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

${\displaystyle {\frac {\partial {f}}{\partial x}}={\frac {\partial {f}}{\partial {y}}}=0}$

and the Hessian determinant is negative:

${\displaystyle {\frac {\partial ^{2}f}{\partial x^{2}}}{\frac {\partial ^{2}f}{\partial y^{2}}}-\left({\frac {\partial ^{2}f}{\partial x~\partial y}}\right)^{2}<0.}$^[5]