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