Crunode

In mathematics, a crunode (archaic; from Latin crux "cross" + node) 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. 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 ∂ f ∂ x = ∂ f ∂ y = 0 {\displaystyle {\frac {\partial {f}}{\partial x}}={\frac {\partial {f}}{\partial {y}}}=0} and the Hessian determinant is negative: ∂ 2 f ∂ x 2 ∂ 2 f ∂ y 2 − ( ∂ 2 f ∂ x ∂ y ) 2 < 0. {\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.}

Point where a curve intersects itself at an angle

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

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

and the Hessian determinant is negative:

∂ 2 f ∂ x 2 ∂ 2 f ∂ y 2 − ( ∂ 2 f ∂ x   ∂ y ) 2 < 0. {\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

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

References

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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.

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