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Why does a flat plane have zero Gaussian curvature?

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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 Gaussian curvature of a surface at any given point is defined as the product of the two principal normal curvatures.12

These principal curvatures represent the maximum and minimum rates at which the surface bends in different directions at that point. For a flat plane, every cross-section is a straight line, and because straight lines have zero curvature, all normal curvatures on a plane are zero. Consequently, when you multiply these principal curvatures together, the result is zero, which is why a flat plane has zero Gaussian curvature.12

Surfaces with constant zero Gaussian curvature are referred to as developable surfaces, as they possess the same intrinsic geometry as a plane and can be flattened without stretching.2