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

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. A crunode is also known as an ordinary double point.^[1]

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

See also

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

References