= Geodesics on a generalized cylinder
{title2=$z=F(y),\quad u(y)=\int^y\sqrt{1+F'(s)^2}ds,\quad u=ax+b$}
For a smooth surface obtained by translating the plane graph $z=F(y)$ along the $x$ direction, the <induced metric> is $ds^2=dx^2+[1+F'(y)^2]dy^2$. The transverse <arc length> coordinate $u(y)=\int^y\sqrt{1+F'(s)^2}ds$ makes it $ds^2=dx^2+du^2$. 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 $\sqrt{1+[1+F'(y)^2]y'^2}$. These straight lines are locally minimizing; on cylinders with a closed generating curve, global identifications can create several different winding paths between two points.
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