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# Crunode

A crunode is a point where a curve intersects itself so that both branches of the curve have distinct tangent lines.

For a plane curve, defined as the locus of points f(x, y) = 0, where f(x, y) is a smooth function of variables x and y ranging over the real numbers, a crunode of the curve is a singularity of the function f, where both partial derivatives $partial fover partial x$ and $partial fover partial y$ vanish. Further the Hessian matrix of second derivatives will be negative definite.