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<div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Geophysical definition</span> </div> </a> <button aria-controls="toc-Geophysical_definition-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Geophysical definition subsection</span> </button> <ul id="toc-Geophysical_definition-sublist" class="vector-toc-list"> <li id="toc-The_plumb_line_and_spirit_level" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_plumb_line_and_spirit_level"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>The plumb line and spirit level</span> </div> </a> <ul id="toc-The_plumb_line_and_spirit_level-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_spherical_Earth" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_spherical_Earth"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>The spherical Earth</span> </div> </a> <ul id="toc-The_spherical_Earth-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_complications" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Further_complications"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Further complications</span> </div> </a> <ul id="toc-Further_complications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Independence_of_horizontal_and_vertical_motions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Independence_of_horizontal_and_vertical_motions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Independence of horizontal and vertical motions</span> </div> </a> <ul id="toc-Independence_of_horizontal_and_vertical_motions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mathematical_definition" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mathematical_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Mathematical definition</span> </div> </a> <button aria-controls="toc-Mathematical_definition-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Mathematical definition subsection</span> </button> <ul id="toc-Mathematical_definition-sublist" class="vector-toc-list"> <li id="toc-In_two_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_two_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>In two dimensions</span> </div> </a> <ul id="toc-In_two_dimensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_three_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_three_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>In three dimensions</span> </div> </a> <ul id="toc-In_three_dimensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_the_classroom" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_the_classroom"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>In the classroom</span> </div> </a> <ul id="toc-In_the_classroom-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Discussion" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Discussion"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Discussion</span> </div> </a> <button aria-controls="toc-Discussion-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Discussion subsection</span> </button> <ul id="toc-Discussion-sublist" class="vector-toc-list"> <li id="toc-Practical_use_in_daily_life" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Practical_use_in_daily_life"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Practical use in daily life</span> </div> </a> <ul id="toc-Practical_use_in_daily_life-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References_and_notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References_and_notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References and notes</span> </div> </a> <ul id="toc-References_and_notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div 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class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Svisl%C3%BD_a_vodorovn%C3%BD" title="Svislý a vodorovný – Czech" lang="cs" hreflang="cs" data-title="Svislý a vodorovný" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Horizontale_et_verticale" title="Horizontale et verticale – French" lang="fr" hreflang="fr" data-title="Horizontale et verticale" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Vertikal_dan_horizontal" title="Vertikal dan horizontal – Indonesian" lang="id" hreflang="id" data-title="Vertikal dan horizontal" 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class="mw-redirect" title="Vertical direction">Vertical direction</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Directional planes</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="/wiki/Vertical_and_horizontal_(disambiguation)" class="mw-redirect mw-disambig" title="Vertical and horizontal (disambiguation)">Vertical and horizontal (disambiguation)</a>.</div> <p>In <a href="/wiki/Astronomy" title="Astronomy">astronomy</a>, <a href="/wiki/Geography" title="Geography">geography</a>, and related sciences and contexts, a <i><a href="/wiki/Direction_(geometry,_geography)" class="mw-redirect" title="Direction (geometry, geography)">direction</a></i> or <i><a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a></i> passing by a given point is said to be <b>vertical</b> if it contains the local <a href="/wiki/Gravity_direction" class="mw-redirect" title="Gravity direction">gravity direction</a> at that point.