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<title>Cross section (geometry)</title>

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

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Cross_section_(drawing)" class="mw-redirect" title="Cross section (drawing)">cross section (drawing)</a>.</div>

<p>In <a href="Geometry" title="Geometry">geometry</a> and <a href="Science" title="Science">science</a>, a <b>cross section</b> is the non-empty <a href="Intersection_(set_theory)" title="Intersection (set theory)">intersection</a> of a solid body in <a href="Three-dimensional_space" title="Three-dimensional space">three-dimensional space</a> with a <a href="Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a>, or the analog in higher-<a href="Dimensional" class="mw-redirect" title="Dimensional">dimensional</a> spaces. Cutting an object into slices creates many parallel cross-sections. The boundary of a cross-section in three-dimensional space that is parallel to two of the <a href="Cartesian_coordinate_system" title="Cartesian coordinate system">axes</a>, that is, parallel to the plane determined by these axes, is sometimes referred to as a <a href="Contour_line" title="Contour line">contour line</a>; for example, if a plane cuts through mountains of a <a href="Raised-relief_map" title="Raised-relief map">raised-relief map</a> parallel to the ground, the result is a contour line in two-dimensional space showing points on the surface of the mountains of equal <a href="Elevation" title="Elevation">elevation</a>.

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="Graphical_projection" class="mw-redirect" title="Graphical projection">Graphical projection</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"></span></td></tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="Planar_projection" title="Planar projection">Planar projections</a></div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left">

<ul><li><a href="Parallel_projection" title="Parallel projection">Parallel projection</a>

<ul><li><a href="Orthographic_projection" title="Orthographic projection">Orthographic projection</a>

<ul><li><a href="Isometric_projection" title="Isometric projection">Isometric projection</a></li></ul></li>

<li><a href="Oblique_projection" title="Oblique projection">Oblique projection</a></li></ul></li>

<li><a href="Perspective_(graphical)" title="Perspective (graphical)">Perspective projection</a>

<ul><li><a href="Curvilinear_perspective" title="Curvilinear perspective">Curvilinear perspective</a></li>

<li><a href="Reverse_perspective" title="Reverse perspective">Reverse perspective</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="color: var(--color-base)">Views</div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left">

<li><a href="Cutaway_drawing" title="Cutaway drawing">Cutaway drawing</a></li>

<li><a href="Exploded-view_drawing" title="Exploded-view drawing">Exploded-view drawing</a></li>

<li><a href="Fisheye_lens" title="Fisheye lens">Fisheye lens</a></li>

<li><a href="Multiview_orthographic_projection" title="Multiview orthographic projection">Multiviews</a></li>

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

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Topics</div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left">

<ul><li><a href="3D_projection" title="3D projection">3D projection</a></li>

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

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

<li><a href="Computer_graphics" title="Computer graphics">Computer graphics</a></li>

<li><a href="Computer-aided_design" title="Computer-aided design">Computer-aided design</a></li>

<li><a href="Descriptive_geometry" title="Descriptive geometry">Descriptive geometry</a></li>

<li><a href="Engineering_drawing" title="Engineering drawing">Engineering drawing</a></li>

<li><a href="Map_projection" title="Map projection">Map projection</a></li>

<li><a href="Picture_plane" title="Picture plane">Picture plane</a></li>

<li><a href="Plan_(drawing)" title="Plan (drawing)">Plan (drawing)</a></li>

<li><a href="Projection_(linear_algebra)" title="Projection (linear algebra)">Projection (linear algebra)</a></li>

<li><a href="Projection_plane" title="Projection plane">Projection plane</a></li>

<li><a href="Projective_geometry" title="Projective geometry">Projective geometry</a></li>

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

<li><a href="Technical_drawing" title="Technical drawing">Technical drawing</a></li>

<li><a href="True_length" title="True length">True length</a></li>

<li><a href="Vanishing_point" title="Vanishing point">Vanishing point</a></li>

<li><a href="Video_game_graphics" title="Video game graphics">Video game graphics</a></li>

<li><a href="Viewing_frustum" title="Viewing frustum">Viewing frustum</a></li></ul></div></div></td>

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<p>In <a href="Technical_drawing" title="Technical drawing">technical drawing</a> a cross-section, being a <a href="Planar_projection" title="Planar projection">projection</a> of an object onto a plane that intersects it, is a common tool used to depict the internal arrangement of a 3-dimensional object in two dimensions. It is traditionally <a href="Crosshatch" class="mw-redirect" title="Crosshatch">crosshatched</a> with the style of crosshatching often indicating the types of materials being used.

