What is "crunode"

Crunode \Cru"node\ (kr?"n?d), n. [Prob. fr. L. crux a cross + E. node.] (Geom.) A point where one branch of a curve crosses another branch. See Double point, under Double,

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n. (context geometry English) A point where one branch of a curve crosses another branch.

In mathematics, a crunode (archaic) or node is a point where a curve intersects itself so that both branches of the curve have distinct tangent lines at the point of intersection. A crunode is also known as an ordinary double point.

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 $\partial f\over \partial x$ and $\partial f\over \partial y$ vanish. Further the Hessian matrix of second derivatives will have both positive and negative eigenvalues.