Crunode

A crunode at the origin of the curve defined by $y^{2}-x^{2}(x+1)=0.$

In mathematics, a crunode (archaic; from Latin crux "cross" + node) 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.

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

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

and the Hessian determinant is negative:

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

See also

• Singular point of a curve
• Acnode
• Cusp
• Tacnode
• Saddle point