Cross section (geometry)

For crosssections in architecture and engineering, see Multiview orthographic projection#Crosssection.
Part of a series on: Graphical projection  Parallel projection
 Orthographic projection
 Multiviews
 Plan, or floor plan
 Section
 Elevation
 Auxiliary
 Axonometric projection (i.e. pictorials)
 Isometric projection
 Dimetric projection
 Trimetric projection
 Multiviews
 Oblique projection
 Cavalier projection
 Cabinet projection
 Orthographic projection
 Perspective projection
 Linear perspective
 Onepoint perspective
 Twopoint perspective
 Threepoint perspective
 Zeropoint perspective
 Curvilinear perspective
 Reverse perspective
 Linear perspective
Views Bird'seye view/Aerial view
 Detail view
 3/4 perspective
 Cutaway drawing
 Exploded view drawing
 Fisheye
 Fixed 3D
 Panorama
 Topdown perspective
 Worm'seye view
 Zoom
In geometry, a crosssection is the intersection of a figure in 2dimensional space with a line, or of a body in 3dimensional space with a plane, etc. More plainly, when cutting an object into slices one gets many parallel crosssections.
Cavalieri's principle states that solids with corresponding crosssections of equal areas have equal volumes.
The crosssectional 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' = πr^{2} when viewed along its central axis, and A' = 3πrh when viewed from an orthogonal direction. A sphere of radius r has A' = πr^{2} when viewed from any angle. More generically, A' can be calculated by evaluating the following surface integral:
where is a unit vector pointing along the viewing direction toward the viewer, is a surface element with outwardpointing normal, and the integral is taken only over the topmost 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 ) and dividing by two:
See also
 Descriptive geometry
 Exploded view drawing
 Graphical projection
 Plans (drawings)
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