Cylinders in Three-Dimensional Space

In three-dimensional space, a cylindrical surface forms when a two-dimensional curve extends along a straight line. This curve is called a profile curve and represents the cross-section of a shape. This process creates a surface made up of parallel lines, called rulings, that follow a fixed direction.

In Cartesian coordinates, a cylindrical surface is often identified when an equation does not include one of the three variables. For example, in y = x², the variable z does not appear.

This equation gives a parabolic curve in the xy-plane, and the absence of z shows that the curve extends infinitely along the z-axis. The surface formed in this way is called a parabolic cylinder.

A similar idea applies to circular profile curves. For example, a circle in the xy-plane can be extended along the z-axis to form a circular cylinder, like a pipe with the same cross-section at every height.

In architecture and engineering, 3D models of structures such as tunnels and arches can be created in parametric modeling software by defining a cross-sectional curve mathematically. This curve is then extended in a chosen direction to form a cylindrical surface.