<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>Conversely, a direction or plane is said to be <b>horizontal</b> (or <i>leveled</i>) if it is perpendicular to the vertical direction. In general, something that is vertical can be drawn from up to down (or down to up), such as the y-axis in the <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Historical_definition">Historical definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertical_and_horizontal&action=edit&section=1" title="Edit section: Historical definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The word <i>horizontal</i> is derived from the Latin <span title="Latin-language text"><i lang="la">horizon</i></span>, which derives from the Greek <span title="Greek-language text"><span lang="el">ὁρῐ́ζων</span></span>, meaning 'separating' or 'marking a boundary'.<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> The word <i>vertical</i> is derived from the late Latin <i><span title="Latin-language text"><i lang="la">verticalis</i></span></i>, which is from the same root as <i>vertex</i>, meaning 'highest point' or more literally the 'turning point' such as in a whirlpool.<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> </p><p><a href="/wiki/Girard_Desargues" title="Girard Desargues">Girard Desargues</a> defined the vertical to be perpendicular to the <a href="/wiki/Horizon" title="Horizon">horizon</a> in his 1636 book <i>Perspective</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Geophysical_definition">Geophysical definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertical_and_horizontal&action=edit&section=2" title="Edit section: Geophysical definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Gravity_of_Earth" title="Gravity of Earth">Gravity of Earth</a> and <a href="/wiki/Figure_of_Earth" class="mw-redirect" title="Figure of Earth">Figure of Earth</a></div> <div class="mw-heading mw-heading3"><h3 id="The_plumb_line_and_spirit_level">The plumb line and spirit level</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertical_and_horizontal&action=edit&section=3" title="Edit section: The plumb line and spirit level"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Bubble_on_a_marble_shelf.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Bubble_on_a_marble_shelf.PNG/220px-Bubble_on_a_marble_shelf.PNG" decoding="async" width="220" height="37" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Bubble_on_a_marble_shelf.PNG/330px-Bubble_on_a_marble_shelf.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Bubble_on_a_marble_shelf.PNG/440px-Bubble_on_a_marble_shelf.PNG 2x" data-file-width="721" data-file-height="121" /></a><figcaption>Spirit level bubble on a marble shelf tests for horizontality</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Plumb_bob.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Plumb_bob.jpg/170px-Plumb_bob.jpg" decoding="async" width="170" height="252" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Plumb_bob.jpg/255px-Plumb_bob.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Plumb_bob.jpg/340px-Plumb_bob.jpg 2x" data-file-width="978" data-file-height="1449" /></a><figcaption>A plumb bob</figcaption></figure> <p>In physics, engineering and construction, the direction designated as vertical is usually that along which a <a href="/wiki/Plumb-bob" class="mw-redirect" title="Plumb-bob">plumb-bob</a> hangs. Alternatively, a <a href="/wiki/Spirit_level" title="Spirit level">spirit level</a> that exploits the buoyancy of an air bubble and its tendency to go vertically upwards may be used to test for horizontality. A <a href="/wiki/Water_level_device" class="mw-redirect" title="Water level device">water level device</a> may also be used to establish horizontality. </p><p>Modern rotary <a href="/wiki/Laser_level" title="Laser level">laser levels</a> that can level themselves automatically are robust sophisticated instruments and work on the same fundamental principle.<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><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> </p> <div class="mw-heading mw-heading3"><h3 id="The_spherical_Earth">The spherical Earth</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertical_and_horizontal&action=edit&section=4" title="Edit section: The spherical Earth"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Local_tangent_plane" class="mw-redirect" title="Local tangent plane">Local tangent plane</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Vertical_wall_are_never_parallel.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Vertical_wall_are_never_parallel.PNG/220px-Vertical_wall_are_never_parallel.PNG" decoding="async" width="220" height="210" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Vertical_wall_are_never_parallel.PNG/330px-Vertical_wall_are_never_parallel.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/a/a2/Vertical_wall_are_never_parallel.PNG 2x" data-file-width="372" data-file-height="355" /></a><figcaption>Strictly, vertical directions are never parallel on the surface of a spherical planet (except at opposite poles, where they are <a href="/wiki/Antiparallel_vectors" class="mw-redirect" title="Antiparallel vectors">anti-parallel</a>).