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Pleas tell me that what a "Kink" is and what this sentence means:

Distance functions have a kink at the interface where $d = 0$ is a local minimum.

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    :-? do u know what is "kink" in general?2012-05-04
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    Hello and welcome to math.SE! Since it's your first time here, I recommend you read the [faq](http://math.stackexchange.com/faq), which will help you ask better questions and thus get better answers. Another tip: it is generally a good idea to leave non-mathematical content out of your questions, such as requesting a prompt answer. It won't prompt users to answer sooner, and may annoy members of the community.2012-05-04
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    Hello.thank you,shure,sorry,but do u know what is "kink"?2012-05-04
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    See my answer below.2012-05-04
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    thank you,can I ask another question here or no?2012-05-04
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    I want know what is relationship between gradiant and curvature?2012-05-04
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    You have to be more precise about what you mean by "curvature". Are you talking about curvature on a surface? What kind of curvature? Gaussian?2012-05-04
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    You should ask that as a separate question and be a little more precise about what you are looking for.2012-05-04

3 Answers 3

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In this case, I believe a "kink" in the function refers to a point at which the function fails to be differentiable. For example, the function $f(x)=|x|$ (which gives the distance between $x$ and $0$) is not differentiable at $x=0$, where the function is $0$ as well.

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    thank you,what is relationship between gradiant and curvature from geometry?2012-05-04
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A "kink" in a curve would be a point where the curve is continuous, yet the first derivative (gradient) is not continuous. The curvature would be infinite at a kink because the direction changes a finite amount in an infinitesimal distance.

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As above you can find it via the web as Cusp (singularity). See the following graphs:

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