How are a plane and a cylinder locally isometric?

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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, a plane and a cylinder are considered locally isometric because they share the same Gaussian curvature of zero.1

Two surfaces are defined as isometric if one can be transformed into the other without stretching, meaning intrinsic distances remain unchanged. For a cylinder, the principal normal curvatures consist of a maximum curvature of $1 / r$ (along horizontal circles) and a minimum curvature of zero (along vertical straight lines); their product, the Gaussian curvature, is therefore zero. Because a plane also has a Gaussian curvature of zero, the two surfaces are locally isometric.1

This relationship is a specific application of Minding’s theorem, which states that smooth surfaces with the same constant Gaussian curvature are locally isometric. Consequently, a cylinder can be cut along a vertical line and flattened into a plane without stretching, a property that classifies such surfaces as "developable".1