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</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="Geometry" title="Geometry">Geometry</a></th></tr><tr><td class="sidebar-image"><div class="sidebar-caption"><a href="Projective_geometry" title="Projective geometry">Projecting</a> a <a href="Sphere" title="Sphere">sphere</a> to a <a href="Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a></div></td></tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="List_of_geometry_topics" class="mw-redirect" title="List of geometry topics">Branches</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">
<ul><li><a href="Euclidean_geometry" title="Euclidean geometry">Euclidean</a></li>
<li><a href="Non-Euclidean_geometry" title="Non-Euclidean geometry">Non-Euclidean</a>
<ul><li><a href="Elliptic_geometry" title="Elliptic geometry">Elliptic</a>
<ul><li><a href="Spherical_geometry" title="Spherical geometry">Spherical</a></li></ul></li>
<li><a href="Hyperbolic_geometry" title="Hyperbolic geometry">Hyperbolic</a></li></ul></li>
<li><a href="Non-Archimedean_geometry" title="Non-Archimedean geometry">Non-Archimedean geometry</a></li>
<li><a href="Projective_geometry" title="Projective geometry">Projective</a></li>
<li><a href="Affine_geometry" title="Affine geometry">Affine</a></li>
<li><a href="Synthetic_geometry" title="Synthetic geometry">Synthetic</a></li>
<li><a href="Analytic_geometry" title="Analytic geometry">Analytic</a></li>
<li><a href="Algebraic_geometry" title="Algebraic geometry">Algebraic</a>
<ul><li><a href="Arithmetic_geometry" title="Arithmetic geometry">Arithmetic</a></li>
<li><a href="Diophantine_geometry" title="Diophantine geometry">Diophantine</a></li></ul></li>
<li><a href="Differential_geometry" title="Differential geometry">Differential</a>
<ul><li><a href="Riemannian_geometry" title="Riemannian geometry">Riemannian</a></li>
<li><a href="Symplectic_geometry" title="Symplectic geometry">Symplectic</a></li>
<li><a href="Discrete_differential_geometry" title="Discrete differential geometry">Discrete differential</a></li></ul></li>
<li><a href="Complex_geometry" title="Complex geometry">Complex</a></li>
<li><a href="Finite_geometry" title="Finite geometry">Finite</a></li>
<li><a href="Discrete_geometry" title="Discrete geometry">Discrete/Combinatorial</a>
<ul><li><a href="Digital_geometry" title="Digital geometry">Digital</a></li></ul></li>
<li><a href="Convex_geometry" title="Convex geometry">Convex</a></li>
<li><a href="Computational_geometry" title="Computational geometry">Computational</a></li>
<li><a href="Fractal" title="Fractal">Fractal</a></li>
<li><a href="Incidence_geometry" title="Incidence geometry">Incidence </a></li>
<li><a href="Noncommutative_geometry" title="Noncommutative geometry">Noncommutative geometry</a>
<ul><li><a href="Noncommutative_algebraic_geometry" title="Noncommutative algebraic geometry">Noncommutative algebraic geometry</a></li></ul></li></ul></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist"><ul><li>Concepts</li><li>Features</li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content hlist"><a href="Dimension_(geometry)" class="mw-redirect" title="Dimension (geometry)">Dimension</a>
<ul><li><a href="Straightedge_and_compass_construction" title="Straightedge and compass construction">Straightedge and compass constructions</a></li></ul>
<ul><li><a href="Angle" title="Angle">Angle</a></li>
<li><a href="Curve" title="Curve">Curve</a></li>
<li><a href="Diagonal" title="Diagonal">Diagonal</a></li>
<li><a href="Orthogonality" title="Orthogonality">Orthogonality</a> (<a href="Perpendicular" title="Perpendicular">Perpendicular</a>)</li>
<li><a href="Parallel_(geometry)" title="Parallel (geometry)">Parallel</a></li>
<li><a href="Vertex_(geometry)" title="Vertex (geometry)">Vertex</a></li></ul>
<ul><li><a href="Congruence_(geometry)" title="Congruence (geometry)">Congruence</a></li>
<li><a href="Similarity_(geometry)" title="Similarity (geometry)">Similarity</a></li>
<li><a href="Symmetry" title="Symmetry">Symmetry</a></li></ul></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Zero-dimensional_space" title="Zero-dimensional space">Zero-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">
<ul><li><a href="Point_(geometry)" title="Point (geometry)">Point</a></li></ul></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="One-dimensional_space" title="One-dimensional space">One-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">
<ul><li><a href="Line_(geometry)" title="Line (geometry)">Line</a>
<ul>
<li><a href="Line_(geometry)#Ray" title="Line (geometry)">ray</a></li></ul></li>
<li><a href="Length" title="Length">Length</a></li></ul></div></div></td>
</tr><tr><td class="sidebar-content-with-subgroup">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Two-dimensional_space" title="Two-dimensional space">Two-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="padding-bottom:0;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist">
