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<title>Parallel (geometry)</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Parallel (geometry)</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about the geometry concept. For other uses, see <a href="Parallel_(disambiguation)" class="mw-redirect mw-disambig" title="Parallel (disambiguation)">Parallel (disambiguation)</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">"Parallel lines" and "Parallel line" redirect here. For other uses, see <a href="Parallel_lines_(disambiguation)" class="mw-disambig" title="Parallel lines (disambiguation)">Parallel lines (disambiguation)</a>.</div>

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<p>In <a href="Geometry" title="Geometry">geometry</a>, <b>parallel lines</b> are <a href="Coplanar" class="mw-redirect" title="Coplanar">coplanar</a> infinite straight <a href="Line_(geometry)" title="Line (geometry)">lines</a> that do not <a href="Intersecting_lines" class="mw-redirect" title="Intersecting lines">intersect</a> at any point. <b>Parallel planes</b> are infinite flat <a href="Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">planes</a> in the same <a href="Three-dimensional_space" title="Three-dimensional space">three-dimensional space</a> that never meet. In three-dimensional Euclidean space, a line and a plane that do not share a point are also said to be parallel. However, two noncoplanar lines are called <i><a href="Skew_lines" title="Skew lines">skew lines</a></i>. <a href="Line_segment" title="Line segment">Line segments</a> and <a href="Euclidean_vectors" class="mw-redirect" title="Euclidean vectors">Euclidean vectors</a> are parallel if they have the same <a href="Direction_(geometry)" title="Direction (geometry)">direction</a> or <a href="Opposite_direction_(geometry)" class="mw-redirect" title="Opposite direction (geometry)">opposite direction</a> (not necessarily the same length).<sup id="cite_ref-HMCS_1-0" class="reference"><a href="#cite_note-HMCS-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>Parallel lines are the subject of <a href="Euclid" title="Euclid">Euclid</a>'s <a href="Parallel_postulate" title="Parallel postulate">parallel postulate</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Parallelism is primarily a property of <a href="Affine_geometry" title="Affine geometry">affine geometries</a> and <a href="Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> is a special instance of this type of geometry.

In some other geometries, such as <a href="Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a>, lines can have analogous properties that are referred to as parallelism.

The concept can also be generalized non-straight <i><a href="Parallel_curve" title="Parallel curve">parallel curves</a></i> and non-flat <i>parallel surfaces</i>, which keep a fixed minimum distance and do not <a href="Tangent" title="Tangent">touch</a> each other or intersect.

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<div class="mw-heading mw-heading2"><h2 id="Symbol">Symbol</h2></div>

<div role="note" class="hatnote navigation-not-searchable">"∥" redirects here. For other uses, see <a href="Vertical_bar_(disambiguation)" class="mw-disambig" title="Vertical bar (disambiguation)">Vertical bar (disambiguation)</a>.</div>

<p>The parallel symbol is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \parallel }">

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<annotation encoding="application/x-tex">{\displaystyle \parallel }</annotation>

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</math></span><img src="./66ed42f2e3eab99383c61f27773eba258aefeaac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.162ex; height:2.843ex;" alt="{\displaystyle \parallel }" loading="lazy"></span>.<sup id="cite_ref-Kersey_1673_3-0" class="reference"><a href="#cite_note-Kersey_1673-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Cajori_1928_4-0" class="reference"><a href="#cite_note-Cajori_1928-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> For example, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AB\parallel CD}">

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<annotation encoding="application/x-tex">{\displaystyle AB\parallel CD}</annotation>

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</math></span><img src="./4604e0c72f8efbce903322cf4cb947e89e11257e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.65ex; height:2.843ex;" alt="{\displaystyle AB\parallel CD}" loading="lazy"></span> indicates that line <i>AB</i> is parallel to line <i>CD</i>.

