Cylinder (geometry)

A cylinder is generated by translating a plane curve along parallel lines transverse to its plane. A circular cylinder uses a circle as its generating plane curve; it is right when the generators are perpendicular to that plane. The term can denote the lateral surface or, for a bounded portion with end caps, the enclosed solid. The circular cylinder curvature formulas describe the smooth lateral surface.

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.

A closed circular cylinder of radius $a>0$ and length $\ell >0$ has volume $\pi a^2\ell$ and total surface area $2\pi a(a+\ell )$, including both end disks. At fixed area $A$, Lagrange multipliers for maximizing the volume give $\ell =2a$, henceEquivalently, eliminate $\ell =A/(2\pi a)-a$ and maximize $(A/2)a-\pi a^3$ on $0<a<\sqrt{A/(2\pi )}$. The derivative changes from positive to negative at the displayed radius, and the volume tends to zero at both ends.

For a cylinder inscribed in a sphere of radius $R$, the constraint is $a^2+{\ell }^2/4=R^2$. Maximizing $a^2+a\ell$ by Lagrange multipliers gives ${\ell }^2+2a\ell =4a^2$, so the positive aspect ratio is $\ell /a=\sqrt{5}-1$. It is independent of $R$. With $q=\ell /a$, the area is $2\pi R^2(1+q)/(1+q^2/4)$; differentiating this expression confirms the unique maximum on $q>0$.