<ul><li><a href="Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">Plane</a></li>
<li><a href="Area" title="Area">Area</a></li>
<li><a href="Polygon" title="Polygon">Polygon</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
<a href="Triangle" title="Triangle">Triangle</a></th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Altitude_(triangle)" title="Altitude (triangle)">Altitude</a></li>
<li><a href="Hypotenuse" title="Hypotenuse">Hypotenuse</a></li>
<li><a href="Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
<a href="Parallelogram" title="Parallelogram">Parallelogram</a></th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Square" title="Square">Square</a></li>
<li><a href="Rectangle" title="Rectangle">Rectangle</a></li>
<li><a href="Rhombus" title="Rhombus">Rhombus</a></li>
<li><a href="Rhomboid" title="Rhomboid">Rhomboid</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
<a href="Quadrilateral" title="Quadrilateral">Quadrilateral</a></th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Trapezoid" title="Trapezoid">Trapezoid</a></li>
<li><a href="Kite_(geometry)" title="Kite (geometry)">Kite</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
<a href="Circle" title="Circle">Circle</a></th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Diameter" title="Diameter">Diameter</a></li>
<li><a href="Circumference" title="Circumference">Circumference</a></li>
<li><a href="Area_of_a_circle" title="Area of a circle">Area</a></li></ul></td>
</tr></tbody></table></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Three-dimensional_space" title="Three-dimensional space">Three-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">
<ul><li><a href="Volume" title="Volume">Volume</a></li></ul>
<ul><li><a href="Cube" title="Cube">Cube</a>
<ul><li><a href="Cuboid" title="Cuboid">cuboid</a></li></ul></li>
<li><a href="Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">Cylinder</a></li>
<li><a href="Dodecahedron" title="Dodecahedron">Dodecahedron</a></li>
<li><a href="Icosahedron" title="Icosahedron">Icosahedron</a></li>
<li><a href="Octahedron" title="Octahedron">Octahedron</a></li>
<li><a href="Pyramid_(geometry)" title="Pyramid (geometry)">Pyramid</a></li>
<li><a href="Platonic_Solid" class="mw-redirect" title="Platonic Solid">Platonic Solid</a></li>
<li><a href="Sphere" title="Sphere">Sphere</a></li>
<li><a href="Tetrahedron" title="Tetrahedron">Tetrahedron</a></li></ul></div></div></td>
</tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Four-dimensional_space" title="Four-dimensional space">Four</a>-/other-dimensional</div></div><div class="sidebar-list-content mw-collapsible-content hlist">
<ul><li><a href="Tesseract" title="Tesseract">Tesseract</a></li>
<li><a href="Hypersphere" class="mw-redirect" title="Hypersphere">Hypersphere</a></li></ul></div></div></td>
</tr><tr><th class="sidebar-heading" style="padding-bottom:0.2em;">
<a href="List_of_geometers" title="List of geometers">Geometers</a></th></tr><tr><td class="sidebar-content">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">by name</div></div><div class="sidebar-list-content mw-collapsible-content hlist">
<ul><li><a href="Yasuaki_Aida" class="mw-redirect" title="Yasuaki Aida">Aida</a></li>
<li><a href="Aryabhata" title="Aryabhata">Aryabhata</a></li>
<li><a href="Ahmes" title="Ahmes">Ahmes</a></li>
<li><a href="Alhazen" class="mw-redirect" title="Alhazen">Alhazen</a></li>
<li><a href="Apollonius_of_Perga" title="Apollonius of Perga">Apollonius</a></li>
<li><a href="Archimedes" title="Archimedes">Archimedes</a></li>
<li><a href="Michael_Atiyah" title="Michael Atiyah">Atiyah</a></li>
<li><a href="Baudhayana" title="Baudhayana">Baudhayana</a></li>
<li><a href="J%C3%A1nos_Bolyai" title="János Bolyai">Bolyai</a></li>
<li><a href="Brahmagupta" title="Brahmagupta">Brahmagupta</a></li>
<li><a href="%C3%89lie_Cartan" title="Élie Cartan">Cartan</a></li>
<li><a href="Shiing-Shen_Chern" title="Shiing-Shen Chern">Chern</a></li>
<li><a href="Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter</a></li>
<li><a href="Ren%C3%A9_Descartes" title="René Descartes">Descartes</a></li>
<li><a href="Euclid" title="Euclid">Euclid</a></li>
<li><a href="Leonhard_Euler" title="Leonhard Euler">Euler</a></li>
<li><a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li>
<li><a href="Mikhail_Leonidovich_Gromov" class="mw-redirect" title="Mikhail Leonidovich Gromov">Gromov</a></li>
<li><a href="David_Hilbert" title="David Hilbert">Hilbert</a></li>
<li><a href="Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li>
<li><a href="Jye%E1%B9%A3%E1%B9%ADhadeva" title="Jyeṣṭhadeva">Jyeṣṭhadeva</a></li>
<li><a href="K%C4%81ty%C4%81yana" title="Kātyāyana">Kātyāyana</a></li>
<li><a href="Omar_Khayy%C3%A1m" class="mw-redirect" title="Omar Khayyám">Khayyám</a></li>
<li><a href="Felix_Klein" title="Felix Klein">Klein</a></li>
