Crunode

{{short description|Point where a curve intersects itself at an angle}}

File:Cubic with double point.svg

In mathematics, a crunode{{cite book |last1=Salmon |first1=George |title=A treatise on the higher plane curves: intended as a sequel to A treatise on conic sections |date=1879 |publisher=Hodges, Foster, & Figgis |location=Dublin |page=24 |url=https://archive.org/details/117724690/page/n49 |access-date=31 January 2025}} (archaic; from Latin crux "cross" + node{{cite encyclopedia |encyclopedia=Oxford English Dictionary |title=crunode (n.) |url=https://www.oed.com/dictionary/crunode_n |doi=10.1093/OED/1018813892}}) 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.{{cite book |last1=Fulton |first1=William |title=Algebraic curves: an introduction to algebraic geometry |date=2008 |page=33 |url=https://dept.math.lsa.umich.edu/~wfulton/CurveBook.pdf |access-date=31 January 2025}}{{cite web|last=Weisstein|first=Eric W.|title=Crunode|url=http://mathworld.wolfram.com/Crunode.html|publisher=Mathworld|accessdate=14 January 2014}}

In the case of a smooth real plane curve {{math|1=f(x, y) = 0 }}, a point is a crunode provided that both first partial derivatives vanish

$\backslash frac\{\backslash partial\{f\}\}\{\backslash partial\; x\}\; =\; \backslash frac\{\backslash partial\{f\}\}\{\backslash partial\{y\}\}\; =\; 0$

and the Hessian determinant is negative:

{{cite book |last1=Hilton |first1=Harold |title=Plane algebraic curves |date=1920 |publisher=Clarendon Press |location=Oxford |page=26 |url=https://archive.org/details/cu31924001544216/page/n45/mode/2up |access-date=31 January 2025}}