Cross section (geometry)
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.
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.
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.
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.
A plane section can be used to visualize the partial derivative of a function with respect to one of its arguments, as shown.
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.
If instead the plane section is taken for a fixed value of the density, the result is an iso-density contour.
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.
For a convex body, each ray through the object from the viewer's perspective crosses just two surfaces.
) 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
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.
Cross-sections are often used in anatomy to illustrate the inner structure of an organ, as shown at the left.