<li><a href="Nikolai_Lobachevsky" title="Nikolai Lobachevsky">Lobachevsky</a></li>
<li><a href="Manava" title="Manava">Manava</a></li>
<li><a href="Hermann_Minkowski" title="Hermann Minkowski">Minkowski</a></li>
<li><a href="Minggatu" title="Minggatu">Minggatu</a></li>
<li><a href="Blaise_Pascal" title="Blaise Pascal">Pascal</a></li>
<li><a href="Pythagoras" title="Pythagoras">Pythagoras</a></li>
<li><a href="Parameshvara" class="mw-redirect" title="Parameshvara">Parameshvara</a></li>
<li><a href="Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li>
<li><a href="Bernhard_Riemann" title="Bernhard Riemann">Riemann</a></li>
<li><a href="Sakabe_K%C5%8Dhan" title="Sakabe Kōhan">Sakabe</a></li>
<li><a href="Sijzi" class="mw-redirect" title="Sijzi">Sijzi</a></li>
<li><a href="Nasir_al-Din_al-Tusi" title="Nasir al-Din al-Tusi">al-Tusi</a></li>
<li><a href="Oswald_Veblen" title="Oswald Veblen">Veblen</a></li>
<li><a href="Virasena" title="Virasena">Virasena</a></li>
<li><a href="Yang_Hui" title="Yang Hui">Yang Hui</a></li>
<li><a href="Ibn_al-Yasamin" title="Ibn al-Yasamin">al-Yasamin</a></li>
<li><a href="Zhang_Heng" title="Zhang Heng">Zhang</a></li>
<li><a href="List_of_geometers" title="List of geometers">List of geometers</a></li></ul></div></div></td>
</tr><tr><td class="sidebar-content-with-subgroup">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">by period</div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="padding-bottom:0;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading">
<a href="Before_Common_Era" class="mw-redirect" title="Before Common Era">BCE</a></th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Ahmes" title="Ahmes">Ahmes</a></li>
<li><a href="Baudhayana" title="Baudhayana">Baudhayana</a></li>
<li><a href="Manava" title="Manava">Manava</a></li>
<li><a href="Pythagoras" title="Pythagoras">Pythagoras</a></li>
<li><a href="Euclid" title="Euclid">Euclid</a></li>
<li><a href="Archimedes" title="Archimedes">Archimedes</a></li>
<li><a href="Apollonius_of_Perga" title="Apollonius of Perga">Apollonius</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
1–1400s</th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Zhang_Heng" title="Zhang Heng">Zhang</a></li>
<li><a href="K%C4%81ty%C4%81yana" title="Kātyāyana">Kātyāyana</a></li>
<li><a href="Aryabhata" title="Aryabhata">Aryabhata</a></li>
<li><a href="Brahmagupta" title="Brahmagupta">Brahmagupta</a></li>
<li><a href="Virasena" title="Virasena">Virasena</a></li>
<li><a href="Alhazen" class="mw-redirect" title="Alhazen">Alhazen</a></li>
<li><a href="Sijzi" class="mw-redirect" title="Sijzi">Sijzi</a></li>
<li><a href="Omar_Khayy%C3%A1m" class="mw-redirect" title="Omar Khayyám">Khayyám</a></li>
<li><a href="Ibn_al-Yasamin" title="Ibn al-Yasamin">al-Yasamin</a></li>
<li><a href="Nasir_al-Din_al-Tusi" title="Nasir al-Din al-Tusi">al-Tusi</a></li>
<li><a href="Yang_Hui" title="Yang Hui">Yang Hui</a></li>
<li><a href="Parameshvara" class="mw-redirect" title="Parameshvara">Parameshvara</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
1400s–1700s</th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Jye%E1%B9%A3%E1%B9%ADhadeva" title="Jyeṣṭhadeva">Jyeṣṭhadeva</a></li>
<li><a href="Ren%C3%A9_Descartes" title="René Descartes">Descartes</a></li>
<li><a href="Blaise_Pascal" title="Blaise Pascal">Pascal</a></li>
<li><a href="Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li>
<li><a href="Minggatu" title="Minggatu">Minggatu</a></li>
<li><a href="Leonhard_Euler" title="Leonhard Euler">Euler</a></li>
<li><a href="Sakabe_K%C5%8Dhan" title="Sakabe Kōhan">Sakabe</a></li>
<li><a href="Yasuaki_Aida" class="mw-redirect" title="Yasuaki Aida">Aida</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
1700s–1900s</th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li>
<li><a href="Nikolai_Lobachevsky" title="Nikolai Lobachevsky">Lobachevsky</a></li>
<li><a href="J%C3%A1nos_Bolyai" title="János Bolyai">Bolyai</a></li>
<li><a href="Bernhard_Riemann" title="Bernhard Riemann">Riemann</a></li>
<li><a href="Felix_Klein" title="Felix Klein">Klein</a></li>
<li><a href="Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li>
<li><a href="David_Hilbert" title="David Hilbert">Hilbert</a></li>
<li><a href="Hermann_Minkowski" title="Hermann Minkowski">Minkowski</a></li>
<li><a href="%C3%89lie_Cartan" title="Élie Cartan">Cartan</a></li>
<li><a href="Oswald_Veblen" title="Oswald Veblen">Veblen</a></li>
<li><a href="Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter</a></li>
<li><a href="Shiing-Shen_Chern" title="Shiing-Shen Chern">Chern</a></li></ul></td>
</tr><tr><th class="sidebar-heading">
Present day</th></tr><tr><td class="sidebar-content hlist">
<ul><li><a href="Michael_Atiyah" title="Michael Atiyah">Atiyah</a></li>
<li><a href="Mikhail_Leonidovich_Gromov" class="mw-redirect" title="Mikhail Leonidovich Gromov">Gromov</a></li></ul></td>
</tr></tbody></table></div></div></td>
