Crunode



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. A crunode is also known as an ordinary double point.

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 $$\tfrac{\partial f}{\partial x}$$ and $$\tfrac{\partial f}{\partial y}$$ vanish. Further the Hessian matrix of second derivatives will have both positive and negative eigenvalues.