![Seal_mechanical_compression](https://upload.wikimedia.org/wikipedia/commons/8/86/Seal_mechanical_compression.png)
In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and
science
Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earliest archeological evidence for ...
, a cross section is the non-empty
intersection of a solid body in
three-dimensional space
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position (geometry), position of an element (i.e., Point (m ...
with a
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
, or the analog in higher-
dimensional
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
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
axes
Axes, plural of '' axe'' and of '' axis'', may refer to
* ''Axes'' (album), a 2005 rock album by the British band Electrelane
* a possibly still empty plot (graphics)
See also
* Axess (disambiguation)
*Axxess (disambiguation) Axxess may refer to ...
, that is, parallel to the plane determined by these axes, is sometimes referred to as a
contour line
A contour line (also isoline, isopleth, or isarithm) of a function of two variables is a curve along which the function has a constant value, so that the curve joins points of equal value. It is a plane section of the three-dimensional grap ...
; for example, if a plane cuts through mountains of a
raised-relief map
A raised-relief map, terrain model or embossed map is a three-dimensional representation, usually of terrain, materialized as a physical artifact. When representing terrain, the vertical dimension is usually exaggerated by a factor between fiv ...
parallel to the ground, the result is a contour line in two-dimensional space showing points on the surface of the mountains of equal
elevation
The elevation of a geographic location is its height above or below a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational surface (see Geodetic datum § Vert ...
.
In
technical drawing
Technical drawing, drafting or drawing, is the act and Academic discipline, discipline of composing Plan (drawing), drawings that Visual communication, visually communicate how something functions or is constructed.
Technical drawing is essent ...
a cross-section, being a
projection
Projection, projections or projective may refer to:
Physics
* Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction
* The display of images by a projector
Optics, graphic ...
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
crosshatch
Hatching (french: hachure) is an artistic technique used to create tonal or shading effects by drawing (or painting or scribing) closely spaced parallel lines. (It is also used in monochromatic representations of heraldry to indicate what the t ...
ed with the style of crosshatching often indicating the types of materials being used.
With
computed axial tomography, computers can construct cross-sections from
x-ray
An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10 picometers to 10 nanometers, corresponding to frequencies in the range 30&nb ...
data.
Definition
If a plane intersects a solid (a 3-dimensional object), then the region common to the plane and the solid is called a cross-section of the solid. A plane containing a cross-section of the solid may be referred to as a ''cutting plane''.
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, 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.
Plane sections
A related concept is that of a plane section, which is the curve of intersection of a plane with a ''surface''. Thus, a plane section is the boundary of a cross-section of a solid in a cutting plane.
If a surface in a three-dimensional space is defined by a function of two variables, i.e., , the plane sections by cutting planes that are parallel to a coordinate plane (a plane determined by two coordinate axes) are called level curves or isolines.
More specifically, cutting planes with equations of the form (planes parallel to the -plane) produce plane sections that are often called contour lines in application areas.
Mathematical examples of cross sections and plane sections
![Conic Sections](https://upload.wikimedia.org/wikipedia/commons/1/11/Conic_Sections.svg)
A cross section of a
polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on th ...
is a
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
.
The
conic sections
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special ...
–
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
s,
ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
s,
parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descript ...
s, and
hyperbolas
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...
– are plane sections of a
cone
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines con ...
with the cutting planes at various different angles, as seen in the diagram at left.
Any cross-section passing through the center of an
ellipsoid
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface; that is, a surface that may be defined as the ...
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
perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
to a symmetry axis. In more generality, the plane sections of a
quadric
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is de ...
are conic sections.
A cross-section of a solid right circular cylinder extending between two bases is a
disk
Disk or disc may refer to:
* Disk (mathematics), a geometric shape
* Disk storage
Music
* Disc (band), an American experimental music band
* ''Disk'' (album), a 1995 EP by Moby
Other uses
* Disk (functional analysis), a subset of a vector sp ...
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
rectangle
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containi ...
(not shown) unless it is just
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
to the cylinder, in which case it is a single
line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
.
The term cylinder can also mean the lateral surface of a solid cylinder (see
cylinder (geometry)
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infini ...
). 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 is a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
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.
A plane section can be used to visualize the
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
of a function with respect to one of its arguments, as shown. Suppose . In taking the partial derivative of with respect to , one can take a plane section of the function at a fixed value of to plot the level curve of solely against ; then the partial derivative with respect to is the slope of the resulting two-dimensional graph.
