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
and the Hessian determinant is negative:
See also
References
- ^ 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.
- ^ "crunode (n.)". Oxford English Dictionary. doi:10.1093/OED/1018813892.
- ^ Fulton, William (2008). Algebraic curves: an introduction to algebraic geometry (PDF). p. 33. Retrieved 31 January 2025.
- ^ Weisstein, Eric W. "Crunode". Mathworld. Retrieved 14 January 2014.
- ^ Hilton, Harold (1920). Plane algebraic curves. Oxford: Clarendon Press. p. 26. Retrieved 31 January 2025.
