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Every formula for curvature I've seen assumes that the curve is given parametrically. (Curves of the form y=f(x) are included since you can take the parameter to be x.) I'm trying to determine if there is a formula for the curvature of an implicit curve f(x, y)=0 at x0, y0, assuming f(x0, y0)=0 and either f1(x0, y0) or f2(x0, y0) is not 0, given in terms of f1, f2, f11, f12, f22. Here fi is the partial derivative of f wrt the ith variable and f is assumed to be "sufficiently nice". If no such formula is possible then there should be f and g which agree up to the second derivatives but where the corresponding curves have different curvatures. --RDBury (talk) 17:24, 21 October 2017 (UTC)[reply]
I assume you mean that my response does not use a partial derivative solution. If that's really the only answer the OP wants, then they have it already, and my reply does not spoil that answer. However, this could be a case of the XY Problem, where they really want any solution, but heard a partial derivative solution exists, so asked for that one. If this is the case, then my response may be helpful. I would ask you to remain civil, in any case. StuRat (talk) 01:22, 22 October 2017 (UTC)[reply]