Cross section (geometry)

For cross-sections in architecture and engineering, see Multiview orthographic projection#Cross-section.

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In geometry, a cross-section is the intersection of a figure in 2-dimensional space with a line, or of a body in 3-dimensional space with a plane, etc. More plainly, when cutting an object into slices one gets many parallel cross-sections.

Cavalieri's principle states that solids with corresponding cross-sections of equal areas have equal volumes.

The cross-sectional area (A') 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 h and radius r has A' = πr² when viewed along its central axis, and A' = 3πrh when viewed from an orthogonal direction. A sphere of radius r has A' = πr² when viewed from any angle. More generically, A' can be calculated by evaluating the following surface integral:

$A' = \iint \limits_\mathrm{top} d\mathbf{A} \cdot \mathbf{\hat{r}},$

where $\mathbf{\hat{r}}$ is a unit vector pointing along the viewing direction toward the viewer, $d\mathbf{A}$ is a surface element with 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 convex body, 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 (A) 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 Divergence Theorem applied to the constant vector field $\mathbf{\hat{r}}$) and dividing by two:

$A' = \frac{1}{2} \iint \limits_A | d\mathbf{A} \cdot \mathbf{\hat{r}}|$

See also

• Descriptive geometry
• Exploded view drawing
• Graphical projection
• Plans (drawings)

This geometry-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Geometry stubs Infographics Elementary geometry Technical drawing Methods of representation Wikimedia Foundation. 2010. Игры ⚽ Нужен реферат? Percival Drayton Beattie Look at other dictionaries: Cross section — may refer to: Cross section (geometry), the intersection of a 3 dimensional body with a plane Cross sectional views in architecture engineering Radar cross section, the unit of measure of how detectable an object is with a radar Cross section… …   Wikipedia Optical cross section — or OCS is a value which describes the maximum amount of optical flux reflected back to the source. The standard unit of measurement is . It is dependent on the geometry and the reflectivity at a particular wavelength of an object. Optical cross… …   Wikipedia Section (mathematics) — In mathematics, section can refer to: * Section (category theory), also in homological algebra, and including: ** Section (fiber bundle), in topology * Section (group theory), a subquotient of a group * Cross section (geometry), the intersection… …   Wikipedia Cross-flow turbine — Cross flow turbine. Image credit; European Communities, Layman s Guidebook (on how to develop a small hydroelectric site) A cross flow turbine, Banki Michell turbine, or Ossberger turbine[1] is a water turbine developed by the Australian Anthony… …   Wikipedia Cross-bedding — Crossbedding of sandstone near Mt. Carmel road, Zion Canyon, indicating wind action and sand dune formation prior to formation of rock (NPS photo by George A. Grant, 1929) …   Wikipedia Section (fiber bundle) — In the mathematical field of topology, a section (or cross section) of a fiber bundle, pi; : E rarr; B , over a topological space, B , is a continuous map, f : B rarr; E , such that pi; ( f ( x ))= x for all x in B .A section is a certain… …   Wikipedia Cross product — This article is about the cross product of two vectors in three dimensional Euclidean space. For other uses, see Cross product (disambiguation). In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on… …   Wikipedia section — n. division of a newspaper 1) the business; classified; news; sports; travel section plane figure (geometry) 2) a conical; cross; vertical section group of instruments 3) a brass; percussion; string; woodwind section 4) in a section (to play in… …   Combinatory dictionary Cylinder (geometry) — A right circular cylinder A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two… …   Wikipedia List of geometry topics — This is list of geometry topics, by Wikipedia page.*Geometric shape covers standard terms for plane shapes *List of mathematical shapes covers all dimensions *List of differential geometry topics *List of geometers *See also list of curves, list… …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”