</tr><tr><td class="sidebar-navbar"><style data-mw-deduplicate="TemplateStyles:r1239400231">
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<p>In <a href="Geometry" title="Geometry">geometry</a>, a <b>line segment</b> is a part of a <a href="Line_(mathematics)" class="mw-redirect" title="Line (mathematics)">straight line</a> that is bounded by two distinct <b>endpoints</b> (its <a href="Extreme_point" title="Extreme point">extreme points</a>), and contains every <a href="Point_(geometry)" title="Point (geometry)">point</a> on the line that is between its endpoints. It is a special case of an <i><a href="Arc_(geometry)" class="mw-redirect" title="Arc (geometry)">arc</a></i>, with zero <a href="Curvature" title="Curvature">curvature</a>. The <a href="Length" title="Length">length</a> of a line segment is given by the <a href="Euclidean_distance" title="Euclidean distance">Euclidean distance</a> between its endpoints. A <b>closed line segment</b> includes both endpoints, while an <b>open line segment</b> excludes both endpoints; a <b>half-open line segment</b> includes exactly one of the endpoints. In <a href="Geometry" title="Geometry">geometry</a>, a line segment is often denoted using an <a href="Overline" title="Overline">overline</a> (<a href="Vinculum_(symbol)" title="Vinculum (symbol)">vinculum</a>) above the symbols for the two endpoints, such as in <span class="texhtml mvar" style="font-style:italic;"><span style="text-decoration:overline;">AB</span></span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p><p>Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a <a href="Polygon" title="Polygon">polygon</a> or <a href="Polyhedron" title="Polyhedron">polyhedron</a>, the line segment is either an <a href="Edge_(geometry)" title="Edge (geometry)">edge</a> (of that polygon or polyhedron) if they are adjacent vertices, or a <a href="Diagonal" title="Diagonal">diagonal</a>. When the end points both lie on a <a href="Curve" title="Curve">curve</a> (such as a <a href="Circle" title="Circle">circle</a>), a line segment is called a <a href="Chord_(geometry)" title="Chord (geometry)">chord</a> (of that curve).
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="In_real_or_complex_vector_spaces">In real or complex vector spaces</h2></div>
<p>If <span class="texhtml mvar" style="font-style:italic;">V</span> is a <a href="Vector_space" title="Vector space">vector space</a> over <span class="nowrap"><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 \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>
</semantics>
</math></span><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"></span></span> or <span class="nowrap"><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 \mathbb {C} ,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ,}</annotation>
</semantics>
</math></span><img src="./c6ff6a3dc2982018ff20f1d2c927afc74a217be6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {C} ,}" loading="lazy"></span></span> and <span class="texhtml mvar" style="font-style:italic;">L</span> is a <a href="Subset" title="Subset">subset</a> of <span class="texhtml mvar" style="font-style:italic;">V</span>, then <span class="texhtml mvar" style="font-style:italic;">L</span> is a <b>line segment</b> if <span class="texhtml mvar" style="font-style:italic;">L</span> can be parameterized as
</p>
<dl><dd><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=\{\mathbf {u} +t\mathbf {v} \mid t\in [0,1]\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">u</mi>
</mrow>
<mo>+</mo>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">v</mi>
</mrow>
<mo>∣<!-- ∣ --></mo>
<mi>t</mi>
<mo>∈<!-- ∈ --></mo>
<mo stretchy="false">[</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo stretchy="false">]</mo>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L=\{\mathbf {u} +t\mathbf {v} \mid t\in [0,1]\}}</annotation>
</semantics>
</math></span><img src="./a7b51ec898964a04faa907aea6fb00df3228c5b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.852ex; height:2.843ex;" alt="{\displaystyle L=\{\mathbf {u} +t\mathbf {v} \mid t\in [0,1]\}}" loading="lazy"></span></dd></dl>
<p>for some vectors <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 \mathbf {u} ,\mathbf {v} \in V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">u</mi>
</mrow>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">v</mi>
</mrow>
<mo>∈<!-- ∈ --></mo>
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} ,\mathbf {v} \in V}</annotation>
</semantics>
</math></span><img src="./55d35c94c8d55735e299fa26c8d298f8c7d9d107.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.558ex; height:2.509ex;" alt="{\displaystyle \mathbf {u} ,\mathbf {v} \in V}" loading="lazy"></span> where <span class="texhtml"><b>v</b></span> is nonzero. The endpoints of <span class="texhtml mvar" style="font-style:italic;">L</span> are then the vectors <span class="texhtml"><b>u</b></span> and <span class="texhtml"><b>u</b> + <b>v</b></span>.
