Cross section (geometry) Information & Cross section (geometry) Links at HealthHaven.com

In geometry, a cross-section is the intersection of a body 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.

A cross-section, or section is also an orthographic projection of a 3-dimensional object from the position of a plane through the object. A floor plan is a section viewed from the top. In such views, the portion of the object in front of the plane is omitted to reveal what lies beyond. In the case of a floor plan, the roof and upper portion of the walls may be omitted. Elevations or roof plans are orthographic projections, but they are not sections as their viewing plane is outside of the object.

With computed axial tomography, computers construct cross-sections from x-ray data.

A cross-section is a common method of depicting the internal arrangement of a 3-dimensional object in two dimensions. It is often used in technical drawing and is traditionally crosshatched. The style of crosshatching indicates the type of material the section passes through.

A 3-D view of a beverage-can stove with a cross-section in yellow.

A 2-D cross-sectional view of a compression seal.

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 $A' = π r 2$ when viewed along its central axis, and $A' = 2 r h$ when viewed from an orthogonal direction. A sphere of radius r has $A' = π r 2$ 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:

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

[edit] See also

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