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Why are principal directions usually perpendicular on a surface?

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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 directions at a point on a surface are defined by the planes containing the surface normal that produce the maximum and minimum normal curvatures.12

These directions are typically perpendicular to one another because they correspond to the eigenvectors of the shape operator (or Weingarten map), which is a symmetric linear operator. In mathematics, the spectral theorem guarantees that the eigenvectors of a symmetric operator associated with distinct eigenvalues are orthogonal. Consequently, for most surfaces where the normal curvatures are not constant, the principal directions are at right angles.12

It is worth noting that on a sphere, where all normal curvatures are identical, every direction is a principal direction, and the concept of distinct, perpendicular principal directions does not apply in the same way.2