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How do the principal curvatures of a cylinder compare with those of a sphere?

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differential geometry An annular strip (the region between two concentric circles) can be cut and bent into a helical strake that follows approximately the contour of a cylinder. Techniques of differential geometry are employed to find the dimensions of the annular strip that will best match the required curvature of the strake.

In differential geometry, the principal curvatures of a surface are the maximum and minimum normal curvatures at a given point. These curvatures differ significantly between a cylinder and a sphere:12

Cylinder: For a right cylinder of radius $r$ , the principal curvatures are not constant in all directions. The vertical cross sections are straight lines, resulting in a minimum principal curvature of $0$ . The horizontal cross sections are circles, resulting in a maximum principal curvature of $1 / r$ . Because one principal curvature is zero, the Gaussian curvature (the product of the principal curvatures) of a cylinder is $0$ .2
Sphere: On a sphere of radius $r$ , all normal curvatures are identical at every point. Consequently, every direction is a principal direction, and the principal curvatures are equal to $1 / r$ . Unlike the cylinder, the sphere has a constant positive Gaussian curvature of $1 / r^{2}$ .2

In summary, while a sphere exhibits uniform curvature in all directions, a cylinder possesses one direction of zero curvature, reflecting its nature as a developable surface.2