</p><p>In the <a href="Unicode" title="Unicode">Unicode</a> 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".<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Euclidean_parallelism">Euclidean parallelism</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Two_lines_in_a_plane">Two lines in a plane</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Conditions_for_parallelism">Conditions for parallelism</h4></div>

<p>Given parallel straight lines <i>l</i> and <i>m</i> in <a href="Euclidean_space" title="Euclidean space">Euclidean space</a>, the following properties are equivalent:

</p>

<ol><li>Every point on line <i>m</i> is located at exactly the same (minimum) distance from line <i>l</i> (<i><a href="Equidistant" title="Equidistant">equidistant</a> lines</i>).</li>

<li>Line <i>m</i> is in the same plane as line <i>l</i> but does not intersect <i>l</i> (recall that lines extend to <a href="Infinity" title="Infinity">infinity</a> in either direction).</li>

<li>When lines <i>m</i> and <i>l</i> are both intersected by a third straight line (a <a href="Transversal_(geometry)" title="Transversal (geometry)">transversal</a>) in the same plane, the <a href="Corresponding_angles" class="mw-redirect" title="Corresponding angles">corresponding angles</a> of intersection with the transversal are <a href="Congruence_(geometry)" title="Congruence (geometry)">congruent</a>.</li></ol>

<p>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.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The other properties are then consequences of <a href="Euclid's_Fifth_Axiom" class="mw-redirect" title="Euclid's Fifth Axiom">Euclid's Parallel Postulate</a>.

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<div class="mw-heading mw-heading4"><h4 id="History">History</h4></div>

<p>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 <a href="Euclid's_Elements" title="Euclid's Elements">Euclid's Elements</a>.<sup id="cite_ref-Euclid_7-0" class="reference"><a href="#cite_note-Euclid-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Alternative definitions were discussed by other Greeks, often as part of an attempt to prove the <a href="Parallel_postulate" title="Parallel postulate">parallel postulate</a>. <a href="Proclus" title="Proclus">Proclus</a> attributes a definition of parallel lines as equidistant lines to <a href="Posidonius" title="Posidonius">Posidonius</a> and quotes <a href="Geminus" title="Geminus">Geminus</a> in a similar vein. <a href="Simplicius_of_Cilicia" title="Simplicius of Cilicia">Simplicius</a> also mentions Posidonius' definition as well as its modification by the philosopher Aganis.<sup id="cite_ref-Euclid_7-1" class="reference"><a href="#cite_note-Euclid-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p><p>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 <a href="Projective_geometry" title="Projective geometry">projective geometry</a> and <a href="Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean geometry</a>, 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.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> These reform texts were not without their critics and one of them, Charles Dodgson (a.k.a. <a href="Lewis_Carroll" title="Lewis Carroll">Lewis Carroll</a>), wrote a play, <i>Euclid and His Modern Rivals</i>, in which these texts are lambasted.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>

</p><p>One of the early reform textbooks was James Maurice Wilson's <i>Elementary Geometry</i> of 1868.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Wilson based his definition of parallel lines on the <a href="Primitive_notion" title="Primitive notion">primitive notion</a> of <i>direction</i>. According to <a href="Wilhelm_Killing" title="Wilhelm Killing">Wilhelm Killing</a><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> the idea may be traced back to <a href="Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Leibniz</a>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> 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 <i>angle</i> between them." <a href="#CITEREFWilson1868">Wilson (1868</a>, p. 2) In definition 15 he introduces parallel lines in this way; "Straight lines which have the <i>same direction</i>, but are not parts of the same straight line, are called <i>parallel lines</i>." <a href="#CITEREFWilson1868">Wilson (1868</a>, p. 12) <a href="Augustus_De_Morgan" title="Augustus De Morgan">Augustus De Morgan</a> 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.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>

</p><p>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 <i>Euclidean Geometry</i> 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.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> The corresponding angles formed by a transversal property, used by W. D. Cooley in his 1860 text, <i>The Elements of Geometry, simplified and explained</i> 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.

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<div class="mw-heading mw-heading4"><h4 id="Construction">Construction</h4></div>

<p>The three properties above lead to three different methods of construction<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> of parallel lines.

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<div class="gallerytext">Property 1: Line <i>m</i> has everywhere the same distance to line <i>l</i>.</div>

</li>

<div class="gallerytext">Property 2: Take a random line through <i>a</i> that intersects <i>l</i> in <i>x</i>. Move point <i>x</i> to infinity.</div>

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<div class="gallerytext">Property 3: Both <i>l</i> and <i>m</i> share a transversal line through <i>a</i> that intersect them at 90°.</div>

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<div class="mw-heading mw-heading4"><h4 id="Distance_between_two_parallel_lines">Distance between two parallel lines</h4></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Distance_between_two_parallel_lines" title="Distance between two parallel lines">Distance between two parallel lines</a></div>