</p><p>With <a href="Computed_axial_tomography" class="mw-redirect" title="Computed axial tomography">computed axial tomography</a>, computers can construct cross-sections from <a href="X-ray" title="X-ray">x-ray</a> data.

</p>

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

<p>If a plane intersects a solid (a 3-dimensional object), then the region common to the plane and the solid is called a <b>cross-section</b> of the solid.<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> A plane containing a cross-section of the solid may be referred to as a <i>cutting plane</i>.

</p><p>The shape of the cross-section of a solid may depend upon the orientation of the cutting plane to the solid. For instance, while all the cross-sections of a ball are disks,<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 cross-sections of a cube depend on how the cutting plane is related to the cube. If the cutting plane is perpendicular to a line joining the centers of two opposite faces of the cube, the cross-section will be a square, however, if the cutting plane is perpendicular to a diagonal of the cube joining opposite vertices, the cross-section can be either a point, a triangle or a hexagon.

</p>

<div class="mw-heading mw-heading3"><h3 id="Plane_sections">Plane sections</h3></div>

<p>A related concept is that of a <b>plane section</b>, which is the curve of intersection of a plane with a <i>surface</i>.<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> Thus, a plane section is the boundary of a cross-section of a solid in a cutting plane.

</p><p>If a surface in a three-dimensional space is defined by a function of two variables, i.e., <span class="texhtml"><i>z</i> = <i>f</i>(<i>x</i>, <i>y</i>)</span>, the plane sections by cutting planes that are parallel to a coordinate plane (a plane determined by two coordinate axes) are called <b>level curves</b> or <b>isolines</b>.<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>

More specifically, cutting planes with equations of the form <span class="texhtml"><i>z</i> = <i>k</i></span> (planes parallel to the <span class="texhtml mvar" style="font-style:italic;">xy</span>-plane) produce plane sections that are often called <b>contour lines</b> in application areas.

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<div class="mw-heading mw-heading2"><h2 id="Mathematical_examples_of_cross_sections_and_plane_sections">Mathematical examples of cross sections and plane sections</h2></div>

<p>A cross section of a <a href="Polyhedron" title="Polyhedron">polyhedron</a> is a <a href="Polygon" title="Polygon">polygon</a>.

</p><p>The <a href="Conic_sections" class="mw-redirect" title="Conic sections">conic sections</a> – <a href="Circle" title="Circle">circles</a>, <a href="Ellipse" title="Ellipse">ellipses</a>, <a href="Parabola" title="Parabola">parabolas</a>, and <a href="Hyperbolas" class="mw-redirect" title="Hyperbolas">hyperbolas</a> – are plane sections of a <a href="Cone_(geometry)" class="mw-redirect" title="Cone (geometry)">cone</a> with the cutting planes at various different angles, as seen in the diagram at left.

</p><p>Any cross-section passing through the center of an <a href="Ellipsoid" title="Ellipsoid">ellipsoid</a> forms an elliptic region, while the corresponding plane sections are ellipses on its surface. These degenerate to disks and circles, respectively, when the cutting planes are <a href="Perpendicular" title="Perpendicular">perpendicular</a> to a symmetry axis. In more generality, the plane sections of a <a href="Quadric" title="Quadric">quadric</a> are conic sections.<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>

<p>A cross-section of a solid right circular cylinder extending between two bases is a <a href="Disk_(mathematics)" title="Disk (mathematics)">disk</a> if the cross-section is parallel to the cylinder's base, or an elliptic region (see diagram at right) if it is neither parallel nor perpendicular to the base. If the cutting plane is perpendicular to the base it consists of a <a href="Rectangle" title="Rectangle">rectangle</a> (not shown) unless it is just <a href="Tangent_line" class="mw-redirect" title="Tangent line">tangent</a> to the cylinder, in which case it is a single <a href="Line_segment" title="Line segment">line segment</a>.

