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I am wondering if I can use some of the numerical operators (gradient, laplacian, divergence, mean curvature...) and apply it to a point on the 0 iso surface of an SDF in order to tell whether it is a saddle point or not.

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  • $\begingroup$ The Hessian will be indefinite if the point is a saddle. The function increases locally along the eigenspace of the positive eigenvalues, decreases locally along the eigenspace of the negative eigenvalues, and remains the same along the eigenspace of the zero eigenvalues. The hessian would be positive (semi-) definite if it is a local minimum. The hessian would be negative (semi-) definite if it is a local maximum. Based on this idea the (signs of the) eigenvalues of the matrix of the implicit equations $(x,1)^TQ(x,1)=0$ of quadratic surfaces are used to classify them. $\endgroup$ Commented Jan 16 at 9:10

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