<p>Because parallel lines in a Euclidean plane are <a href="Equidistant" title="Equidistant">equidistant</a> there is a unique distance between the two parallel lines. Given the equations of two non-vertical, non-horizontal parallel lines,

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</math></span><img src="./35b689641493203ec74490249169faa2c00f5078.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.903ex; height:2.509ex;" alt="{\displaystyle y=mx+b_{1}\,}" loading="lazy"></span></dd>

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<p>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 <i>m</i>, a common perpendicular would have slope −1/<i>m</i> and we can take the line with equation <i>y</i> = −<i>x</i>/<i>m</i> as a common perpendicular. Solve the linear systems

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<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}y=mx+b_{1}\\y=-x/m\end{cases}}}</annotation>

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</math></span><img src="./d7f42d0562ac1d8b33fa4c2a97424e238cd78599.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.011ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}y=mx+b_{1}\\y=-x/m\end{cases}}}" loading="lazy"></span></dd></dl>

<p>and

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<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}y=mx+b_{2}\\y=-x/m\end{cases}}}</annotation>

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

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<annotation encoding="application/x-tex">{\displaystyle \left(x_{1},y_{1}\right)\ =\left({\frac {-b_{1}m}{m^{2}+1}},{\frac {b_{1}}{m^{2}+1}}\right)\,}</annotation>

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</math></span><img src="./9322bb1bc703f209c09823de2570842b5d8a5103.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.196ex; height:6.176ex;" alt="{\displaystyle \left(x_{1},y_{1}\right)\ =\left({\frac {-b_{1}m}{m^{2}+1}},{\frac {b_{1}}{m^{2}+1}}\right)\,}" loading="lazy"></span></dd></dl>

<p>and

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<annotation encoding="application/x-tex">{\displaystyle \left(x_{2},y_{2}\right)\ =\left({\frac {-b_{2}m}{m^{2}+1}},{\frac {b_{2}}{m^{2}+1}}\right).}</annotation>

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</math></span><img src="./6b321e156620ac02fb39a7739384057852ed2ab4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.843ex; height:6.176ex;" alt="{\displaystyle \left(x_{2},y_{2}\right)\ =\left({\frac {-b_{2}m}{m^{2}+1}},{\frac {b_{2}}{m^{2}+1}}\right).}" loading="lazy"></span></dd></dl>

<p>These formulas still give the correct point coordinates even if the parallel lines are horizontal (i.e., <i>m</i> = 0). The distance between the points is

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<annotation encoding="application/x-tex">{\displaystyle 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}}}\,,}</annotation>

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</math></span><img src="./1fb3a1586f9c227af9e0584fe0ff4abc3cc2d571.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:68.083ex; height:7.676ex;" alt="{\displaystyle 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}}}\,,}" loading="lazy"></span></dd></dl>

<p>which reduces to

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<annotation encoding="application/x-tex">{\displaystyle d={\frac {|b_{2}-b_{1}|}{\sqrt {m^{2}+1}}}\,.}</annotation>

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</math></span><img src="./976dfcbb79c4d09396b2b65277135bf1324085b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:15.606ex; height:7.009ex;" alt="{\displaystyle d={\frac {|b_{2}-b_{1}|}{\sqrt {m^{2}+1}}}\,.}" loading="lazy"></span></dd></dl>

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

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<annotation encoding="application/x-tex">{\displaystyle ax+by+c_{1}=0\,}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle ax+by+c_{2}=0,\,}</annotation>

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</math></span><img src="./e5e16a81812ecbb18dcd3f209a9e1aa656ed6def.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.749ex; height:2.509ex;" alt="{\displaystyle ax+by+c_{2}=0,\,}" loading="lazy"></span></dd></dl>

<p>their distance can be expressed as

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<annotation encoding="application/x-tex">{\displaystyle d={\frac {|c_{2}-c_{1}|}{\sqrt {a^{2}+b^{2}}}}.}</annotation>

</semantics>

</math></span><img src="./accd1aa172d44c7b35228ed52f4c99db274104ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:15.297ex; height:7.009ex;" alt="{\displaystyle d={\frac {|c_{2}-c_{1}|}{\sqrt {a^{2}+b^{2}}}}.}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Two_lines_in_three-dimensional_space">Two lines in three-dimensional space</h3></div>

<p>Two lines in the same <a href="Three-dimensional_space" title="Three-dimensional space">three-dimensional space</a> that do not intersect need not be parallel. Only if they are in a common plane are they called parallel; otherwise they are called <a href="Skew_lines" title="Skew lines">skew lines</a>.

