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proof fixes

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Pomax
2020-08-22 10:38:10 -07:00
parent 557a931705
commit 8e2c6a4c51
5 changed files with 11 additions and 9 deletions
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4
docs/chapters/curvature/content.en-GB.md
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@@ -6,13 +6,13 @@ Problem solved!
But, if we think about this a little more, this cannot possible work, because of something that you may have noticed in the section on [reordering curves](#reordering): what a curve looks like, and the function that draws that curve, are not in some kind of universal, fixed, one-to-one relation. If we have some quadratic curve, then simply by raising the curve order we can get corresponding cubic, quartic, and higher and higher mathematical expressions that all draw the _exact same curve_ but with wildly different derivatives. So: if we want to make a transition from one curve to the next look good, and we want to use the derivative, then we suddenly need to answer the question: "Which derivative?".
How would you even decide? What makes the cubic derivatives better or less suited than, say, quintic derivatives? Wouldn't it be nicer if we could use something that was inherent to the curve, without being tied to the functions that yield that curve? And (of course) as it turns out, there is a way to define curvature in such a way that it only relies on what the curve actually looks like, and given where this section is in the larger body of this Primer, it should hopefully not be surprising that we thing we can use to define curvature is the thing we talked about in the previous section: arc length.
How would you even decide? What makes the cubic derivatives better or less suited than, say, quintic derivatives? Wouldn't it be nicer if we could use something that was inherent to the curve, without being tied to the functions that yield that curve? And (of course) as it turns out, there is a way to define curvature in such a way that it only relies on what the curve actually looks like, and given where this section is in the larger body of this Primer, it should hopefully not be surprising that thee thing we can use to define curvature is the thing we talked about in the previous section: arc length.
Intuitively, this should make sense, even if we have no idea what the maths would look like: if we travel some fixed distance along some curve, then the point at that distance is simply the point at that distance. It doesn't matter what function we used to draw the curve: once we know what the curve looks like, the function(s) used to draw it become irrelevant: a point a third along the full distance of the curve is simply the point a third along the distance of the curve.
You might think that in order to find the curvature of a curve, we now need to find and then solve the arc length function, and that would be a problem because we just saw that there is no way to actually do that: don't worry, we don't. We do need to know the _form_ of the arc length function, which we saw above, but it's not the thing we're actually interested in, and we're going to be rewriting it in a way that makes most of the crazy complex things about it just... disappear.
In fact, after [running through the steps necessary](http://mathworld.wolfram.com/Curvature.html) to determine what we're left with if we use the arclength function's derivative (with another run-through of the maths [here](https://math.stackexchange.com/a/275324/71940)), rather than the curve's original function's derivative, then the integral disappears entirely (because of the [fundamental therem of calculus](https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus)), and we're left with 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.
In fact, after [running through the steps necessary](http://mathworld.wolfram.com/Curvature.html) to determine what we're left with if we use the arclength function's derivative (with another run-through of the maths [here](https://math.stackexchange.com/a/275324/71940)), rather than the curve's original function's derivative, then the integral disappears entirely (because of the [fundamental theorem of calculus](https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus)), and we're left with 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 just highlight that before we move on: we calculate the curvature of a curve using the arc length function derivative, because the original function's derivative is entirely unreliable, and in doing so we end up with a formula that expresses curvature in terms of the original function's derivatives.

4
docs/index.html
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@@ -4749,7 +4749,7 @@ if (roots.length > 0) {
as it turns out, there is a way to define curvature in such a way
that it only relies on what the curve actually looks like, and given
where this section is in the larger body of this Primer, it should
hopefully not be surprising that we thing we can use to define
hopefully not be surprising that thee thing we can use to define
curvature is the thing we talked about in the previous section: arc
length.
</p>
@@ -4785,7 +4785,7 @@ if (roots.length &gt; 0) {
integral disappears entirely (because of the
<a
href="https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus"
>fundamental therem of calculus</a
>fundamental theorem of calculus</a
>), and we're left with 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

4
docs/ja-JP/index.html
View File

@@ -4461,7 +4461,7 @@ if (roots.length &gt; 0) {
as it turns out, there is a way to define curvature in such a way
that it only relies on what the curve actually looks like, and given
where this section is in the larger body of this Primer, it should
hopefully not be surprising that we thing we can use to define
hopefully not be surprising that thee thing we can use to define
curvature is the thing we talked about in the previous section: arc
length.
</p>
@@ -4497,7 +4497,7 @@ if (roots.length &gt; 0) {
integral disappears entirely (because of the
<a
href="https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus"
>fundamental therem of calculus</a
>fundamental theorem of calculus</a
>), and we're left with 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

4
docs/js/custom-element/api/types/bezier.js
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@@ -101,7 +101,9 @@ class Bezier extends Original {
api.circle(p.x, p.y, 5);
if (labels) {
api.setFill(`black`);
api.text(`(${p.x},${p.y})`, p.x + 10, p.y + 10);
let x = p.x|0;
let y = p.y|0;
api.text(`(${x},${y})`, x + 10, y + 10);
}
});
ctx.restoreStyle();

4
docs/zh-CN/index.html
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@@ -4471,7 +4471,7 @@ if (roots.length &gt; 0) {
as it turns out, there is a way to define curvature in such a way
that it only relies on what the curve actually looks like, and given
where this section is in the larger body of this Primer, it should
hopefully not be surprising that we thing we can use to define
hopefully not be surprising that thee thing we can use to define
curvature is the thing we talked about in the previous section: arc
length.
</p>
@@ -4507,7 +4507,7 @@ if (roots.length &gt; 0) {
integral disappears entirely (because of the
<a
href="https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus"
>fundamental therem of calculus</a
>fundamental theorem of calculus</a
>), and we're left with 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