Crunode

Point where a curve intersects itself at an angle

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

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

References

^ Weisstein, Eric W. "Crunode". Mathworld. Retrieved 14 January 2014.

Topics in algebraic curves

Rational curves

Five points determine a conic
Projective line
Rational normal curve
Riemann sphere
Twisted cubic

Elliptic curves

Analytic theory	Elliptic function Elliptic integral Fundamental pair of periods Modular form
Arithmetic theory	Counting points on elliptic curves Division polynomials Hasse's theorem on elliptic curves Mazur's torsion theorem Modular elliptic curve Modularity theorem Mordell–Weil theorem Nagell–Lutz theorem Supersingular elliptic curve Schoof's algorithm Schoof–Elkies–Atkin algorithm
Applications	Elliptic curve cryptography Elliptic curve primality

Higher genus

Plane curves

Riemann surfaces

Constructions

Structure of curves

Divisors on curves	Abel–Jacobi map Brill–Noether theory Clifford's theorem on special divisors Gonality of an algebraic curve Jacobian variety Riemann–Roch theorem Weierstrass point Weil reciprocity law
Moduli	ELSV formula Gromov–Witten invariant Hodge bundle Moduli of algebraic curves Stable curve
Morphisms	Hasse–Witt matrix Riemann–Hurwitz formula Prym variety Weber's theorem (Algebraic curves)
Singularities	Acnode Crunode Cusp Delta invariant Tacnode
Vector bundles	Birkhoff–Grothendieck theorem Stable vector bundle Vector bundles on algebraic curves