</p><p>Two distinct lines <i>l</i> and <i>m</i> in three-dimensional space are parallel <a href="If_and_only_if" title="If and only if">if and only if</a> the distance from a point <i>P</i> on line <i>m</i> to the nearest point on line <i>l</i> is independent of the location of <i>P</i> on line <i>m</i>. This never holds for skew lines.

</p>

<div class="mw-heading mw-heading3"><h3 id="A_line_and_a_plane">A line and a plane</h3></div>

<p>A line <i>m</i> and a plane <i>q</i> in three-dimensional space, the line not lying in that plane, are parallel if and only if they do not intersect.

</p><p>Equivalently, they are parallel if and only if the distance from a point <i>P</i> on line <i>m</i> to the nearest point in plane <i>q</i> is independent of the location of <i>P</i> on line <i>m</i>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Two_planes">Two planes</h3></div>

<p>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.

</p><p>Two distinct planes <i>q</i> and <i>r</i> are parallel if and only if the distance from a point <i>P</i> in plane <i>q</i> to the nearest point in plane <i>r</i> is independent of the location of <i>P</i> in plane <i>q</i>. This will never hold if the two planes are not in the same three-dimensional space.

</p>

<div class="mw-heading mw-heading2"><h2 id="In_non-Euclidean_geometry">In non-Euclidean geometry</h2></div>

<p>In <a href="Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean geometry</a>, the concept of a straight line is replaced by the more general concept of a <a href="Geodesic" title="Geodesic">geodesic</a>, a curve which is <a href="Local_property" title="Local property">locally</a> straight with respect to the <a href="Metric_tensor" title="Metric tensor">metric</a> (definition of distance) on a <a href="Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a>, a surface (or higher-dimensional space) which may itself be curved. In <a href="General_relativity" title="General relativity">general relativity</a>, particles not under the influence of external forces follow geodesics in <a href="Spacetime" title="Spacetime">spacetime</a>, a four-dimensional manifold with 3 spatial dimensions and 1 time dimension.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup>

</p><p>In non-Euclidean geometry (<a href="Elliptic_geometry" title="Elliptic geometry">elliptic</a> or <a href="Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a>) the three Euclidean properties mentioned above are not equivalent and only the second one (Line m is in the same plane as line l but does not intersect l) is useful in non-Euclidean geometries, since it involves no measurements. In general geometry the three properties above give three different types of curves, <b>equidistant curves</b>, <b>parallel geodesics</b> and <b>geodesics sharing a common perpendicular</b>, respectively.

</p>

<div class="mw-heading mw-heading3"><h3 id="Hyperbolic_geometry">Hyperbolic geometry</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Hyperbolic_geometry" title="Hyperbolic geometry">Hyperbolic geometry</a></div>

<p>While in Euclidean geometry two geodesics can either intersect or be parallel, in hyperbolic geometry, there are three possibilities. Two geodesics belonging to the same plane can either be:

</p>

<ol><li><b>intersecting</b>, if they intersect in a common point in the plane,</li>

<li><b>parallel</b>, if they do not intersect in the plane, but converge to a common limit point at infinity (<a href="Ideal_point" title="Ideal point">ideal point</a>), or</li>

<li><b>ultra parallel</b>, if they do not have a common limit point at infinity.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup></li></ol>

<p>In the literature <i>ultra parallel</i> geodesics are often called <i>non-intersecting</i>. <i>Geodesics intersecting at infinity</i> are called <i><a href="Limiting_parallel" title="Limiting parallel">limiting parallel</a></i>.

</p><p>As in the illustration through a point <i>a</i> not on line <i>l</i> there are two <a href="Limiting_parallel" title="Limiting parallel">limiting parallel</a> lines, one for each direction <a href="Ideal_point" title="Ideal point">ideal point</a> of line l. They separate the lines intersecting line l and those that are ultra parallel to line <i>l</i>.

</p><p>Ultra parallel lines have single common perpendicular (<a href="Ultraparallel_theorem" title="Ultraparallel theorem">ultraparallel theorem</a>), and diverge on both sides of this common perpendicular.

