# Crunode

A crunode at the origin of the curve defined by y2x2(x+1)=0

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.

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.