</p><p>Sometimes, one needs to distinguish between "open" and "closed" line segments. In this case, one would define a <b>closed line segment</b> as above, and an <b>open line segment</b> as a subset <span class="texhtml mvar" style="font-style:italic;">L</span> that can be parametrized as
</p>
<dl><dd><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=\{\mathbf {u} +t\mathbf {v} \mid t\in (0,1)\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">u</mi>
</mrow>
<mo>+</mo>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">v</mi>
</mrow>
<mo>∣<!-- ∣ --></mo>
<mi>t</mi>
<mo>∈<!-- ∈ --></mo>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L=\{\mathbf {u} +t\mathbf {v} \mid t\in (0,1)\}}</annotation>
</semantics>
</math></span><img src="./26a480e5815d99cf3a1cb1d569157cd92528f11f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.368ex; height:2.843ex;" alt="{\displaystyle L=\{\mathbf {u} +t\mathbf {v} \mid t\in (0,1)\}}" loading="lazy"></span></dd></dl>
<p>for some vectors <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 \mathbf {u} ,\mathbf {v} \in V.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">u</mi>
</mrow>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">v</mi>
</mrow>
<mo>∈<!-- ∈ --></mo>
<mi>V</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} ,\mathbf {v} \in V.}</annotation>
</semantics>
</math></span><img src="./8ae373e6ca50a1140c8b9b84a3c6cae12136eaea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.205ex; height:2.509ex;" alt="{\displaystyle \mathbf {u} ,\mathbf {v} \in V.}" loading="lazy"></span>
</p><p>Equivalently, a line segment is the <a href="Convex_hull" title="Convex hull">convex hull</a> of two points. Thus, the line segment can be expressed as a <a href="Convex_combination" title="Convex combination">convex combination</a> of the segment's two end points.
</p><p>In <a href="Geometry" title="Geometry">geometry</a>, one might define point <span class="texhtml mvar" style="font-style:italic;">B</span> to be between two other points <span class="texhtml mvar" style="font-style:italic;">A</span> and <span class="texhtml mvar" style="font-style:italic;">C</span>, if the distance <span class="texhtml mvar" style="font-style:italic;">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">AB</span>|</span> added to the distance <span class="texhtml mvar" style="font-style:italic;">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">BC</span>|</span> is equal to the distance <span class="texhtml mvar" style="font-style:italic;">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">AC</span>|</span>. Thus in <span class="nowrap"><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 \mathbb {R} ^{2},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2},}</annotation>
</semantics>
</math></span><img src="./d349b099a2e00103b347c5f640a30e0af2a6ee18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.379ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} ^{2},}" loading="lazy"></span></span> the line segment with endpoints <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 A=(a_{x},a_{y})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A=(a_{x},a_{y})}</annotation>
</semantics>
</math></span><img src="./b93951475037e75f61db107dbe62c7a2405e7f70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.366ex; height:3.009ex;" alt="{\displaystyle A=(a_{x},a_{y})}" loading="lazy"></span> and <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 C=(c_{x},c_{y})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C=(c_{x},c_{y})}</annotation>
</semantics>
</math></span><img src="./3481c79948ed3f0350e8ad2a598e33c5edcc47c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.944ex; height:3.009ex;" alt="{\displaystyle C=(c_{x},c_{y})}" loading="lazy"></span> is the following collection of points:
</p>
<dl><dd><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 {\Biggl \{}(x,y)\mid {\sqrt {(x-c_{x})^{2}+(y-c_{y})^{2}}}+{\sqrt {(x-a_{x})^{2}+(y-a_{y})^{2}}}={\sqrt {(c_{x}-a_{x})^{2}+(c_{y}-a_{y})^{2}}}{\Biggr \}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.470em" minsize="2.470em">{</mo>
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</mrow>
<mo stretchy="false">(</mo>
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<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>∣<!-- ∣ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
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</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<mo stretchy="false">(</mo>
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<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
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<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