</p><p><br>

</p>

<div class="mw-heading mw-heading3"><h3 id="Spherical_or_elliptic_geometry">Spherical or elliptic geometry</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Spherical_geometry" title="Spherical geometry">Spherical geometry</a> and <a href="Elliptic_geometry" title="Elliptic geometry">Elliptic geometry</a></div>

<p>In <a href="Spherical_geometry" title="Spherical geometry">spherical geometry</a>, all geodesics are <a href="Great_circles" class="mw-redirect" title="Great circles">great circles</a>. Great circles divide the sphere in two equal <a href="Sphere" title="Sphere">hemispheres</a> and all great circles intersect each other. Thus, there are no parallel geodesics to a given geodesic, as all geodesics intersect. Equidistant curves on the sphere are called <b>parallels of latitude</b> analogous to the <a href="Latitude" title="Latitude">latitude</a> lines on a globe. Parallels of latitude can be generated by the intersection of the sphere with a plane parallel to a plane through the center of the sphere.

</p>

<div class="mw-heading mw-heading2"><h2 id="Reflexive_variant">Reflexive variant</h2></div>

<p>If <i>l, m, n</i> are three distinct lines, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l\parallel m\ \land \ m\parallel n\ \implies \ l\parallel n.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle l\parallel m\ \land \ m\parallel n\ \implies \ l\parallel n.}</annotation>

</semantics>

</math></span><img src="./69c5be6db1f6bdaa60a8110750d6bf5d3f693942.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.553ex; height:2.843ex;" alt="{\displaystyle l\parallel m\ \land \ m\parallel n\ \implies \ l\parallel n.}" loading="lazy"></span>

</p><p>In this case, parallelism is a <a href="Transitive_relation" title="Transitive relation">transitive relation</a>. However, in case <i>l</i> = <i>n</i>, the superimposed lines are <i>not</i> considered parallel in Euclidean geometry. The <a href="Binary_relation" title="Binary relation">binary relation</a> between parallel lines is evidently a <a href="Symmetric_relation" title="Symmetric relation">symmetric relation</a>. According to Euclid's tenets, parallelism is <i>not</i> a <a href="Reflexive_relation" title="Reflexive relation">reflexive relation</a> and thus <i>fails</i> to be an <a href="Equivalence_relation" title="Equivalence relation">equivalence relation</a>. Nevertheless, in <a href="Affine_geometry" title="Affine geometry">affine geometry</a> a <a href="Pencil_of_parallel_lines" class="mw-redirect" title="Pencil of parallel lines">pencil of parallel lines</a> is taken as an <a href="Equivalence_class" title="Equivalence class">equivalence class</a> in the set of lines where parallelism is an equivalence relation.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup>

</p><p>To this end, <a href="Emil_Artin" title="Emil Artin">Emil Artin</a> (1957) adopted a definition of parallelism where two lines are parallel if they have all or none of their points in common.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup>

Then a line <i>is</i> parallel to itself so that the reflexive and transitive properties belong to this type of parallelism, creating an equivalence relation on the set of lines. In the study of <a href="Incidence_geometry" title="Incidence geometry">incidence geometry</a>, this variant of parallelism is used in the <a href="Affine_plane_(incidence_geometry)" title="Affine plane (incidence geometry)">affine plane</a>.

</p>

<ul><li><a href="Clifford_parallel" title="Clifford parallel">Clifford parallel</a></li>

<li><a href="Collinearity" title="Collinearity">Collinearity</a></li>

<li><a href="Concurrent_lines" title="Concurrent lines">Concurrent lines</a></li>

<li><a href="Limiting_parallel" title="Limiting parallel">Limiting parallel</a></li>

<li><a href="Parallel_curve" title="Parallel curve">Parallel curve</a></li>

<li><a href="Ultraparallel_theorem" title="Ultraparallel theorem">Ultraparallel theorem</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><cite id="CITEREFHarrisStöcker1998" class="citation book cs1">Harris, John W.; Stöcker, Horst (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DnKLkOb_YfIC&pg=PA332"><i>Handbook of mathematics and computational science</i></a>. Birkhäuser. Chapter 6, p. 332. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-94746-9</bdi>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Although this postulate only refers to when lines meet, it is needed to prove the uniqueness of parallel lines in the sense of <a href="Playfair's_axiom" title="Playfair's axiom">Playfair's axiom</a>.</span>

