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Cross section (geometry)
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In geometry and science, a cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional 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, that is, parallel to the plane determined by these axes, is sometimes referred to as a contour line; for example, if a plane cuts through mountains of a raised-relief map 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.
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In technical drawing a cross-section, being a projection 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 crosshatched 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 data.
Definition
[edit]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.[1] 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,[2] 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
[edit]A related concept is that of a plane section, which is the curve of intersection of a plane with a surface.[3] 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., z = f(x, y), 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.[4] More specifically, cutting planes with equations of the form z = k (planes parallel to the xy-plane) produce plane sections that are often called contour lines in application areas.
Mathematical examples of cross sections and plane sections
[edit]
A cross section of a polyhedron is a polygon.
The conic sections – circles, ellipses, parabolas, and hyperbolas – are plane sections of a cone 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 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 to a symmetry axis. In more generality, the plane sections of a quadric are conic sections.[5]
A cross-section of a solid right circular cylinder extending between two bases is a disk 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 (not shown) unless it is just tangent to the cylinder, in which case it is a single line segment.
The term cylinder can also mean the lateral surface of a solid cylinder (see cylinder (geometry)). 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[6] is a circle 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 of a function with respect to one of its arguments, as shown. Suppose z = f(x, y). In taking the partial derivative of f(x, y) with respect to x, one can take a plane section of the function f at a fixed value of y to plot the level curve of z solely against x; then the partial derivative with respect to x is the slope of the resulting two-dimensional graph.
In related subjects
[edit]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, these contours are ellipses.
In economics, a production function f(x, y) specifies the output that can be produced by various quantities x and y 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 xy-plane, the result is an isoquant 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 y—that is, parallel to the xz-plane—then the result is a two-dimensional graph showing how much output can be produced at each of various values of usage x of one input combined with the fixed value of the other input y.
Also in economics, a cardinal or ordinal utility function u(w, v) gives the degree of satisfaction of a consumer obtained by consuming quantities w and v 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 showing various alternative combinations of consumed amounts w and v of the two goods all of which give the specified level of utility.
Area and volume
[edit]Cavalieri's principle 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, 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 applied to the constant vector field ) and dividing by two:
In higher dimensions
[edit]In analogy with the cross-section of a solid, the cross-section of an n-dimensional body in an n-dimensional space is the non-empty intersection of the body with a hyperplane (an (n − 1)-dimensional subspace). This concept has sometimes been used to help visualize aspects of higher dimensional spaces.[7] 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
[edit]
In geology, the structure of the interior of a planet 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 at right.
Cross-sections are often used in anatomy to illustrate the inner structure of an organ, as shown at the left.
A cross-section of a tree trunk, as shown at left, reveals growth rings that can be used to find the age of the tree and the temporal properties of its environment.
See also
[edit]
- Descriptive geometry
- Exploded-view drawing
- Graphical projection
- Plans (drawings)
- Profile gauge
- Section lining; representation of materials
- Secant plane
Notes
[edit]- ^ Swokowski 1983, p. 296
- ^ in more technical language, the cross-sections of a 3-ball are 2-balls
- ^ Albert 2016, p. 38
- ^ Swokowski 1983, p. 716
- ^ Albert 2016, p. 117
- ^ these cylinders are open, they do not contain their bases
- ^ Stewart 2001, p. 59
References
[edit]- Albert, Abraham Adrian (2016) [1949], Solid Analytic Geometry, Dover, ISBN 978-0-486-81026-3
- Stewart, Ian (2001), Flatterland / like flatland, only more so, Persus Publishing, ISBN 0-7382-0675-X
