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some polish

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2020-09-11 16:30:05 -07:00
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6
docs/chapters/arclength/content.en-GB.md
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@@ -12,11 +12,11 @@ or, more commonly written using Leibnitz notation as:
length = \int_{0}^{z}\sqrt{ \left (dx/dt \right )^2+\left (dy/dt \right )^2} dt
\]
This formula says that the length of a parametric curve is in fact equal to the **area** underneath a function that looks a remarkable amount like Pythagoras' rule for computing the diagonal of a straight angled triangle. This sounds pretty simple, right? Sadly, it's far from simple... cutting straight to after the chase is over: for quadratic curves, this formula generates an [unwieldy computation](http://www.wolframalpha.com/input/?i=antiderivative+for+sqrt((2*(1-t)*t*B+%2B+t%5E2*C)%27%5E2+%2B+(2*(1-t)*t*E)%27%5E2)&incParTime=true), and we're simply not going to implement things that way. For cubic Bézier curves, things get even more fun, because there is no "closed form" solution, meaning that due to the way calculus works, there is no generic formula that allows you to calculate the arc length. Let me just repeat this, because it's fairly crucial: ***for cubic and higher Bézier curves, there is no way to solve this function if you want to use it "for all possible coordinates"***.
This formula says that the length of a parametric curve is in fact equal to the **area** underneath a function that looks a remarkable amount like Pythagoras' rule for computing the diagonal of a straight angled triangle. This sounds pretty simple, right? Sadly, it's far from simple... cutting straight to after the chase is over: for quadratic curves, this formula generates an [unwieldy computation](https://www.wolframalpha.com/input/?i=antiderivative+for+sqrt((2*(1-t)*t*B+%2B+t%5E2*C)%27%5E2+%2B+(2*(1-t)*t*E)%27%5E2)&incParTime=true), and we're simply not going to implement things that way. For cubic Bézier curves, things get even more fun, because there is no "closed form" solution, meaning that due to the way calculus works, there is no generic formula that allows you to calculate the arc length. Let me just repeat this, because it's fairly crucial: ***for cubic and higher Bézier curves, there is no way to solve this function if you want to use it "for all possible coordinates"***.
Seriously: [It cannot be done](https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem).
So we turn to numerical approaches again. The method we'll look at here is the [Gauss quadrature](http://www.youtube.com/watch?v=unWguclP-Ds&feature=BFa&list=PLC8FC40C714F5E60F&index=1). This approximation is a really neat trick, because for any *n<sup>th</sup>* degree polynomial it finds approximated values for an integral really efficiently. Explaining this procedure in length is way beyond the scope of this page, so if you're interested in finding out why it works, I can recommend the University of South Florida video lecture on the procedure, linked in this very paragraph. The general solution we're looking for is the following:
So we turn to numerical approaches again. The method we'll look at here is the [Gauss quadrature](https://www.youtube.com/watch?v=unWguclP-Ds&feature=BFa&list=PLC8FC40C714F5E60F&index=1). This approximation is a really neat trick, because for any *n<sup>th</sup>* degree polynomial it finds approximated values for an integral really efficiently. Explaining this procedure in length is way beyond the scope of this page, so if you're interested in finding out why it works, I can recommend the University of South Florida video lecture on the procedure, linked in this very paragraph. The general solution we're looking for is the following:
\[
\int_{-1}^{1}\sqrt{ \left (dx/dt \right )^2+\left (dy/dt \right )^2} dt
@@ -73,7 +73,7 @@ Note that one requirement for the approach we'll use is that the integral must r
That may look a bit more complicated, but the fraction involving *z* is a fixed number, so the summation, and the evaluation of the *f(t)* values are still pretty simple.
So, what do we need to perform this calculation? For one, we'll need an explicit formula for *f(t)*, because that derivative notation is handy on paper, but not when we have to implement it. We'll also need to know what these *C<sub>i</sub>* and *t<sub>i</sub>* values should be. Luckily, that's less work because there are actually many tables available that give these values, for any *n*, so if we want to approximate our integral with only two terms (which is a bit low, really) then [these tables](legendre-gauss.html) would tell us that for *n=2* we must use the following values:
So, what do we need to perform this calculation? For one, we'll need an explicit formula for *f(t)*, because that derivative notation is handy on paper, but not when we have to implement it. We'll also need to know what these *C<sub>i</sub>* and *t<sub>i</sub>* values should be. Luckily, that's less work because there are actually many tables available that give these values, for any *n*, so if we want to approximate our integral with only two terms (which is a bit low, really) then [these tables](./legendre-gauss.html) would tell us that for *n=2* we must use the following values:
\[
\begin{array}{l}