Extensive Definition
A crunode is a point where a curve intersects itself so that
both branches of the curve have distinct tangent lines.
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 be negative
definite.