Crunode

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

In mathematics, a crunode ^[1] (archaic; from Latin crux "cross" + node ^[2] ) or node of an algebraic curve is a type of singular point at which the 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. ^{[3] [4]}

In the case of a smooth real plane curve f(x, y) = 0, a point is a crunode provided that both first partial derivatives vanish

${\displaystyle {\frac {\partial {f}}{\partial x}}={\frac {\partial {f}}{\partial {y}}}=0}$

and the Hessian determinant is negative:

${\displaystyle {\frac {\partial ^{2}f}{\partial x^{2}}}{\frac {\partial ^{2}f}{\partial y^{2}}}-\left({\frac {\partial ^{2}f}{\partial x~\partial y}}\right)^{2}<0.}$ ^[5]

See also

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

References

1. Salmon, George (1879). A treatise on the higher plane curves: intended as a sequel to A treatise on conic sections. Dublin: Hodges, Foster, & Figgis. p. 24. Retrieved 31 January 2025.
2. "crunode (n.)" . Oxford English Dictionary. doi:10.1093/OED/1018813892.
3. Fulton, William (2008). Algebraic curves: an introduction to algebraic geometry (PDF). p. 33. Retrieved 31 January 2025.
4. Weisstein, Eric W. "Crunode". Mathworld. Retrieved 14 January 2014.
5. Hilton, Harold (1920). Plane algebraic curves. Oxford: Clarendon Press. p. 26. Retrieved 31 January 2025.