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[edit] Saddle points and Hessians
The function locus-graphed in the picture has a saddle point at the origin, so I believe its Hessian matrix must be indefinite. There exist crunodes that are local extrema of the locus function (consider f(x,y) = (x − y)2(x + y)2), so I'm not sure that there is anything to say about the Hessian here. --Tardis 16:51, 16 January 2007 (UTC)
- Yes you are right about the indefinite hessian. Whether (x-y)^2(x+y)^2 should be considered a crunode is an interesting question. If you take a classification of singularities, you find that x^2-y^2 and (x-y)^2(x+y)^2 have different types, the most important type being the simpler case. I'm not at all clear wherther the more complex case should really be called a curnode or not.
- Taking the simplest case I think the defining characteristic is that the determinant of the hessian is negative, that is the quadratic form is hyperbolic. Acnodes have elliptics quadratic forms and cusps have parabolic forms. --Salix alba (talk) 21:26, 16 January 2007 (UTC)