</figcaption></figure> <p>When the curvature of the Earth is taken into account, the concepts of vertical and horizontal take on yet another meaning. On the surface of a smoothly spherical, homogenous, non-rotating planet, the plumb bob picks out as vertical the radial direction. Strictly speaking, it is now no longer possible for vertical walls to be parallel: <i>all</i> verticals intersect. This fact has real practical applications in construction and civil engineering, e.g., the tops of the towers of a suspension bridge are further apart than at the bottom. <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> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Horizontal_planes_can_intersect.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Horizontal_planes_can_intersect.PNG/220px-Horizontal_planes_can_intersect.PNG" decoding="async" width="220" height="195" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Horizontal_planes_can_intersect.PNG/330px-Horizontal_planes_can_intersect.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/2/2a/Horizontal_planes_can_intersect.PNG 2x" data-file-width="360" data-file-height="319" /></a><figcaption>On a spherical planet, horizontal planes intersect. In the example shown, the blue line represents the horizontal tangent plane at the North pole, the red the horizontal tangent plane at an equatorial point. The two intersect at a right angle.</figcaption></figure> <p>Also, horizontal planes can intersect when they are <a href="/wiki/Tangent" title="Tangent">tangent</a> planes to separated points on the surface of the Earth. In particular, a plane tangent to a point on the equator intersects the plane tangent to the <a href="/wiki/North_Pole" title="North Pole">North Pole</a> at a <a href="/wiki/Right_angle" title="Right angle">right angle</a>. (See diagram). Furthermore, the <a href="/wiki/Equator" title="Equator">equatorial</a> plane is parallel to the tangent plane at the North Pole and as such has claim to be a horizontal plane. But it is. at the same time, a vertical plane for points on the equator. In this sense, a plane can, arguably, be <i>both</i> horizontal and vertical, horizontal <i>at one place</i>, and vertical <i>at another</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Further_complications">Further complications</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertical_and_horizontal&action=edit&section=5" title="Edit section: Further complications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Vertical_deflection" title="Vertical deflection">Vertical deflection</a></div> <p>For a spinning earth, the plumb line deviates from the radial direction as a function of latitude.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Only on the equator and at the North and South Poles does the plumb line align with the local radius. The situation is actually even more complicated because Earth is not a <a href="/wiki/Homogeneous" class="mw-redirect" title="Homogeneous">homogeneous</a> smooth sphere. It is a non homogeneous, non spherical, knobby planet in motion, and the vertical not only need not lie along a radial, it may even be curved and be varying with time. On a smaller scale, a mountain to one side may deflect the plumb bob away from the true <a href="/wiki/Zenith" title="Zenith">zenith</a>.<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> </p><p>On a larger scale the gravitational field of the Earth, which is at least approximately radial near the Earth, is not radial when it is affected by the Moon at higher altitudes.<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><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> </p> <div class="mw-heading mw-heading3"><h3 id="Independence_of_horizontal_and_vertical_motions">Independence of horizontal and vertical motions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertical_and_horizontal&action=edit&section=6" title="Edit section: Independence of horizontal and vertical motions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Neglecting the curvature of the earth, horizontal and vertical motions of a projectile moving under gravity are independent of each other.<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> Vertical displacement of a projectile is not affected by the horizontal component of the launch velocity, and, conversely, the horizontal displacement is unaffected by the vertical component. The notion dates at least as far back as Galileo.<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> </p><p>When the curvature of the Earth is taken into account, the independence of the two motion does <i>not</i> hold. For example, even a projectile fired in a horizontal direction (i.e., with a zero vertical component) may leave the surface of the spherical Earth and indeed escape altogether.<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> <div class="mw-heading mw-heading2"><h2 id="Mathematical_definition">Mathematical definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertical_and_horizontal&action=edit&section=7" title="Edit section: Mathematical definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="In_two_dimensions">In two dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertical_and_horizontal&action=edit&section=8" title="Edit section: In two dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Vertical_horizontal.