</p><p>The term cylinder can also mean the lateral surface of a solid cylinder (see <a href="Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">cylinder (geometry)</a>). If a cylinder is used in this sense, the above paragraph would read as follows: A plane section of a right circular cylinder of finite length<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> is a <a href="Circle" title="Circle">circle</a> if the cutting plane is perpendicular to the cylinder's axis of symmetry, or an ellipse if it is neither parallel nor perpendicular to that axis. If the cutting plane is parallel to the axis the plane section consists of a pair of parallel line segments unless the cutting plane is tangent to the cylinder, in which case, the plane section is a single line segment.

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</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:254px;max-width:254px"><div class="trow"><div class="tsingle" style="width:252px;max-width:252px"><div class="thumbimage"><span typeof="mw:File"></span></div><div class="thumbcaption">A graph of <span class="texhtml"><i>z</i> = <i>x</i><sup>2</sup> + <i>xy</i> + <i>y</i><sup>2</sup></span>. For the partial derivative at <span class="nowrap">(1, 1, 3)</span> that leaves <span class="texhtml"><i>y</i></span> constant, the corresponding <a href="Tangent" title="Tangent">tangent</a> line is parallel to the <span class="texhtml"><i>xz</i></span>-plane.</div></div></div><div class="trow"><div class="tsingle" style="width:252px;max-width:252px"><div class="thumbimage"><span typeof="mw:File"></span></div><div class="thumbcaption">A plane section of the above graph showing the level curve in the <span class="texhtml"><i>xz</i></span>-plane at <span class="texhtml"><i>y</i>= 1</span></div></div></div></div></div>

<p>A plane section can be used to visualize the <a href="Partial_derivative" title="Partial derivative">partial derivative</a> of a function with respect to one of its arguments, as shown. Suppose <span class="texhtml"><i>z</i> = <i>f</i>(<i>x</i>, <i>y</i>)</span>. In taking the partial derivative of <span class="texhtml"><i>f</i>(<i>x</i>, <i>y</i>)</span> with respect to <span class="texhtml"><i>x</i></span>, one can take a plane section of the function <span class="texhtml"><i>f</i></span> at a fixed value of <span class="texhtml"><i>y</i></span> to plot the level curve of <span class="texhtml"><i>z</i></span> solely against <span class="texhtml"><i>x</i></span>; then the partial derivative with respect to <span class="texhtml"><i>x</i></span> is the slope of the resulting two-dimensional graph.

</p>

<div class="mw-heading mw-heading3"><h3 id="In_related_subjects">In related subjects</h3></div>

<p>A plane section of a <a href="Probability_density_function#Densities_associated_with_multiple_variables" title="Probability density function">probability density function of two random variables</a> in which the cutting plane is at a fixed value of one of the variables is a <a href="Conditional_probability_distribution" title="Conditional probability distribution">conditional density function</a> of the other variable (conditional on the fixed value defining the plane section). If instead the plane section is taken for a fixed value of the density, the result is an iso-density contour. For the <a href="Normal_distribution" title="Normal distribution">normal distribution</a>, these contours are ellipses.

</p><p>In <a href="Economics" title="Economics">economics</a>, a <a href="Production_function" title="Production function">production function</a> <span class="texhtml"><i>f</i>(<i>x</i>, <i>y</i>)</span> specifies the output that can be produced by various quantities <span class="texhtml"><i>x</i></span> and <span class="texhtml"><i>y</i></span> of inputs, typically labor and physical capital. The production function of a firm or a society can be plotted in three-dimensional space. If a plane section is taken parallel to the <span class="texhtml"><i>xy</i></span>-plane, the result is an <a href="Isoquant" title="Isoquant">isoquant</a> showing the various combinations of labor and capital usage that would result in the level of output given by the height of the plane section. Alternatively, if a plane section of the production function is taken at a fixed level of <span class="texhtml"><i>y</i></span>—that is, parallel to the <span class="texhtml"><i>xz</i></span>-plane—then the result is a two-dimensional graph showing how much output can be produced at each of various values of usage <span class="texhtml"><i>x</i></span> of one input combined with the fixed value of the other input <span class="texhtml"><i>y</i></span>.