- Swokowski, Earl W. (1983), Calculus with analytic geometry (Alternate ed.), Prindle, Weber & Schmidt, ISBN 0-87150-341-7
Cross section (geometry)
View on GrokipediaFundamentals
Definition
In geometry, a cross section is the figure formed by the intersection of a plane with a three-dimensional solid, resulting in a two-dimensional plane figure that represents a slice through the object.[7] This intersection is non-empty and bounded by the curve where the plane meets the surface of the solid. More generally, the concept extends to higher-dimensional analogs, where a cross section is the intersection of a hyperplane with a higher-dimensional object, yielding a lower-dimensional figure.[8] The terms "section" and "cross section" are often used interchangeably in geometric contexts, though a section broadly refers to the act of cutting a solid along a plane, while a cross section specifically denotes the resulting two-dimensional shape exposed by that cut.[2] The notion of cross sections originates from the principles of descriptive geometry, systematized in the late 18th century by Gaspard Monge, a French mathematician who developed methods for accurately representing three-dimensional forms and their plane intersections in two dimensions to aid engineering and construction.[9] Monge's work laid the foundation for visualizing and constructing such sections without physical models. Cross sections rely on fundamental prerequisites from Euclidean geometry, particularly the concept of a plane as a flat surface that lies evenly with the straight lines drawn upon it, extending infinitely in all directions without curvature.[10] The intersection principle states that when such a plane cuts through a solid, it produces a closed curve or polygonal boundary defining the cross-sectional figure, assuming the plane does not merely touch the surface tangentially. This process conceptually represents slicing an object to reveal its internal geometry, akin to diagrammatic views that aid in understanding spatial relationships, though specific shapes depend on the object's form and the plane's orientation—details explored in plane sections.[7]Plane Sections
A plane section, or cross section, arises from the intersection of a plane with a solid figure in three-dimensional space, where the plane slices through the solid to reveal a two-dimensional figure defined by the common boundary points. The mechanics of this intersection depend fundamentally on the plane's orientation relative to the solid's surfaces and edges. When the plane is positioned to cut through the interior of the solid, it intersects the bounding faces or surfaces along lines, forming a closed curve or polygon that lies entirely within the plane; the specific shape emerges from how these intersection lines connect at vertices or along curves.[11][12] Plane sections are classified by the orientation of the cutting plane: parallel, perpendicular, or oblique. In parallel sections, the plane runs parallel to a principal face or generating direction of the solid, resulting in a uniform shape that mirrors the intersected feature across multiple such cuts, maintaining consistent dimensions and form. Perpendicular sections occur when the plane is normal to a reference axis or face, often producing sections with maximal extent in certain directions, such as right sections where the cut aligns orthogonally to the solid's height. Oblique sections involve a plane at an acute or obtuse angle to the solid's elements, leading to distorted or sheared shapes that elongate or foreshorten features compared to perpendicular cuts.[13][12] Geometrically, the cross section is invariably a plane figure—typically a polygon for polyhedral solids or a curve for curved surfaces—with its boundaries corresponding directly to the lines of intersection on the solid's faces; these edges or arcs connect at points where the plane meets edges or rulings of the solid. The figure's true size and shape are preserved only when viewed perpendicular to the section plane, as projections in other directions introduce foreshortening or distortion.[12] To construct and visualize plane sections, descriptive geometry employs orthographic projections and auxiliary views. In orthographic projection, the solid and plane are represented in multiple views (e.g., front, top, side), where intersection points are located by tracing projectors from the plane's lines to the solid's edges, revealing the section as a series of connected line segments. Auxiliary views, created parallel to the section plane, provide the true shape by eliminating perspective distortion, allowing accurate measurement of lengths and angles within the figure. These techniques, rooted in parallel projection principles, ensure the section's boundaries are delineated precisely without physical slicing.[13][12]Examples
Geometric Solids