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Vertical_horizontal.PNG/220px-Vertical_horizontal.PNG" decoding="async" width="220" height="171" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Vertical_horizontal.PNG/330px-Vertical_horizontal.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/e/e9/Vertical_horizontal.PNG 2x" data-file-width="343" data-file-height="266" /></a><figcaption>In two dimensions. 1. The vertical direction is designated. 2. The horizontal is perpendicular to the vertical. Through any point P, there is exactly one vertical and exactly one horizontal. Alternatively, one can start by designating the horizontal direction.</figcaption></figure> <p>In the context of a 1-dimensional orthogonal <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a> on a Euclidean plane, to say that a line is horizontal or vertical, an initial designation has to be made. One can start off by designating the vertical direction, usually labelled the Y direction.<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 horizontal direction, usually labelled the X direction,<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> is then automatically determined. Or, one can do it the other way around, i.e., nominate the <i>x</i>-axis, in which case the <i>y</i>-axis is then automatically determined. There is no special reason to choose the horizontal over the vertical as the initial designation: the two directions are on par in this respect. </p><p>The following hold in the two-dimensional case: </p> <ol><li>Through any point P in the plane, there is one and only one <i>vertical line</i> within the plane and one and only one <i>horizontal line</i> within the plane. This symmetry breaks down as one moves to the three-dimensional case.</li> <li>A <i>vertical line</i> is any line parallel to the vertical direction. A <i>horizontal line</i> is any line normal to a vertical line.</li> <li>Horizontal lines do not cross each other.</li> <li>Vertical lines do not cross each other.</li></ol> <p>Not all of these elementary geometric facts are true in the 3-D context. </p> <div class="mw-heading mw-heading3"><h3 id="In_three_dimensions">In three dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertical_and_horizontal&action=edit&section=9" title="Edit section: In three dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the three-dimensional case, the situation is more complicated as now one has horizontal and vertical planes in addition to horizontal and vertical lines. Consider a point P and designate a direction through P as vertical. A plane which contains P and is normal to the designated direction is the <i><a href="/wiki/Horizontal_plane" class="mw-redirect" title="Horizontal plane">horizontal plane</a></i> at P. Any plane going through P, normal to the horizontal plane is a <i><a href="/wiki/Vertical_plane" class="mw-redirect" title="Vertical plane">vertical plane</a></i> at P. Through any point P, there is one and only one horizontal plane but a <i>multiplicity</i> of vertical planes. This is a new feature that emerges in three dimensions. The symmetry that exists in the two-dimensional case no longer holds. </p> <div class="mw-heading mw-heading3"><h3 id="In_the_classroom">In the classroom</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertical_and_horizontal&action=edit&section=10" title="Edit section: In the classroom"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Mystery_of_the_horizontal_y-axis.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Mystery_of_the_horizontal_y-axis.PNG/220px-Mystery_of_the_horizontal_y-axis.PNG" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Mystery_of_the_horizontal_y-axis.PNG/330px-Mystery_of_the_horizontal_y-axis.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/Mystery_of_the_horizontal_y-axis.PNG/440px-Mystery_of_the_horizontal_y-axis.PNG 2x" data-file-width="614" data-file-height="460" /></a><figcaption>The <i>y</i>-axis on the wall is vertical, but the one on the table is horizontal.</figcaption></figure> <p>In the 2-dimension case, as mentioned already, the usual designation of the vertical coincides with the <a href="/wiki/Y-axis" class="mw-redirect" title="Y-axis">y-axis</a> in co-ordinate geometry. This convention can cause confusion in the classroom. For the teacher, writing perhaps on a white board, the <i>y</i>-axis really is vertical in the sense of the plumbline verticality but for the student the axis may well lie on a horizontal table. </p> <div class="mw-heading mw-heading2"><h2 id="Discussion">Discussion</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertical_and_horizontal&action=edit&section=11" title="Edit section: Discussion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Cleanup_merge plainlinks metadata ambox ambox-move" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Split-arrows.svg/50px-Split-arrows.svg.png" decoding="async" width="50" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Split-arrows.svg/75px-Split-arrows.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Split-arrows.svg/100px-Split-arrows.svg.png 2x" data-file-width="60" data-file-height="20" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span"><b>This article may need to be cleaned up.</b> It has been merged from <i><a href="/wiki/Horizontal_plane" class="mw-redirect" title="Horizontal plane">Horizontal plane</a></i>.