</p><p>Also in economics, a <a href="Utility#Cardinal_and_ordinal_utility" title="Utility">cardinal or ordinal utility function</a> <span class="texhtml"><i>u</i>(<i>w</i>, <i>v</i>)</span> gives the degree of satisfaction of a consumer obtained by consuming quantities <span class="texhtml"><i>w</i></span> and <span class="texhtml"><i> v</i></span> of two goods. If a plane section of the utility function is taken at a given height (level of utility), the two-dimensional result is an <a href="Indifference_curve" title="Indifference curve">indifference curve</a> showing various alternative combinations of consumed amounts <span class="texhtml"><i>w</i></span> and <span class="texhtml"><i>v</i></span> of the two goods all of which give the specified level of utility.

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<div class="mw-heading mw-heading2"><h2 id="Area_and_volume">Area and volume</h2></div>

<p><a href="Cavalieri's_principle" title="Cavalieri's principle">Cavalieri's principle</a> states that solids with corresponding cross-sections of equal areas have equal volumes.

</p><p>The cross-sectional area (<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'}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./98a12527148d6ed68adc91d9b419eb4b92d58ef6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:2.509ex;" alt="{\displaystyle A'}" loading="lazy"></span>) of an object when viewed from a particular angle is the total area of the orthographic projection of the object from that angle. For example, a cylinder of height <i>h</i> and radius <i>r</i> has <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'=\pi r^{2}}">

<msup>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A'=\pi r^{2}}</annotation>

</semantics>

</math></span><img src="./aede342545c4b5b5d1b76e82c33bed1805f52b9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.961ex; height:2.676ex;" alt="{\displaystyle A'=\pi r^{2}}" loading="lazy"></span> when viewed along its central axis, 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 A'=2rh}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A'=2rh}</annotation>

</semantics>

</math></span><img src="./314d07a4bcc7e02850f7f0740933659aa7bf0181.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.076ex; height:2.509ex;" alt="{\displaystyle A'=2rh}" loading="lazy"></span> when viewed from an orthogonal direction. A sphere of radius <i>r</i> has <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'=\pi r^{2}}">

<msup>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A'=\pi r^{2}}</annotation>

</semantics>

</math></span><img src="./aede342545c4b5b5d1b76e82c33bed1805f52b9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.961ex; height:2.676ex;" alt="{\displaystyle A'=\pi r^{2}}" loading="lazy"></span> when viewed from any angle. More generically, <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'}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</p>

<msup>

</msup>

</mrow>

</munder>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A'=\iint \limits _{\mathrm {top} }d\mathbf {A} \cdot \mathbf {\hat {r}} ,}</annotation>

</semantics>

</math></span><img src="./5ef7909768d5b6d5510aede11029d73533800baf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:16.231ex; height:7.509ex;" alt="{\displaystyle A'=\iint \limits _{\mathrm {top} }d\mathbf {A} \cdot \mathbf {\hat {r}} ,}" loading="lazy"></span></dd></dl>

<p>where <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 {\hat {r}} }">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {r}} }</annotation>

</semantics>

</math></span><img src="./7fe52dfe80c9a6604b3a46b24d65eb02c92c59e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.343ex;" alt="{\displaystyle \mathbf {\hat {r}} }" loading="lazy"></span> is the unit vector pointing along the viewing direction toward the viewer, <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 d\mathbf {A} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d\mathbf {A} }</annotation>

</semantics>

</math></span><img src="./f5fdd48cc46dd596e66ef7a4ad469cb909127f4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.235ex; height:2.176ex;" alt="{\displaystyle d\mathbf {A} }" loading="lazy"></span> is a surface element with an outward-pointing normal, and the integral is taken only over the top-most surface, that part of the surface that is "visible" from the perspective of the viewer. For a <a href="Convex_body" title="Convex body">convex body</a>, each ray through the object from the viewer's perspective crosses just two surfaces. For such objects, the integral may be taken over the entire surface (<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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span>) by taking the absolute value of the integrand (so that the "top" and "bottom" of the object do not subtract away, as would be required by the <a href="Divergence_Theorem" class="mw-redirect" title="Divergence Theorem">Divergence Theorem</a> applied to the constant vector field <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 {\hat {r}} }">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {r}} }</annotation>

</semantics>

</p>

<msup>

</msup>

<mfrac>

</mfrac>

</mrow>

</mrow>

</munder>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A'={\frac {1}{2}}\iint \limits _{A}|d\mathbf {A} \cdot \mathbf {\hat {r}} |}</annotation>