Cross sections of geometric solids provide fundamental examples of how a plane intersects a three-dimensional figure to produce two-dimensional shapes. These intersections depend on the orientation of the plane relative to the solid's axes, bases, and vertices. Common solids such as prisms, cylinders, cones, spheres, and pyramids illustrate a variety of resulting shapes, from polygons to conic sections.[7] For prisms, which consist of two parallel polygonal bases connected by rectangular lateral faces in right prisms or parallelogram faces in oblique prisms, the cross section varies with the plane's angle. A plane parallel to the bases yields a cross section congruent to the base polygon, such as a rectangle for a rectangular prism or a triangle for a triangular prism. A plane perpendicular to the bases produces a rectangle for right prisms or a parallelogram for oblique prisms, reflecting the dimensions of the lateral faces. An oblique plane, neither parallel nor perpendicular, results in a parallelogram, as the intersection traces slanted lines across the lateral faces.[14][15] Cylinders, formed by two parallel circular bases connected by a curved surface, exhibit cross sections influenced by alignment with the axis. A plane perpendicular to the axis intersects the curved surface in a circle identical to the bases. An oblique plane, tilted relative to the axis, produces an ellipse, with the shape elongating as the tilt increases. A plane parallel to the axis, such as a longitudinal cut, yields a rectangle, capturing the height and diameter of the cylinder.[14] Cones, with a circular base and a single apex, generate the full family of conic sections depending on the plane's angle relative to the axis and nappes. A plane perpendicular to the axis, parallel to the base, produces a circle. A tilted plane intersecting a single nappe at an angle less steep than the cone's side yields an ellipse. A plane parallel to the cone's side generates a parabola. If the plane intersects both nappes, the cross section is a hyperbola. These outcomes arise from the double-napped cone structure, where the plane's orientation determines the curve's eccentricity.[16] Spheres, as perfectly symmetric solids, always yield circular cross sections regardless of plane orientation. The circle's radius decreases with the plane's distance from the center; planes through the center produce great circles with the sphere's full radius, while offset planes form smaller circles. This uniformity stems from the sphere's equidistant points from the center.[17] Pyramids, featuring a polygonal base and triangular faces meeting at an apex, produce cross sections based on the plane's position relative to the base and apex. A plane parallel to the base creates a smaller polygon similar to the base, such as a square for a square pyramid. Planes intersecting the lateral faces perpendicular to the base often yield triangles. Depending on the cut's location—such as slicing through the apex and base edges or across multiple faces—the cross section can be a triangle, quadrilateral, or other polygon.[14] The following table summarizes representative cross section shapes for these solids based on common plane orientations:| Solid | Plane Orientation | Cross Section Shape |
|---|---|---|
| Prism | Parallel to bases | Polygon (base shape) |
| Prism | Perpendicular to bases | Rectangle (right) or parallelogram (oblique) |
| Prism | Oblique | Parallelogram |
| Cylinder | Perpendicular to axis | Circle |
| Cylinder | Oblique to axis | Ellipse |
| Cylinder | Parallel to axis | Rectangle |
| Cone | Perpendicular to axis | Circle |
| Cone | Tilted, single nappe | Ellipse |
| Cone | Parallel to side | Parabola |
| Cone | Intersects both nappes | Hyperbola |
| Sphere | Any orientation | Circle |
| Pyramid | Parallel to base | Polygon (similar to base) |
| Pyramid | Perpendicular to base | Triangle (through apex) or trapezoid/polygon |
| Pyramid | Through apex and edges | Triangle or polygon |
In Related Mathematical Fields
In three-dimensional Euclidean space, the cross section of a space curve with a plane consists of the discrete points where the curve intersects the plane, determined by solving the parametric equations of the curve simultaneously with the plane equation.[18] For a generic plane not containing the curve, these intersections are finite points unless the curve lies entirely within the plane, in which case the cross section is the curve itself.[18] If the plane is tangent to the curve at a point, the intersection may include that point with multiplicity, reflecting local geometric contact.[18] Cross sections of surfaces extend this concept, where a plane intersects a surface in a curve. For ruled surfaces, such as the hyperboloid of one sheet, planes containing a ruling line yield straight line segments as cross sections, highlighting the surface's linear generators.[19] More generally, horizontal planes parallel to the xy-plane intersect the hyperboloid in ellipses, while vertical planes parallel to the xz- or yz-planes produce hyperbolas.[19] Quadric surfaces like the ellipsoid exhibit plane cross sections that are always ellipses, regardless of the plane's orientation, due to the surface's quadratic form preserving elliptical traces.