</div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Essay-like plainlinks metadata ambox ambox-style ambox-essay-like" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>is written like a <a href="/wiki/Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_publisher_of_original_thought" title="Wikipedia:What Wikipedia is not">personal reflection, personal essay, or argumentative essay</a></b> that states a Wikipedia editor's personal feelings or presents an original argument about a topic.<span class="hide-when-compact"> Please <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Vertical_and_horizontal&action=edit">help improve it</a> by rewriting it in an <a href="/wiki/Wikipedia:Writing_better_articles#Information_style_and_tone" title="Wikipedia:Writing better articles">encyclopedic style</a>.</span> <span class="date-container"><i>(<span class="date">July 2018</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Bubble_on_a_marble_shelf.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Bubble_on_a_marble_shelf.PNG/220px-Bubble_on_a_marble_shelf.PNG" decoding="async" width="220" height="37" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Bubble_on_a_marble_shelf.PNG/330px-Bubble_on_a_marble_shelf.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Bubble_on_a_marble_shelf.PNG/440px-Bubble_on_a_marble_shelf.PNG 2x" data-file-width="721" data-file-height="121" /></a><figcaption>A spirit level on a shelf</figcaption></figure> <p>Although the word horizontal is commonly used in daily life and language (see below), it is subject to many misconceptions. </p> <ul><li>The concept of horizontality only makes sense in the context of a clearly measurable gravity field, i.e., in the 'neighborhood' of a planet, star, etc. When the gravity field becomes very weak (the masses are too small or too distant from the point of interest), the notion of being horizontal loses its meaning.</li></ul> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Verticals_at_separate_points.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Verticals_at_separate_points.PNG/220px-Verticals_at_separate_points.PNG" decoding="async" width="220" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Verticals_at_separate_points.PNG/330px-Verticals_at_separate_points.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/8/8a/Verticals_at_separate_points.PNG 2x" data-file-width="358" data-file-height="379" /></a><figcaption>Verticals at two separate points are not parallel. The same holds for their associated horizontal planes.</figcaption></figure> <ul><li>A plane is horizontal only at the chosen point. Horizontal planes at two separate points are not parallel, they intersect.</li> <li>In general, a horizontal plane will only be perpendicular to a <a href="/wiki/Vertical_direction" class="mw-redirect" title="Vertical direction">vertical direction</a> if both are specifically defined with respect to the same point: a direction is only vertical at the point of reference. Thus both horizontality and verticality are strictly speaking local concepts, and it is always necessary to state to which location the direction or the plane refers to. (1) the same restriction applies to the straight lines contained within the plane: they are horizontal only at the point of reference, and (2) those straight lines contained in the plane but not passing by the reference point are not necessarily horizontal anywhere.</li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gravitational_field_due_to_knobbly_planet.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Gravitational_field_due_to_knobbly_planet.PNG/220px-Gravitational_field_due_to_knobbly_planet.PNG" decoding="async" width="220" height="148" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Gravitational_field_due_to_knobbly_planet.PNG/330px-Gravitational_field_due_to_knobbly_planet.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/43/Gravitational_field_due_to_knobbly_planet.PNG/440px-Gravitational_field_due_to_knobbly_planet.PNG 2x" data-file-width="462" data-file-height="310" /></a><figcaption>Field lines for a non-homogeneous knobbly planet in motion may be curved. The white, red and blue bits illustrate the planet's heterogeneity.</figcaption></figure> <ul><li>In reality, the gravity field of a heterogeneous planet such as <a href="/wiki/Earth" title="Earth">Earth</a> is deformed due to the inhomogeneous spatial distribution of materials with different <a href="/wiki/Density" title="Density">densities</a>. Actual horizontal planes are thus not even parallel even if their reference points are along the same vertical line, since a vertical line is slightly curved.</li> <li>At any given location, the total gravitational force is not quite constant over <a href="/wiki/Time" title="Time">time</a>, because the objects that generate the gravity are moving. For instance, on Earth the horizontal plane at a given point (as determined by a pair of <a href="/wiki/Spirit_level" title="Spirit level">spirit levels</a>) changes with the position of the Moon (air, sea and land <a href="/wiki/Tide" title="Tide">tides</a>).