Geodesics on a generalized cylinder ($z=F(y),u(y)={\int }^y\sqrt{1+F^{\prime }(s{)}^2}ds,u=ax+b$)

For a smooth surface obtained by translating the plane graph $z=F(y)$ along the $x$ direction, the induced metric is $d{s}^2=d{x}^2+[1+F^{\prime }(y{)}^2]d{y}^2$. The transverse arc length coordinate $u(y)={\int }^y\sqrt{1+F^{\prime }(s{)}^2}ds$ makes it $d{s}^2=d{x}^2+d{u}^2$. This is a local isometry to the Euclidean plane, extending the intrinsic flatness of a circular cylinder to arbitrary smooth generating graphs.

Its geodesics therefore lift to straight lines in $(x,u)$ coordinates. Those that are graphs over $x$ satisfy $u=ax+b$, while the additional family $x=\text{constant}$ cannot be written as $y(x)$. The same graph formula follows from the Beltrami identity for the arc-length integrand $\sqrt{1+[1+F^{\prime }(y{)}^2]y^{\prime 2}}$. These straight lines are locally minimizing; on cylinders with a closed generating curve, global identifications can create several different winding paths between two points.