mirror image of ''c'').
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| In the geometries where the concept of a line is a [[primitive notion]], as may be the case in some [[synthetic geometry|synthetic geometries]], other methods of determining collinearity are needed.
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| === Types of lines === | |
| In a sense,<ref>Technically, the [[collineation group]] acts [[transitive action|transitively]] on the set of lines.</ref> all lines in Euclidean geometry are equal, in that, without coordinates, one can not tell them apart from one another. However, lines may play special roles with respect to other objects in the geometry and be divided into types according to that relationship. For instance, with respect to a [[Conic section|conic]], lines can be:
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| * [[Tangent line]]s,
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| * [[Secant line]]s,
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| * Exterior lines, which do not meet the conic at any point of the Euclidean plane, or a more specialized
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| * [[Directrix of a conic section|directrix]].
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| For more general [[algebraic curve]]s, lines could also be:
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| * ''i''-secant lines, meeting the curve in ''i'' points counted without multiplicity, or
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| * [[asymptote]]s.
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| With respect to [[Euclidean triangle|triangle]]s we have:
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| * the [[Euler line]], and
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| * the [[Simson line]]s.
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| For a [[hexagon]] with vertices lying on a conic we have the [[Pascal line]] and, in the special case where the conic is a pair of lines, we have the [[Pappus's hexagon theorem|Pappus line]].
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| ==Projective geometry== | |
| {{main|Projective geometry}}
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| In many models of [[projective geometry]], the representation of a line rarely conforms to the notion of the "straight curve" as it is visualised in Euclidean geometry. In [[Elliptic geometry]] we see a typical example of this.<ref>Faber, Part III, p. 108.</ref> In the spherical representation of elliptic geometry, lines are represented by [[great circle]]s of a sphere with diametrically opposite points identified. In a different model of elliptic geometry, lines are represented by Euclidean [[plane (geometry)|planes]] passing through the origin. Even though these representations are visually distinct, they satisfy all the properties (such as, two points determining a unique line) that make them suitable representations for lines in this geometry.
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| ==Geodesics==
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| The "straightness" of a line, interpreted as the property that it minimizes distances between its points, can be generalized and leads to the concept of [[geodesic]]s in [[metric space]]s.
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| ==See also==
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| * [[Affine function]]
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| * [[Distance from a point to a line]]
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| * [[Five points determine a conic]], just as two points determine a line
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| * [[Glossary of Riemannian and metric geometry#R]] for its meaning in [[Riemannian geometry]].
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| * [[Incidence (geometry)]]
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| * [[Line drawing algorithm]]
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| * [[Line segment]]
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| * [[Robotics conventions|Minimal line representation]]
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| * [[Number line]]
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| * [[Plane (geometry)]], including [[Plane (geometry)#Distance from a point to a plane]], which generalizes the distance from a point to a line.
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| * [[Plücker coordinates]]
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| * [[Real line]]
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| * [[Ridge detection]] and [[Hough transform]] for algorithms for detecting lines in digital images
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| ==Notes==
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| <references/> | |
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| ==References==
| |
| {{Wikisource1911Enc|Line}}
| |
| *{{citation|last=Coxeter|first=H.S.M|title=Introduction to Geometry|edition=2nd|publisher=John Wiley & Sons|year=1969|place=New York|isbn=0-471-18283-4}}
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| *{{cite book |title=Foundations of Euclidean and Non-Euclidean Geometry |last=Faber |first=Richard L. |year=1983 |publisher=Marcel Dekker |location=New York|isbn=0-8247-1748-1 }}
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| *{{citation|last=Pedoe|first=Dan|title=Geometry: A Comprehensive Course|year=1988|publisher= Dover|place=Mineola, NY|isbn=0-486-65812-0}}
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| *{{citation|last=Wylie, Jr.|first=C. R.|title=Foundations of Geometry|publisher=McGraw-Hill|place=New York|year=1964|isbn=07-072191-2}}
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| ==External links==
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| * {{MathWorld |urlname=Line |title=Line}}
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| * [http://www.cut-the-knot.org/Curriculum/Calculus/StraightLine.shtml Equations of the Straight Line] at [[Cut-the-Knot]]
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| * [http://en.citizendium.org/wiki/Line_(geometry) Citizendium]
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| {{DEFAULTSORT:Line (Geometry)}}
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| [[Category:Elementary geometry]]
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| [[Category:Analytic geometry]]
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| [[Category:Mathematical concepts]]
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| {{Link FA|pl}}
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