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I was reading a paper related to convex curves, and encountered the following quantity:

Let $\gamma$ be a convex curve and $k(\cdot)$ be its curvature (with respect to arc length). The author is using the following highly ambiguous (at least to me) notation: $$\int_\gamma k(s)^\frac{1}{3} ds$$ and calling this quantity the affine curvature. My question is:

$\textbf{What does this notation mean?}$

First of all, what exactly is $k(s)?$ Secondly, what does it mean to integrate $k(s)$ with respect to $s$ along the curve $\gamma$? Any help will be greatly appreciated!

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  • $\begingroup$ $k(s)$ probably stands for the curvature as a function of the arc length $s$. Integrating over $ds$ means summing up all the quantity over the curve itself. $\endgroup$
    – embedded_dev
    Commented Nov 18, 2017 at 2:21
  • $\begingroup$ Is there a rigorous definition of the integral? $\endgroup$
    – abcd
    Commented Nov 18, 2017 at 2:25
  • $\begingroup$ It's just the usual line integral of a scalar function... $\endgroup$
    – Anthony Carapetis
    Commented Nov 18, 2017 at 4:52

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As stated by me and Anthony Carapetis in the comments, $k(s)$ is just some scalar function and integrating it over $ds$ means that you are integrating along the curve. Hence, this is a line integral, more rigorously speaking.

Essentially what you are doing is summing up all cubic roots of the curvature $k(s)$ alongside the curve $s$.

Notice that to calculate the integral you would need a parametrization $s(t)$ and then integrate $k(s)$ wrt $t$.

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