Curvature approximation of 3D manifolds in 4D space (Q5906682)

To approximate the principal curvatures of a triangulated 3-manifold in 4-space at a point \(x\), the author proposes to construct a Dupin quadric that passes through all vertices of the tetrahedra for which \(x\) is a vertex and then to find the eigenvalues of the corresponding quadratic form. \{The details of the implementation are somewhat weird; it is much better to find the eigenvalues by QR or the roots of a cubic equation by the trigonometric formulas. Since the metric determines the principal curvatures (cf. the reviewer, Differential geometry (1963; Zbl 0116.134), p. 214), they can be found immediately from any spline representation with much less error\}.