In related subjects
A plane section of a
probability density function of two random variables in which the cutting plane is at a fixed value of one of the variables is a
conditional density function 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
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
, these contours are ellipses.
In
economics
Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and intera ...
, a
production function
In economics, a production function gives the technological relation between quantities of physical inputs and quantities of output of goods. The production function is one of the key concepts of mainstream neoclassical theories, used to define ...
specifies the output that can be produced by various quantities and 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 -plane, the result is an
isoquant
An isoquant (derived from quantity and the Greek word iso, meaning equal), in microeconomics, is a contour line drawn through the set of points at which the same quantity of output is produced while changing the quantities of two or more inputs. T ...
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 —that is, parallel to the -plane—then the result is a two-dimensional graph showing how much output can be produced at each of various values of usage of one input combined with the fixed value of the other input .
Also in economics, a
cardinal or ordinal utility function gives the degree of satisfaction of a consumer obtained by consuming quantities and 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
indifference curve
In economics, an indifference curve connects points on a graph representing different quantities of two goods, points between which a consumer is ''indifferent''. That is, any combinations of two products indicated by the curve will provide the c ...
showing various alternative combinations of consumed amounts and of the two goods all of which give the specified level of utility.
Area and volume
Cavalieri's principle
In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows:
* 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that pl ...
states that solids with corresponding cross-sections of equal areas have equal volumes.
The cross-sectional area (
) 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
when viewed along its central axis, and
when viewed from an orthogonal direction. A sphere of radius ''r'' has
when viewed from any angle. More generically,
can be calculated by evaluating the following surface integral:
:
where
is the unit vector pointing along the viewing direction toward the viewer,
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
convex body
In mathematics, a convex body in n-dimensional Euclidean space \R^n is a compact convex set with non-empty interior.
A convex body K is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point x lies in ...
, 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 (
) 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
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
applied to the constant vector field
) and dividing by two:
:
In higher dimensions
In analogy with the cross-section of a solid, the cross-section of an -dimensional body in an -dimensional space is the non-empty intersection of the body with a hyperplane (an -dimensional subspace). This concept has sometimes been used to help visualize aspects of higher dimensional spaces.
For instance, if a
four-dimensional object 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.
Examples in science
![Pinus taeda crossx7358](https://upload.wikimedia.org/wikipedia/commons/2/22/Pinus_taeda_crossx7358.JPG)
In
geology
Geology () is a branch of natural science concerned with Earth and other astronomical objects, the features or rocks of which it is composed, and the processes by which they change over time. Modern geology significantly overlaps all other Ear ...
, the structure of the interior of a
planet
A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
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
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
at right.
Cross-sections are often used in
anatomy
Anatomy () is the branch of biology concerned with the study of the structure of organisms and their parts. Anatomy is a branch of natural science that deals with the structural organization of living things. It is an old science, having its ...
to illustrate the inner structure of an organ, as shown at the left.
A cross-section of a
tree
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...
trunk, as shown at left, reveals
growth rings
Dendrochronology (or tree-ring dating) is the scientific method of dating tree rings (also called growth rings) to the exact year they were formed. As well as dating them, this can give data for dendroclimatology, the study of climate and atmos ...
that can be used to find the age of the tree and the temporal properties of its environment.
See also
*
Descriptive geometry
*
Exploded-view drawing
*
Graphical projection
A 3D projection (or graphical projection) is a design technique used to display a three-dimensional (3D) object on a two-dimensional (2D) surface. These projections rely on visual perspective and aspect analysis to project a complex object fo ...
*
Plans (drawings)
*
Profile gauge
A profile gauge or contour gauge is a tool for recording the cross-sectional shape of a surface. Contour gauges consist of a set of steel or plastic pins that are set tightly against one another in a frame which keeps them in the same plane an ...
*
Section lining; representation of materials
Notes
References
*
*
* {{citation, first=Earl W., last=Swokowski, title=Calculus with analytic geometry, edition=Alternate, year=1983, publisher=Prindle, Weber & Schmidt, isbn=0-87150-341-7, url-access=registration, url=https://archive.org/details/calculuswithanal00swok
Infographics
Elementary geometry
Technical drawing
Methods of representation