Cross-Section

Cross-Section

In solid geometry, the cross-section of a three-dimensional object is a two-dimensional figure obtained by slicing the object perpendicular to its axis and viewing it end on. Thus, a sausage has a circular cross-section, a 4× 4 fence post has a square cross-section, and a football has a circular cross-section when sliced one way and an elliptical cross-section when sliced another. More formally, a cross-section is the locus of points obtained when a plane intersects an object at right angles to one of its axes, which are taken to be the axes of the associated rectangular coordinate system. Since we are free to associate a coordinate system relative to an object in any way we please, and because every cross-section is one dimension less than the object from which it is obtained, a careful choice of axes provides a cross-section containing nearly as much information about the object from which it is obtained, a careful choice of axes provides a cross-section containing nearly as much information about the object as a full-dimensional view.

Often choosing an axis of symmetry provides the most useful cross-section. An axis of symmetry is a line segment about which the object is symmetric, defined as a line segment passing through the object in such a way that every line segment drawn perpendicular to the axis having endpoints on the surface of the object is bisected by the axis. Examples of three-dimensional solids with an axis of symmetry include right parallelepipeds (most ordinary cardboard boxes), which have rectangular cross-sections; spheres (basketballs, baseballs, etc.), which have circular cross-sections; and pyramids with square bases (such as those found in Egypt), which have square cross-sections.

Other times, the most useful cross-section is obtained by choosing an axis parallel to the axis of symmetry. In this case, the plane that intersects the object will contain the axis of symmetry. This is useful for picturing such things as fancy parfait glasses in two dimensions.

Finally, there are innumerable objects of interest that have no axis of symmetry. In this case, take care to choose the cross-section that provides the most detail.

The great usefulness of a properly chosen cross-section comes in the representation of three-dimensional objects using two-dimensional media, such as paper and pencil or flat computer screens. The same idea helps in the study of objects with four or more dimensions. A three-dimensional object represents the cross-section of one or more four-dimensional objects. For instance, a cube is the cross-section of a four-dimensional hypercube. In general, one way to define the cross-section of any N-dimensional object as the locus of points obtained when any (N– 1) dimensional”surface“intersects an N-dimensional”solid“perpendicular to one of the solid’s axes. Again, the axes of an N-dimensional object are the N axes of the associated rectangular coordinate system. While this concept is impossible to represent geometrically, it is easily dealt with algebraically, using vectors and matrices.