Cylinder

A cylinder with height h and radius r

A cylinder is a solid object that has a circular base and a height. If a cylinder is cut parallel to the base the resulting shape (called a cross-section) would be a circle of the same shape. Examples of cylinders are a tin can and a spool of thread.

This surface can be generalised into the generalised cylinder by replacing the circle at the base with any curve and extending it in any direction, whereas with the standard cylinder the extension is vertical.

More formally, if $\gamma (u)$ is a curve in 3-space and a is a vector then the generalised cylinder of $\gamma$ in direction a is $\sigma(u,v) = \gamma(u) + v\textbf{a}$. This surface is regular if $\gamma (u)$ is regular and a is never parallel to a tangent vector of $\gamma (u)$.