{{short description|Point where a curve intersects itself at an angle}}
thumb|right|300px|A crunode at the origin of the curve defined by <math>y^2 - x^2(x+1)=0.</math>
In mathematics, a '''crunode'''<ref>{{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}}</ref> (archaic; from Latin ''crux'' "cross" + ''node''<ref>{{cite encyclopedia |encyclopedia=Oxford English Dictionary |title=crunode (''n.'') |url=https://www.oed.com/dictionary/crunode_n |doi=10.1093/OED/1018813892|url-access=subscription }}</ref>) 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''.<ref>{{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}}</ref><ref>{{cite web|last=Weisstein|first=Eric W.|title=Crunode|url=http://mathworld.wolfram.com/Crunode.html|publisher=Mathworld|accessdate=14 January 2014}}</ref>
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
<math class="block">\frac{\partial{f}}{\partial x} = \frac{\partial{f}}{\partial{y}} = 0</math>
and the Hessian determinant is negative:
<math class="block">\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.</math><ref>{{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}}</ref>
==See also== *Singular point of a curve *Acnode *Cusp *Tacnode *Saddle point
==References== {{reflist}}
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