</li>

<li id="cite_note-Kersey_1673-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kersey_1673_3-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFKersey_(the_elder)1673" class="citation book cs1 cs1-prop-long-vol"><a href="John_Kersey_the_elder" title="John Kersey the elder">Kersey (the elder), John</a> (1673). <i>Algebra</i>. Vol. Book IV. London. p. 177.</cite></span>

</li>

<li id="cite_note-Cajori_1928-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Cajori_1928_4-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFCajori1993" class="citation book cs1"><a href="Florian_Cajori" title="Florian Cajori">Cajori, Florian</a> (1993) [September 1928]. <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00cajo_0/page/193">"§ 184, § 359, § 368"</a>. <i>A History of Mathematical Notations - Notations in Elementary Mathematics</i>. Vol. 1 (two volumes in one unaltered reprint ed.). Chicago, US: <a href="Open_court_publishing_company" class="mw-redirect" title="Open court publishing company">Open court publishing company</a>. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00cajo_0/page/193">193, <span class="nowrap">402–</span>403, 411–412</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-67766-4</bdi>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/93-29211">93-29211</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-07-22</span></span>. <q>§359. […] ∥ for parallel occurs in <a href="William_Oughtred" title="William Oughtred">Oughtred</a>'s <i>Opuscula mathematica hactenus inedita</i> (1677) [p. 197], a posthumous work (§ 184) […] §368. Signs for parallel lines. […] when <a href="Robert_Recorde" title="Robert Recorde">Recorde</a>'s sign of equality won its way upon the Continent, vertical lines came to be used for parallelism. We find ∥ for "parallel" in <a href="John_Kersey_the_elder" title="John Kersey the elder">Kersey</a>,[14] <a href="John_Caswell" title="John Caswell">Caswell</a>, <a href="William_Jones_(mathematician)" title="William Jones (mathematician)">Jones</a>,[15] Wilson,[16] <a href="William_Emerson_(mathematician)" title="William Emerson (mathematician)">Emerson</a>,[17] Kambly,[18] and the writers of the last fifty years who have been already quoted in connection with other pictographs. Before about 1875 it does not occur as often […] Hall and Stevens[1] use "par[1] or ∥" for parallel […] [14] <a href="John_Kersey_the_elder" title="John Kersey the elder">John Kersey</a>, <i>Algebra</i> (London, 1673), Book IV, p. 177. [15] <a href="William_Jones_(mathematician)" title="William Jones (mathematician)">W. Jones</a>, <i>Synopsis palmarioum matheseos</i> (London, 1706). [16] John Wilson, <i>Trigonometry</i> (Edinburgh, 1714), characters explained. [17] <a href="William_Emerson_(mathematician)" title="William Emerson (mathematician)">W. Emerson</a>, <i>Elements of Geometry</i> (London, 1763), p. 4. [18] L. Kambly, <i>Die Elementar-Mathematik</i>, Part 2: <i>Planimetrie</i>, 43. edition (Breslau, 1876), p. 8. […] [1] H. S. Hall and F. H. Stevens, <i>Euclid's Elements</i>, Parts I and II (London, 1889), p. 10. […]</q></cite> <a rel="nofollow" class="external autonumber" href="https://monoskop.org/images/2/21/Cajori_Florian_A_History_of_Mathematical_Notations_2_Vols.pdf">[1]</a></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.unicode.org/charts/PDF/U2200.pdf">"Mathematical Operators – Unicode Consortium"</a> <span class="cs1-format">(PDF)</span><span class="reference-accessdate">. Retrieved <span class="nowrap">2013-04-21</span></span>.</cite></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFWylie1964">Wylie 1964</a>, pp. 92—94</span>

</li>

<li id="cite_note-Euclid-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-Euclid_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Euclid_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFHeath1956">Heath 1956</a>, pp. 190–194</span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="#CITEREFRichards1988">Richards 1988</a>, Chap. 4: Euclid and the English Schoolchild. pp. 161–200</span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite id="CITEREFCarroll2009" class="citation cs2">Carroll, Lewis (2009) [1879], <i>Euclid and His Modern Rivals</i>, Barnes & Noble, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4351-2348-9</bdi></cite></span>