</msqrt>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
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</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo>−<!-- − --></mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle {\Biggl \{}(x,y)\mid {\sqrt {(x-c_{x})^{2}+(y-c_{y})^{2}}}+{\sqrt {(x-a_{x})^{2}+(y-a_{y})^{2}}}={\sqrt {(c_{x}-a_{x})^{2}+(c_{y}-a_{y})^{2}}}{\Biggr \}}.}</annotation>
</semantics>
</math></span><img src="./95c064b868daebf79e70f1da68569acd2f9857d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:90.118ex; height:7.509ex;" alt="{\displaystyle {\Biggl \{}(x,y)\mid {\sqrt {(x-c_{x})^{2}+(y-c_{y})^{2}}}+{\sqrt {(x-a_{x})^{2}+(y-a_{y})^{2}}}={\sqrt {(c_{x}-a_{x})^{2}+(c_{y}-a_{y})^{2}}}{\Biggr \}}.}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>
<ul><li>A line segment is a <a href="Connected_set" class="mw-redirect" title="Connected set">connected</a>, <a href="Non-empty" class="mw-redirect" title="Non-empty">non-empty</a> <a href="Set_(mathematics)" title="Set (mathematics)">set</a>.</li>
<li>If <span class="texhtml mvar" style="font-style:italic;">V</span> is a <a href="Topological_vector_space" title="Topological vector space">topological vector space</a>, then a closed line segment is a <a href="Closed_set" title="Closed set">closed set</a> in <span class="texhtml mvar" style="font-style:italic;">V</span>. However, an open line segment is an <a href="Open_subset" class="mw-redirect" title="Open subset">open set</a> in <span class="texhtml mvar" style="font-style:italic;">V</span> <a href="If_and_only_if" title="If and only if">if and only if</a> <span class="texhtml mvar" style="font-style:italic;">V</span> is <a href="One-dimensional_space" title="One-dimensional space">one-dimensional</a>.</li>
<li>More generally than above, the concept of a line segment can be defined in an <a href="Ordered_geometry" title="Ordered geometry">ordered geometry</a>.</li>
<li>A pair of line segments can be any one of the following: <a href="Intersection_(geometry)" title="Intersection (geometry)">intersecting</a>, <a href="Parallel_(geometry)" title="Parallel (geometry)">parallel</a>, <a href="Skew_lines" title="Skew lines">skew</a>, or none of these. The last possibility is a way that line segments differ from lines: if two nonparallel lines are in the same Euclidean plane then they must cross each other, but that need not be true of segments.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="In_proofs">In proofs</h2></div>
<p>In an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or defined in terms of an <a href="Isometry" title="Isometry">isometry</a> of a line (used as a coordinate system).
</p><p>Segments play an important role in other theories. For example, in a <i><a href="Convex_set" title="Convex set">convex set</a></i>, the segment that joins any two points of the set is contained in the set. This is important because it transforms some of the analysis of convex sets, to the analysis of a line segment. The <i><a href="Segment_addition_postulate" title="Segment addition postulate">segment addition postulate</a></i> can be used to add congruent segment or segments with equal lengths, and consequently substitute other segments into another statement to make segments congruent.
</p>
<div class="mw-heading mw-heading2"><h2 id="As_a_degenerate_ellipse">As a degenerate ellipse</h2></div>
<p>A line segment can be viewed as a <a href="Degenerate_conic" title="Degenerate conic">degenerate case</a> of an <a href="Ellipse#Line_segment_as_a_type_of_degenerate_ellipse" title="Ellipse">ellipse</a>, in which the semiminor axis goes to zero, the <a href="Focus_(geometry)" title="Focus (geometry)">foci</a> go to the endpoints, and the eccentricity goes to one. A standard definition of an ellipse is the set of points for which the sum of a point's distances to two foci is a constant; if this constant equals the distance between the foci, the line segment is the result. A complete orbit of this ellipse traverses the line segment twice. As a degenerate orbit, this is a <a href="Elliptic_orbit#Radial_elliptic_trajectory" title="Elliptic orbit">radial elliptic trajectory</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="In_other_geometric_shapes">In other geometric shapes</h2></div>
<p>In addition to appearing as the edges and <a href="Diagonal" title="Diagonal">diagonals</a> of <a href="Polygon" title="Polygon">polygons</a> and <a href="Polyhedron" title="Polyhedron">polyhedra</a>, line segments also appear in numerous other locations relative to other <a href="Geometric_shape" class="mw-redirect" title="Geometric shape">geometric shapes</a>.