</li> <li>On a rotating planet such as Earth, the strictly gravitational pull of the planet (and other celestial objects such as the Moon, the <a href="/wiki/Sun" title="Sun">Sun</a>, etc.) is different from the apparent net force (e.g., on a free-falling object) that can be measured in the laboratory or in the field. This difference is the <a href="/wiki/Centrifugal_force_(fictitious)" class="mw-redirect" title="Centrifugal force (fictitious)">centrifugal force</a> associated with the planet's rotation. This is a <a href="/wiki/Fictitious_force" title="Fictitious force">fictitious force</a>: it only arises when calculations or experiments are conducted in <a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">non-inertial frames of reference</a>, such as the surface of the Earth.</li></ul> <p>In general or in practice, something that is horizontal can be <a href="/wiki/Drawing" title="Drawing">drawn</a> from left to right (or right to left), such as the x-axis in the <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (July 2018)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Practical_use_in_daily_life">Practical use in daily life</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertical_and_horizontal&action=edit&section=12" title="Edit section: Practical use in daily life"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Mystery_of_the_horizontal_y-axis.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Mystery_of_the_horizontal_y-axis.PNG/220px-Mystery_of_the_horizontal_y-axis.PNG" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Mystery_of_the_horizontal_y-axis.PNG/330px-Mystery_of_the_horizontal_y-axis.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/Mystery_of_the_horizontal_y-axis.PNG/440px-Mystery_of_the_horizontal_y-axis.PNG 2x" data-file-width="614" data-file-height="460" /></a><figcaption>The y-axis on the wall is vertical, but the one on the table is horizontal.</figcaption></figure> <p>The concept of a horizontal plane is thus anything but simple, although, in practice, most of these effects and variations are rather small: they are measurable and can be predicted with great accuracy, but they may not greatly affect our daily life. </p><p>This dichotomy between the apparent simplicity of a concept and an actual complexity of defining (and measuring) it in scientific terms arises from the fact that the typical linear scales and dimensions of relevance in daily life are 3 <a href="/wiki/Orders_of_magnitude" class="mw-redirect" title="Orders of magnitude">orders of magnitude</a> (or more) smaller than the size of the Earth. Hence, the world appears to be flat locally, and horizontal planes in nearby locations appear to be parallel. Such statements are nevertheless approximations; whether they are acceptable in any particular context or application depends on the applicable requirements, in particular in terms of accuracy. In graphical contexts, such as <a href="/wiki/Technical_drawing" title="Technical drawing">drawing and drafting</a> and Co-ordinate geometry on rectangular paper, it is very common to associate one of the dimensions of the paper with a horizontal, even though the entire sheet of paper is standing on a flat horizontal (or slanted) table. In this case, the horizontal direction is typically from the left side of the paper to the right side. This is purely conventional (although it is somehow 'natural' when drawing a natural scene as it is seen in reality), and may lead to misunderstandings or misconceptions, especially in an educational context. </p> <div style="clear:left;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertical_and_horizontal&action=edit&section=13" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 23em;"> <ul><li><a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a></li> <li><a href="/wiki/Coordinate_system" title="Coordinate system">Coordinate system</a></li> <li><a href="/wiki/Horizon" title="Horizon">Horizon</a></li> <li><a href="/wiki/Horizontal_angle" class="mw-redirect" title="Horizontal angle">Horizontal angle</a></li> <li><a href="/wiki/Horizontal_coordinate_system" title="Horizontal coordinate system">Horizontal coordinate system</a></li> <li><a href="/wiki/Horizontal_position_representation" title="Horizontal position representation">Horizontal position representation</a></li> <li><a href="/wiki/Local_tangent_plane" class="mw-redirect" title="Local tangent plane">Local tangent plane</a></li> <li><a href="/wiki/Northing_and_easting" class="mw-redirect" title="Northing and easting">Northing and easting</a></li> <li><a href="/w/index.php?title=Tilted_plane&action=edit&redlink=1" class="new" title="Tilted plane (page does not exist)">Tilted plane</a></li> <li><a href="/wiki/Transverse_plane" title="Transverse plane">Transverse plane</a></li> <li><a href="/wiki/Vertical_and_horizontal_(radio_propagation)" title="Vertical and horizontal (radio propagation)">Vertical and horizontal (radio propagation)</a></li> <li><a href="/wiki/Vertical_circle" title="Vertical circle">Vertical circle</a></li> <li><a href="/wiki/Vertical_position" title="Vertical position">Vertical position</a></li> <li><a href="/wiki/Zenith" title="Zenith">Zenith</a> <ul><li><a href="/wiki/Zenith_angle" class="mw-redirect" title="Zenith angle">Zenith angle</a> (vertical angle)</li></ul></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References_and_notes">References and notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertical_and_horizontal&action=edit&section=14" title="Edit section: References and notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><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">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHofmann-WellenhofMoritz2006" class="citation book cs1">Hofmann-Wellenhof, B.; Moritz, H. (2006). <i>Physical Geodesy</i> (2nd ed.). Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-211-33544-4" title="Special:BookSources/978-3-211-33544-4"><bdi>978-3-211-33544-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physical+Geodesy&rft.edition=2nd&rft.pub=Springer&rft.date=2006&rft.isbn=978-3-211-33544-4&rft.aulast=Hofmann-Wellenhof&rft.aufirst=B.&rft.au=Moritz%2C+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertical+and+horizontal" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReference-OED-horizontal" class="citation encyclopaedia cs1"><span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://www.oed.com/search/dictionary/?q=horizontal">"horizontal"</a></span>. <i><a href="/wiki/Oxford_English_Dictionary" title="Oxford English Dictionary">Oxford English Dictionary</a></i> (Online ed.). <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=horizontal&rft.btitle=Oxford+English+Dictionary&rft.edition=Online&rft.pub=Oxford+University+Press&rft_id=https%3A%2F%2Fwww.oed.com%2Fsearch%2Fdictionary%2F%3Fq%3Dhorizontal&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertical+and+horizontal" class="Z3988"></span> <span style="font-size:0.95em; font-size:95%; color: var( --color-subtle, #555 )">(Subscription or <a rel="nofollow" class="external text" href="https://www.oed.com/public/login/loggingin#withyourlibrary">participating institution membership</a> required.)</span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReference-OED-vertical" class="citation encyclopaedia cs1"><span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://www.oed.com/search/dictionary/?q=vertical">"vertical"</a></span>. <i><a href="/wiki/Oxford_English_Dictionary" title="Oxford English Dictionary">Oxford English Dictionary</a></i> (Online ed.). <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=vertical&rft.btitle=Oxford+English+Dictionary&rft.edition=Online&rft.pub=Oxford+University+Press&rft_id=https%3A%2F%2Fwww.oed.com%2Fsearch%2Fdictionary%2F%3Fq%3Dvertical&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertical+and+horizontal" class="Z3988"></span> <span style="font-size:0.95em; font-size:95%; color: var( --color-subtle, #555 )">(Subscription or <a rel="nofollow" class="external text" href="https://www.oed.com/public/login/loggingin#withyourlibrary">participating institution membership</a> required.)</span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.laser-level.co.uk/products/stabila-rotary-lasers/stabila-lar200.html">"Laser Levels"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Laser+Levels&rft_id=http%3A%2F%2Fwww.laser-level.co.uk%2Fproducts%2Fstabila-rotary-lasers%2Fstabila-lar200.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertical+and+horizontal" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.physicsforums.com/threads/how-does-a-spirit-level-work.562730/">"How Does a Spirit Level Work?"</a>. <i>Physics Forums | Science Articles, Homework Help, Discussion</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Physics+Forums+%26%23124%3B+Science+Articles%2C+Homework+Help%2C+Discussion&rft.atitle=How+Does+a+Spirit+Level+Work%3F&rft_id=https%3A%2F%2Fwww.physicsforums.com%2Fthreads%2Fhow-does-a-spirit-level-work.562730%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertical+and+horizontal" class="Z3988"></span></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">Encyclopedia.com. In very long bridges, it may be necessary to take the Earth's curvature into account when designing the towers. For example, in the New York's Verrazano Narrows rivers, the towers, which are 700 ft (215 m) tall and stand 4260 ft (298 m) apart, are about 1.75 in (4.5 cm) farther apart at the top than they are at the bottom.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20170906224122/http://faculty.trinityvalleyschool.org/hoseltom/handouts/Chap%2014-Ax-Plumb-bob%20Deviation%20w%20Latitude-34PbyE.pdf">"Working in the Rotating Reference Frame of the Earth"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="http://faculty.trinityvalleyschool.org/hoseltom/handouts/Chap%2014-Ax-Plumb-bob%20Deviation%20w%20Latitude-34PbyE.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2017-09-06<span class="reference-accessdate">. Retrieved <span class="nowrap">2013-03-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Working+in+the+Rotating+Reference+Frame+of+the+Earth&rft_id=http%3A%2F%2Ffaculty.trinityvalleyschool.org%2Fhoseltom%2Fhandouts%2FChap%252014-Ax-Plumb-bob%2520Deviation%2520w%2520Latitude-34PbyE.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertical+and+horizontal" class="Z3988"></span></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">Such a deflection was measured by <a href="/wiki/Nevil_Maskelyne" title="Nevil Maskelyne">Nevil Maskelyne</a>. See Maskelyne, N. (1775). "An Account of Observations Made on the Mountain Schiehallion for Finding Its Attraction". Phil. Trans. Royal Soc. 65 (0): 500–542. doi:10.1098/rstl.1775.0050. <a href="/wiki/Charles_Hutton" title="Charles Hutton">Charles Hutton</a> used the observed value to determine the density of the Earth.</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCornish" class="citation web cs1">Cornish, Neil J. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150907090116/http://www.physics.montana.edu/faculty/cornish/lagrange.pdf">"The Lagrangian Points"</a> <span class="cs1-format">(PDF)</span>. <a href="/wiki/Montana_State_University_%E2%80%93_Bozeman" class="mw-redirect" title="Montana State University – Bozeman">Montana State University</a> – Department of Physics. Archived from <a rel="nofollow" class="external text" href="http://www.physics.montana.edu/faculty/cornish/lagrange.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 7 September 2015<span class="reference-accessdate">. Retrieved <span class="nowrap">29 July</span> 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Lagrangian+Points&rft.pub=Montana+State+University+%E2%80%93+Department+of+Physics&rft.aulast=Cornish&rft.aufirst=Neil+J.&rft_id=http%3A%2F%2Fwww.physics.montana.edu%2Ffaculty%2Fcornish%2Flagrange.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertical+and+horizontal" class="Z3988"></span></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">For an example of curved field lines, see <i>The gravitational field of a cube</i> by James M. Chappell, Mark J. Chappell, Azhar Iqbal, Derek Abbott for an example of curved gravitational field. arXiv:1206.3857 [physics.class-ph] (or arXiv:1206.3857v1 [physics.class-ph] for this version)</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">Salters Hornerns Advanced Physics Project, As Student Book, Edexcel Pearson, London, 2008, p. 48.</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">See Galileo's discussion of how bodies rise and fall under gravity on a moving ship in his <i>Dialogue Concerning the Two Chief World Systems</i>(trans. S. Drake). University of California Press, Berkeley, 1967, pp. 186–187.</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">See Harris Benson <i>University Physics</i>, New York 1991, page 268.</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.mathsteacher.com.au/year8/ch15_graphs/05_hor/ver.htm">"Horizontal and Vertical Lines"</a>. <i>www.mathsteacher.com.au</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.mathsteacher.com.au&rft.atitle=Horizontal+and+Vertical+Lines&rft_id=http%3A%2F%2Fwww.mathsteacher.com.au%2Fyear8%2Fch15_graphs%2F05_hor%2Fver.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVertical+and+horizontal" class="Z3988"></span></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"><a rel="nofollow" class="external text" href="http://www.icoachmath.com/math_dictionary/Horizontal_Axis.html">For a definition of "Horizontal axis" see Math Dictionary at www.icoachmath.com</a></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertical_and_horizontal&action=edit&section=15" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Brennan, David A.; Esplen, Matthew F.; Gray, Jeremy J. (1998), <i>Geometry</i>, Cambridge: Cambridge University Press, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-59787-0" title="Special:BookSources/0-521-59787-0">0-521-59787-0</a></li> <li>Murray R Spiegel, (1987), <i>Theory and Problems of Theoretical Mechanics</i>, Singapore, Mcgraw Hill's: Schaum's, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-084357-0" title="Special:BookSources/0-07-084357-0">0-07-084357-0</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Vertical_and_horizontal&action=edit&section=16" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=nA8CCqA-JlY">Video explanations of the terms</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Vertical_and_horizontal&oldid=1188964764">https://en.wikipedia.org/w/index.php?title=Vertical_and_horizontal&oldid=1188964764</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Geometry" title="Category:Geometry">Geometry</a></li><li><a href="/wiki/Category:Coordinate_systems" title="Category:Coordinate systems">Coordinate systems</a></li><li><a href="/wiki/Category:Geodesy" title="Category:Geodesy">Geodesy</a></li><li><a href="/wiki/Category:Surveying" title="Category:Surveying">Surveying</a></li><li><a href="/wiki/Category:Orientation_(geometry)" title="Category:Orientation (geometry)">Orientation (geometry)</a></li><li><a href="/wiki/Category:Position" title="Category:Position">Position</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Articles_containing_Latin-language_text" title="Category:Articles containing Latin-language text">Articles containing Latin-language text</a></li><li><a href="/wiki/Category:Articles_containing_Greek-language_text" title="Category:Articles containing Greek-language text">Articles containing Greek-language text</a></li><li><a href="/wiki/Category:Wikipedia_articles_with_style_issues_from_July_2018" title="Category:Wikipedia articles with style issues from July 2018">Wikipedia articles with style issues from July 2018</a></li><li><a href="/wiki/Category:All_articles_with_style_issues" title="Category:All articles with style issues">All articles with style issues</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_July_2018" title="Category:Articles with unsourced statements from July 2018">Articles with unsourced statements from July 2018</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 8 December 2023, at 20:45<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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