Cross section (geometry)

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.

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

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 of 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.