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some polish

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2020-09-11 16:30:05 -07:00
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docs/chapters/curvature/content.en-GB.md
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@@ -14,7 +14,7 @@ So what we really want is some kind of expression that's not based on any partic
We've seen this before... that's the arc length function.
So you might think that in order to find the curvature of a curve, we now need to solve the arc length function itself, and that this would be quite a problem because we just saw that there is no way to actually do that. Thankfully, we don't. We only need to know the _form_ of the arc length function, which we saw above and is fairly simple, rather than needing to _solve_ the arc length function. If we start with the arc length expression and the [run through the steps necessary](http://mathworld.wolfram.com/Curvature.html) to determine _its_ derivative (with an alternative, shorter demonstration of how to do this found [over on Stackexchange](https://math.stackexchange.com/a/275324/71940)), then the integral that was giving us so much problems in solving the arc length function disappears entirely (because of the [fundamental theorem of calculus](https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus)), and what we're left with us some surprisingly simple maths that relates curvature (denoted as κ, "kappa") to—and this is the truly surprising bit—a specific combination of derivatives of our original function.
So you might think that in order to find the curvature of a curve, we now need to solve the arc length function itself, and that this would be quite a problem because we just saw that there is no way to actually do that. Thankfully, we don't. We only need to know the _form_ of the arc length function, which we saw above and is fairly simple, rather than needing to _solve_ the arc length function. If we start with the arc length expression and the [run through the steps necessary](https://mathworld.wolfram.com/Curvature.html) to determine _its_ derivative (with an alternative, shorter demonstration of how to do this found [over on Stackexchange](https://math.stackexchange.com/questions/275248/deriving-curvature-formula/275324#275324)), then the integral that was giving us so much problems in solving the arc length function disappears entirely (because of the [fundamental theorem of calculus](https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus)), and what we're left with us some surprisingly simple maths that relates curvature (denoted as κ, "kappa") to—and this is the truly surprising bit—a specific combination of derivatives of our original function.
Let me highlight what just happened, because it's pretty special: