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Pomax
2020-08-27 21:46:01 -07:00
parent 17d71c7d70
commit 4e34774afb
23 changed files with 278 additions and 215 deletions
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4
docs/chapters/curvature/content.en-GB.md
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@@ -8,9 +8,9 @@ What we want is to ensure that the [curvature](https://en.wikipedia.org/wiki/Cur
Problem solved!
However, there's a problem with this approach: if we think about this a little more, we realise that "what a curve looks like" and its derivative values are pretty much entirely unrelated. After all, the section on [reordering curves](#reordering) showed us that the same looking curve can have an infinite number of curve expressions of arbitraryly high Bezier degree, and each of those will have _widly_ different derivative values.
However, there's a problem with this approach: if we think about this a little more, we realise that "what a curve looks like" and its derivative values are pretty much entirely unrelated. After all, the section on [reordering curves](#reordering) showed us that the same looking curve can have an infinite number of curve expressions of arbitraryly high Bézier degree, and each of those will have _widly_ different derivative values.
So what we really want is some kind of expression that's not based on any particular expression of `t`, but is based on something that is invariant to the _kind_ of function(s) we use to draw our curve. And the prime candidate for this is our curve expression, reparameterised for distance: no matter what order of Bezier curve we use, if we were able to rewrite it as a function of distance-along-the-curve, all those different degree Bezier functions would end up being _the same_ function for "coordinate at some distance D along the curve".
So what we really want is some kind of expression that's not based on any particular expression of `t`, but is based on something that is invariant to the _kind_ of function(s) we use to draw our curve. And the prime candidate for this is our curve expression, reparameterised for distance: no matter what order of Bézier curve we use, if we were able to rewrite it as a function of distance-along-the-curve, all those different degree Bézier functions would end up being _the same_ function for "coordinate at some distance D along the curve".
We've seen this before... that's the arc length function.

5
docs/chapters/curvature/curvature.js
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@@ -3,8 +3,9 @@ let q, c;
setup() {
q = new Bezier(this, 60,55, 125,160, 365,165);
c = new Bezier(this, 385,165, 645,165, 645,70, 750,165);
setSlider(`.slide-control`, `position`, 0);
if (this.parameters.omni) {
setSlider(`.slide-control`, `position`, 0);
}
setMovable(q.points.concat(c.points));
}