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How do planes passing through a surface normal reveal a surface’s 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 curvature of a surface at a specific point is analyzed by examining cross sections created by planes that contain the surface normal—the line perpendicular to the tangent plane at that point.12

These cross sections produce curves whose curvatures are known as normal curvatures. By evaluating these sections, two specific planes emerge:
* Principal Directions: One plane yields the section with the maximum normal curvature, while the other yields the minimum. These two directions are typically perpendicular to each other.12
* Principal Curvatures: The values of these maximum and minimum curvatures are the principal curvatures of the surface at that point.12

These measurements provide a comprehensive understanding of the surface's shape. For instance, the product of these two principal curvatures defines the Gaussian curvature, a fundamental value that reveals intrinsic geometric properties, such as whether a surface can be flattened without stretching.12