Crunode

In mathematics, a crunode[1] (archaic; from Latin crux "cross" + node[2]) or node of an algebraic curve is a type of singular point at which the 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.

[3][4] In the case of a smooth real plane curve f(x, y) = 0, a point is a crunode provided that both first partial derivatives vanish

∂

f

and the Hessian determinant is negative:

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A crunode at the origin of the curve defined by