</p>
<div class="mw-heading mw-heading3"><h3 id="Triangles">Triangles</h3></div>
<p>Some very frequently considered segments in a <a href="Triangle" title="Triangle">triangle</a> to include the three <a href="Altitude_(geometry)" class="mw-redirect" title="Altitude (geometry)">altitudes</a> (each <a href="Perpendicular" title="Perpendicular">perpendicularly</a> connecting a side or its <a href="Extended_side" title="Extended side">extension</a> to the opposite <a href="Vertex_(geometry)" title="Vertex (geometry)">vertex</a>), the three <a href="Median_(geometry)" title="Median (geometry)">medians</a> (each connecting a side's <a href="Midpoint" title="Midpoint">midpoint</a> to the opposite vertex), the <a href="Perpendicular_bisector" class="mw-redirect" title="Perpendicular bisector">perpendicular bisectors</a> of the sides (perpendicularly connecting the midpoint of a side to one of the other sides), and the <a href="Angle_bisector" class="mw-redirect" title="Angle bisector">internal angle bisectors</a> (each connecting a vertex to the opposite side). In each case, there are various <a href="Equality_(mathematics)" title="Equality (mathematics)">equalities</a> relating these segment lengths to others (discussed in the articles on the various types of segment), as well as <a href="List_of_triangle_inequalities" title="List of triangle inequalities">various inequalities</a>.
</p><p>Other segments of interest in a triangle include those connecting various <a href="Triangle_center" title="Triangle center">triangle centers</a> to each other, most notably the <a href="Incenter" title="Incenter">incenter</a>, the <a href="Circumcenter" class="mw-redirect" title="Circumcenter">circumcenter</a>, the <a href="Nine-point_center" title="Nine-point center">nine-point center</a>, the <a href="Centroid" title="Centroid">centroid</a> and the <a href="Orthocenter" title="Orthocenter">orthocenter</a>.
</p>
<div class="mw-heading mw-heading3"><h3 id="Quadrilaterals">Quadrilaterals</h3></div>
<p>In addition to the sides and diagonals of a <a href="Quadrilateral" title="Quadrilateral">quadrilateral</a>, some important segments are the two <a href="Quadrilateral#Special_line_segments" title="Quadrilateral">bimedians</a> (connecting the midpoints of opposite sides) and the four <a href="Quadrilateral#Special_line_segments" title="Quadrilateral">maltitudes</a> (each perpendicularly connecting one side to the midpoint of the opposite side).
</p>
<div class="mw-heading mw-heading3"><h3 id="Circles_and_ellipses">Circles and ellipses</h3></div>
<p>Any straight line segment connecting two points on a <a href="Circle" title="Circle">circle</a> or <a href="Ellipse" title="Ellipse">ellipse</a> is called a <a href="Chord_(geometry)" title="Chord (geometry)">chord</a>. Any chord in a circle which has no longer chord is called a <a href="Diameter" title="Diameter">diameter</a>, and any segment connecting the circle's <a href="Center_(geometry)" class="mw-redirect" title="Center (geometry)">center</a> (the midpoint of a diameter) to a point on the circle is called a <a href="Radius" title="Radius">radius</a>.
</p><p>In an ellipse, the longest chord, which is also the longest <a href="Diameter#Ellipse" title="Diameter">diameter</a>, is called the <i>major axis</i>, and a segment from the midpoint of the major axis (the ellipse's center) to either endpoint of the major axis is called a <i>semi-major axis</i>. Similarly, the shortest diameter of an ellipse is called the <i>minor axis</i>, and the segment from its midpoint (the ellipse's center) to either of its endpoints is called a <i>semi-minor axis</i>. The chords of an ellipse which are <a href="Perpendicular" title="Perpendicular">perpendicular</a> to the major axis and pass through one of its <a href="Focus_(geometry)" title="Focus (geometry)">foci</a> are called the <a href="Latus_rectum" class="mw-redirect" title="Latus rectum">latera recta</a> of the ellipse. The <i>interfocal segment</i> connects the two foci.
</p>
<div class="mw-heading mw-heading2"><h2 id="Directed_line_segment">Directed line segment</h2></div>
<style data-mw-deduplicate="TemplateStyles:r1236090951">
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</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Orientation_(vector_space)#On_a_line" title="Orientation (vector space)">Orientation (vector space) § On a line</a></div>
<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Relative_position" class="mw-redirect" title="Relative position">Relative position</a></div>
<p>When a line segment is given an <a href="Orientation_(vector_space)" title="Orientation (vector space)">orientation</a> (<a href="Direction_(geometry)" title="Direction (geometry)">direction</a>) it is called a <b>directed line segment</b> or <b>oriented line segment</b>. It suggests a <a href="Translation_(geometry)" title="Translation (geometry)">translation</a> or <a href="Displacement_(geometry)" title="Displacement (geometry)">displacement</a> (perhaps caused by a <a href="Force" title="Force">force</a>). The magnitude and direction are indicative of a potential change. Extending a directed line segment semi-infinitely produces a <i><a href="Directed_half-line" class="mw-redirect" title="Directed half-line">directed half-line</a></i> and infinitely in both directions produces a <i><a href="Directed_line" class="mw-redirect" title="Directed line">directed line</a></i>. This suggestion has been absorbed into <a href="Mathematical_physics" title="Mathematical physics">mathematical physics</a> through the concept of a <a href="Euclidean_vector" title="Euclidean vector">Euclidean vector</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><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The collection of all directed line segments is usually reduced by making <a href="Equipollent_(geometry)" class="mw-redirect" title="Equipollent (geometry)">equipollent</a> any pair having the same length and orientation.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> This application of an <a href="Equivalence_relation" title="Equivalence relation">equivalence relation</a> was introduced by <a href="Giusto_Bellavitis" title="Giusto Bellavitis">Giusto Bellavitis</a> in 1835.
