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Saddle points and Hessians

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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 ), so I'm not sure that there is anything to say about the Hessian here. --Tardis 16:51, 16 January 2007 (UTC)[reply]

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)[reply]

Crunode?

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I've never heard the term crunode in my life. Is there a relation to node? The picture depicts what most people would call a node, not a crunode. The stuff about the Hessian contradicts the picture. This article is in serious trouble.--345Kai (talk) 07:33, 7 March 2010 (UTC)[reply]

Yes the info about the Hessian was wrong, somehow it didn't get corrected last time it was pointed out. I think the term is a little dated, but it is well sourced in certain parts of the literature.--Salix (talk): 17:05, 7 March 2010 (UTC)[reply]

Image quality

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There is a vector image available Media:Cubic_with_double_point.svg. Is it not enough?Electron Kid (talk) 23:59, 3 December 2010 (UTC)[reply]