</semantics>

</math></span><img src="./503889b5bce4b5801e944dd54566a70842a477b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:19.264ex; height:7.343ex;" alt="{\displaystyle A'={\frac {1}{2}}\iint \limits _{A}|d\mathbf {A} \cdot \mathbf {\hat {r}} |}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="In_higher_dimensions">In higher dimensions</h2></div>

<p>In analogy with the cross-section of a solid, the cross-section of an <span class="texhtml mvar" style="font-style:italic;">n</span>-dimensional body in an <span class="texhtml mvar" style="font-style:italic;">n</span>-dimensional space is the non-empty intersection of the body with a hyperplane (an <span class="texhtml">(<i>n</i> − 1)</span>-dimensional subspace). This concept has sometimes been used to help visualize aspects of higher dimensional spaces.<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> For instance, if a <a href="Four-dimensional_space" title="Four-dimensional space">four-dimensional object</a> passed through our three-dimensional space, we would see a three-dimensional cross-section of the four-dimensional object. In particular, a 4-ball (hypersphere) passing through 3-space would appear as a 3-ball that increased to a maximum and then decreased in size during the transition. This dynamic object (from the point of view of 3-space) is a sequence of cross-sections of the 4-ball.

</p>

<div class="mw-heading mw-heading2"><h2 id="Examples_in_science">Examples in science</h2></div>

<p>In <a href="Geology" title="Geology">geology</a>, the structure of the interior of a <a href="Planet" title="Planet">planet</a> is often illustrated using a diagram of a cross-section of the planet that passes through the planet's center, as in the cross-section of <a href="Earth" title="Earth">Earth</a> at right.

</p><p>Cross-sections are often used in <a href="Anatomy" title="Anatomy">anatomy</a> to illustrate the inner structure of an organ, as shown at the left.

</p><p>A cross-section of a <a href="Tree" title="Tree">tree</a> trunk, as shown at left, reveals <a href="Tree_ring" class="mw-redirect" title="Tree ring">growth rings</a> that can be used to find the age of the tree and the temporal properties of its environment.

</p>

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<div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Cross_sections" class="extiw external" title="commons:Category:Cross sections">Cross sections</a></span>.</div></div>

</div>

<ul><li><a href="Descriptive_geometry" title="Descriptive geometry">Descriptive geometry</a></li>

<li><a href="Exploded-view_drawing" title="Exploded-view drawing">Exploded-view drawing</a></li>

<li><a href="Graphical_projection" class="mw-redirect" title="Graphical projection">Graphical projection</a></li>

<li><a href="Plans_(drawings)" class="mw-redirect" title="Plans (drawings)">Plans (drawings)</a></li>

<li><a href="Profile_gauge" title="Profile gauge">Profile gauge</a></li>

<li><a href="Hatching#Representation_of_materials" title="Hatching">Section lining</a>; representation of materials</li>

<li><a href="Secant_plane" title="Secant plane">Secant plane</a></li></ul>

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

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFSwokowski1983">Swokowski 1983</a>, p. 296</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">in more technical language, the cross-sections of a 3-ball are 2-balls</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"><a href="#CITEREFAlbert2016">Albert 2016</a>, p. 38</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"><a href="#CITEREFSwokowski1983">Swokowski 1983</a>, p. 716</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"><a href="#CITEREFAlbert2016">Albert 2016</a>, p. 117</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">these cylinders are <i>open</i>, they do not contain their bases</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"><a href="#CITEREFStewart2001">Stewart 2001</a>, p. 59</span>

</li>

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

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

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</style><cite id="CITEREFAlbert2016" class="citation cs2">Albert, Abraham Adrian (2016) [1949], <i>Solid Analytic Geometry</i>, Dover, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-81026-3</bdi></cite></li>

<li><cite id="CITEREFStewart2001" class="citation cs2">Stewart, Ian (2001), <i>Flatterland / like flatland, only more so</i>, Persus Publishing, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7382-0675-X</bdi></cite></li>

<li><cite id="CITEREFSwokowski1983" class="citation cs2">Swokowski, Earl W. (1983), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/calculuswithanal00swok"><i>Calculus with analytic geometry</i></a></span> (Alternate ed.), Prindle, Weber & Schmidt, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-87150-341-7</bdi></cite></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2024-12-16" href="https://en.wikipedia.org/wiki/?title=Cross_section_(geometry)&oldid=1263389852">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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