Principal curvatures postview8/18/2023 ![]() Is there a better way to compute the principal curvatures when working with local coordinates? Using the formulas for the Gaussian and mean curvature is pretty straight forward, which I like, but it does involve a lot of computation. Why is the formula different? Is it a different context? As far as I could tell everything else matches what I am working with. Now when looking through the Wikipedia page about Differential geometry of surfaces I saw a different formula (in the bottom row of the table I have linked): Euler’s Theorem: Planes of principal curvature are orthogonal and independent of parameterization. Where the $k_i$ denote the two principal curvatures, K the Gaussian curvature and H the mean curvature. In a separate paper, Takagi 30 proved that in the case g 4, if one of the principal curvatures of M has multiplicity one, then M must be homogeneous. Since I have formulas for the Gaussian and the mean curvature, I used them to derive: This list included some examples with 6 principal curvatures, as well as those with 1,2,3 or 4 distinct principal curvatures. The Gaussian curvature K and mean curvature H are related to kappa1 and kappa2 by K kappa1kappa2 (1) H 1/2(kappa1+kappa2). ![]() The principal curvatures measure the maximum and minimum bending of a regular surface at each point. Principal curvatures (K) for each surface (s1 or s2). The maximum and minimum of the normal curvature kappa1 and kappa2 at a given point on a surface are called the principal curvatures. I was wondering how to compute the principal curvatures when working with a local parametrization. A tool in PostView identified these regions and calculated the surface area in each paired set of articular regions.
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