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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. Mo ...
s at the point of intersection. A crunode is also known as an ''ordinary double point''.
For a plane curve
In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic ...
, defined as the locus of points , where is a smooth function of variables and ranging over the real numbers, a crunode of the curve is a singularity of the function , where both partial derivatives 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.
See also
*Singular point of a curve
In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied.
Algebraic curves in the plane
Algebraic cur ...
*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
A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth.
Cusp or CUSP may also refer to:
Mathematics
* Cusp (singularity), a singular point of a curve
* Cusp catastrophe, a branch of bifurc ...
*Tacnode
In classical algebraic geometry, a tacnode (also called a point of osculation or double cusp). is a kind of singular point of a curve. It is defined as a point where two (or more) osculating circles to the curve at that point are tangent
I ...
* Saddle point
References
Curves Algebraic curves {{differential-geometry-stub es:Punto singular de una curva#Crunodos