[20] In algebraic geometry, cross sections of algebraic varieties with hyperplanes (affine or projective planes) reduce the dimension by one, yielding subvarieties whose intersections are governed by Bézout's theorem.[21] For plane curves—zero sets of polynomials in the projective plane—this theorem states that two curves of degrees and without common components intersect in exactly points, counted with multiplicity, providing a foundational count for cross-sectional intersections.[21] The multiplicity at an intersection point measures the order of contact, analogous to tangent or higher-order coincidences in classical geometry, and extends to higher-dimensional varieties where hyperplane sections reveal degrees and structures.[21] Differential geometry examines plane cross sections of curves and surfaces embedded in Euclidean space to study local properties like curvature. For surfaces, such sections are plane curves whose curvature is given by in parametric form, revealing intrinsic bending independent of the embedding.[18] In the context of manifolds restricted to Euclidean space, these planar cuts facilitate computation of Gaussian curvature through principal curvatures of the section curves, though the focus remains on extrinsic geometry via traces in coordinate planes.[18] Historically, Archimedes employed cross sections in his method of exhaustion to compute the volume of the sphere, balancing infinitesimal slices of the sphere against those of a circumscribed cylinder and inscribed cone.[22] By considering parallel cross-sectional planes, he showed that the sphere's volume equals two-thirds that of the cylinder with the same base radius and height equal to the diameter, using equilibrium arguments on these sections to avoid direct summation.[22] This approach prefigured integral calculus by exhausting the solid through approximating polygonal sections, rigorously bounding the volume between inscribed and circumscribed figures.[22]Properties
Area Calculations
The area of a cross section in geometry is determined by the intersection of a plane with a solid, resulting in a two-dimensional figure whose area can be calculated using integration over the plane or established geometric formulas derived from the solid's properties. For solids featuring uniform cross sections, such as prisms and right cylinders, the two-dimensional form of Cavalieri's principle applies: if two plane figures between parallel lines have corresponding cross strips of equal length at every width, the figures possess equal areas. This principle facilitates verification of area constancy in parallel sections without recomputation. For a sphere of radius , a plane perpendicular to a diameter at distance from the center intersects in a circle of radius . The area follows directly as derived from the Pythagorean theorem in the right triangle formed by the radius, distance , and cross-section radius. This formula quantifies how the area diminishes from the maximum at to zero at .[23] In a right circular cylinder of radius , a plane perpendicular to the axis yields a circular cross section with area , independent of position along the axis due to uniformity. For an oblique plane intersecting the axis at angle (where ), the cross section is an ellipse with semi-minor axis and semi-major axis . The area is thus reflecting elongation along the major axis from the inclination; this derives from projecting the circular generators onto the plane, preserving the minor axis as the diameter while stretching the major by the secant of the complementary angle.[24][25] For a right circular cone of base radius and height , a plane parallel to the base at distance from the apex (where ) produces a similar circle scaled by , with area . An oblique plane yields an ellipse, whose area (with semi-axes and ) can be derived from the cone's parametric equations , , (for , ), by substituting the plane equation and solving the resulting quadratic form, which confirms the elliptical boundary and axes lengths via eigenvalue decomposition or direct integration.[26][27] Cross sections of prisms and pyramids, being polygons, have areas computable from vertex coordinates in the plane using the shoelace formula. For a polygon with vertices for to (with ), the area is originating from summing signed areas of triangles from the origin; in prisms, perpendicular sections match the base polygon's area, while in pyramids, oblique or parallel sections scale accordingly by similarity. This method applies generally to irregular polygonal cross sections. The following table summarizes key area formulas for common solids and plane orientations:| Solid | Plane Type | Shape | Area Formula | Variables |
|---|---|---|---|---|
| Sphere | Perpendicular to diameter | Circle | : radius, : distance from center | |
| Cylinder | Perpendicular to axis | Circle | : radius | |
| Cylinder | Oblique (angle to axis) | Ellipse | : radius, : angle to axis | |
| Cone | Parallel to base | Circle | : base radius, : scale from apex | |
| Cone | Oblique | Ellipse | : semi-major, semi-minor axes | |
| Prism/Pyramid | Arbitrary (polygon vertices known) | Polygon | Shoelace formula | Coordinates |