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In mathematics, a crunode (archaic) or node is a point where a
In mathematics, a crunode (archaic) or node is a point where a curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
intersects itself so that both branches of the curve have distinct tangent line
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
s 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 , where is a smooth function
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
of variables and ranging over the real numbers, a crunode of the curve is a singularity of the function , where both partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s and vanish. Further the Hessian matrix
In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed ...
of second derivatives will have both positive and negative eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
.
See also
* Singular point of a curve *Acnode
An acnode is an isolated point in the solution set of a polynomial equation in two real variables. Equivalent terms are " isolated point or hermit point".
For example the equation
:f(x,y)=y^2+x^2-x^3=0
has an acnode at the origin, because it is ...
* Cusp
* Tacnode
* Saddle point
References
Curves Algebraic curves {{differential-geometry-stub es:Punto singular de una curva#Crunodos