</p>
<div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2></div>
<p>Analogous to <a href="Straight_line" class="mw-redirect" title="Straight line">straight line</a> segments above, one can also define <a href="Arc_(geometry)" class="mw-redirect" title="Arc (geometry)">arcs</a> as segments of a <a href="Curve" title="Curve">curve</a>.
</p><p>In one-dimensional space, a <i><a href="Ball_(mathematics)" title="Ball (mathematics)">ball</a></i> is a line segment.
</p><p>An <a href="Oriented_plane_segment" class="mw-redirect" title="Oriented plane segment">oriented plane segment</a> or <i><a href="Bivector" title="Bivector">bivector</a></i> generalizes the directed line segment.
</p><p>Beyond Euclidean geometry, <a href="Geodesic_segment" class="mw-redirect" title="Geodesic segment">geodesic segments</a> play the role of line segments.
</p><p>A line segment is a one-dimensional <i><a href="Simplex" title="Simplex">simplex</a></i>; a two-dimensional simplex is a triangle.
</p>
<div class="mw-heading mw-heading2"><h2 id="Types_of_line_segments">Types of line segments</h2></div>
<ul><li><a href="Chord_(geometry)" title="Chord (geometry)">Chord (geometry)</a></li>
<li><a href="Diameter" title="Diameter">Diameter</a></li>
<li><a href="Radius" title="Radius">Radius</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Polygonal_chain" title="Polygonal chain">Polygonal chain</a></li>
<li><a href="Interval_(mathematics)" title="Interval (mathematics)">Interval (mathematics)</a></li>
<li><a href="Line_segment_intersection" class="mw-redirect" title="Line segment intersection">Line segment intersection</a>, the algorithmic problem of finding intersecting pairs in a collection of line segments</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>
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<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathopenref.com/linesegment.html">"Line Segment Definition - Math Open Reference"</a>. <i>www.mathopenref.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-09-01</span></span>.</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">Harry F. Davis & Arthur David Snider (1988) <i>Introduction to Vector Analysis</i>, 5th edition, page 1, Wm. C. Brown Publishers <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-697-06814-5</bdi></span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Matiur Rahman & Isaac Mulolani (2001) <i>Applied Vector Analysis</i>, pages 9 & 10, <a href="CRC_Press" title="CRC Press">CRC Press</a> <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8493-1088-1</bdi></span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Eutiquio C. Young (1978) <i>Vector and Tensor Analysis</i>, pages 2 & 3, <a href="Marcel_Dekker" title="Marcel Dekker">Marcel Dekker</a> <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8247-6671-7</bdi></span>
</li>
</ol></div></div>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<ul><li><a href="David_Hilbert" title="David Hilbert">David Hilbert</a> <i>The Foundations of Geometry</i>. The Open Court Publishing Company 1950, p. 4</li></ul>
<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>
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<div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Line_segment" class="extiw external" title="commons:Line segment"><span style="font-style:italic; font-weight:bold;">Line segment</span></a>.</div></div>
</div>
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<div class="side-box-flex">
<div class="side-box-image"><span class="noviewer" typeof="mw:File"></span></div>
<div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/line_segment" class="extiw external" title="wiktionary:line segment">line segment</a></b></i> in Wiktionary, the free dictionary.</div></div>
</div>
<ul><li><span class="citation mathworld" id="Reference-Mathworld-Line_segment"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/LineSegment.html">"Line segment"</a>. <i><a href="MathWorld" title="MathWorld">MathWorld</a></i>.</cite></span></li>
<li><a rel="nofollow" class="external text" href="https://planetmath.org/linesegment">Line Segment</a> at <a href="PlanetMath" title="PlanetMath">PlanetMath</a></li>
<li><a rel="nofollow" class="external text" href="http://www.mathopenref.com/constcopysegment.html">Copying a line segment with compass and straightedge</a></li>
<li><a rel="nofollow" class="external text" href="http://www.mathopenref.com/constdividesegment.html">Dividing a line segment into N equal parts with compass and straightedge</a> Animated demonstration</li></ul>
<p><i>This article incorporates material from Line segment on <a href="PlanetMath" title="PlanetMath">PlanetMath</a>, which is licensed under the Creative Commons Attribution/Share-Alike License.</i>
</p>
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