</li>

<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a href="#CITEREFWilson1868">Wilson 1868</a></span>

</li>

<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><i>Einführung in die Grundlagen der Geometrie, I</i>, p. 5</span>

</li>

<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="#CITEREFHeath1956">Heath 1956</a>, p. 194</span>

</li>

<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a href="#CITEREFRichards1988">Richards 1988</a>, pp. 180–184</span>

</li>

<li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><a href="#CITEREFHeath1956">Heath 1956</a>, p. 194</span>

</li>

<li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Only the third is a straightedge and compass construction, the first two are infinitary processes (they require an "infinite number of steps".)</span>

</li>

<li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><cite id="CITEREFChurch2022" class="citation web cs1">Church, Benjamin (2022-12-03). <a rel="nofollow" class="external text" href="https://web.stanford.edu/~bvchurch/assets/files/talks/GR.pdf">"A Not So Gentle Introduction to General Relativity"</a> <span class="cs1-format">(PDF)</span>.</cite></span>

</li>

<li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.libretexts.org/Bookshelves/Geometry/An_IBL_Introduction_to_Geometries_(Mark_Fitch)/05:_Hyperbolic_Geometry/5.03:_New_Page">"5.3: Theorems of Hyperbolic Geometry"</a>. <i>Mathematics LibreTexts</i>. 2021-10-30<span class="reference-accessdate">. Retrieved <span class="nowrap">2024-08-22</span></span>.</cite></span>

</li>

<li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><a href="H._S._M._Coxeter" class="mw-redirect" title="H. S. M. Coxeter">H. S. M. Coxeter</a> (1961) <i>Introduction to Geometry</i>, p 192, <a href="John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a></span>

</li>

<li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><a href="Wanda_Szmielew" title="Wanda Szmielew">Wanda Szmielew</a> (1983) <i>From Affine to Euclidean Geometry</i>, p 17, <a href="D._Reidel" title="D. Reidel">D. Reidel</a> <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>90-277-1243-3</bdi></span>

</li>

<li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">Andy Liu (2011) "Is parallelism an equivalence relation?", <a href="The_College_Mathematics_Journal" title="The College Mathematics Journal">The College Mathematics Journal</a> 42(5):372</span>

</li>

<li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><a href="Emil_Artin" title="Emil Artin">Emil Artin</a> (1957) <a rel="nofollow" class="external text" href="https://archive.org/details/geometricalgebra033556mbp/page/n63/mode/2up?view=theater"><i>Geometric Algebra</i>, page 52</a> via <a href="Internet_Archive" title="Internet Archive">Internet Archive</a></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFHeath1956" class="citation cs2"><a href="T._L._Heath" class="mw-redirect" title="T. L. Heath">Heath, Thomas L.</a> (1956), <i>The Thirteen Books of Euclid's Elements</i> (2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925] ed.), New York: Dover Publications</cite></li></ul>

<dl><dd>(3 vols.): <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-60088-2</bdi> (vol. 1), <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-60089-0</bdi> (vol. 2), <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-60090-4</bdi> (vol. 3). Heath's authoritative translation plus extensive historical research and detailed commentary throughout the text.</dd></dl>

<ul><li><cite id="CITEREFRichards1988" class="citation cs2"><a href="Joan_L._Richards" title="Joan L. Richards">Richards, Joan L.</a> (1988), <i>Mathematical Visions: The Pursuit of Geometry in Victorian England</i>, Boston: Academic Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-587445-6</bdi></cite></li>

<li><cite id="CITEREFWilson1868" class="citation cs2">Wilson, James Maurice (1868), <i>Elementary Geometry</i> (1st ed.), London: Macmillan and Co.</cite></li>

<li><cite id="CITEREFWylie1964" class="citation cs2">Wylie, C. R. Jr. (1964), <i>Foundations of Geometry</i>, McGraw–Hill</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFPapadopoulosThéret2014" class="citation cs2">Papadopoulos, Athanase; Théret, Guillaume (2014), <i>La théorie des parallèles de Johann Heinrich Lambert : Présentation, traduction et commentaires</i>, Paris: Collection Sciences dans l'histoire, Librairie Albert Blanchard, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-2-85367-266-5</bdi></cite></li></ul></div><div><div class="zim-footer">

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