Wikipedia:Reference desk/Archives/Mathematics/2017 October 21

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October 21

Curvature from partial derivatives

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 f_i 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)

Implicit curve#Curvature says:

Due to clarity of the formulas the arguments $(x_{0},y_{0})$ are omitted:

$\kappa ={\frac {-F_{y}^{2}F_{xx}+2F_{x}F_{y}F_{xy}-F_{x}^{2}F_{yy}}{(F_{x}^{2}+F_{y}^{2})^{3/2}}}$ is the curvature at a regular point.

Loraof (talk) 18:45, 21 October 2017 (UTC)

An alternative is to use numerical methods. That is, use 3 sample points close enough together so that the average curvature for that region will be an acceptable approximation of the instantaneous curvature at the middle point, but not so close that round-off error will throw off the results. If calculating the curvature on computer, this may be the easier way to go. StuRat (talk) 20:58, 21 October 2017 (UTC)

Off topic.
The following discussion has been closed. Please do not modify it.
Oh FFS did you even read the question? Please stop with the self-indulgent ramblings. --JBL (talk) 21:09, 21 October 2017 (UTC) 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)

Thanks, seems like it should be in the Curvature article but I didn't see it. --RDBury (talk) 00:18, 22 October 2017 (UTC)