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2023-08-15 15:28:56 +00:00
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@@ -38,7 +38,7 @@
<meta property="og:locale" content="en-GB" />
<meta property="og:type" content="article" />
<meta property="og:published_time" content="2013-06-13T12:00:00+00:00" />
<meta property="og:updated_time" content="2023-07-24T16:25:38+00:00" />
<meta property="og:updated_time" content="2023-08-15T15:28:19+00:00" />
<meta property="og:author" content="Mike 'Pomax' Kamermans" />
<meta property="og:section" content="Bézier Curves" />
<meta property="og:tag" content="Bézier Curves" />
@@ -8149,56 +8149,35 @@ O(t) = B(t) + d · N(t)
</p>
<!--
╭ B'(t) ╮
N(t) \bot │ ────────── │
╰ || B'(t)|| ╯
╭ B'(t) ╮
N(t) = \bot │ ────────── │
╰ || B'(t)|| ╯
-->
<img
class="LaTeX SVG"
src="./images/chapters/offsetting/57e62f3f2f7526b2cf7c1b276c17e472.svg"
width="120px"
src="./images/chapters/offsetting/e7e8e1f5727387079dbc5770181187c2.svg"
width="141px"
height="40px"
loading="lazy"
/>
<p>
Determining the length requires computing an arc length, and this is where things get Tricky with a capital T. First off, to compute arc
length from some start <code>a</code> to end <code>b</code>, we must use the formula we saw earlier. Noting that "length" is usually
denoted with double vertical bars:
</p>
<p>The magnitude of <code>B'(t)</code>, usually denoted with double vertical bars, is given by the following formula:</p>
<!--
┌───────────┐
╭ b │ 2 2
|| f(x,y)|| = | │f ' + f '
╯ a ⟍│ x y
┌─────────────────┐
│ 2 2
|| B'(t)|| = │B '(t) + B '(t)
⟍│ x y
-->
<img
class="LaTeX SVG"
src="./images/chapters/offsetting/cf8e602eb0595cf4d9b851c6bda741af.svg"
width="169px"
height="36px"
src="./images/chapters/offsetting/7fd3e895c0eea0965470dd619450b679.svg"
width="188px"
height="25px"
loading="lazy"
/>
<p>
So if we want the length of the tangent, we plug in <code>B'(t)</code>, with <code>t = 0</code> as start and <code>t = 1</code> as end:
</p>
<!--
┌───────────────────┐
╭ 1 │ 2 2
|| B'(t)|| = | │B ''(t) + B ''(t)
╯ 0 ⟍│ x y
-->
<img
class="LaTeX SVG"
src="./images/chapters/offsetting/af4b584bb280cc941603255f62c9cc1a.svg"
width="209px"
height="36px"
loading="lazy"
/>
<p>
And that's where things go wrong. It doesn't even really matter what the second derivative for <code>B(t)</code> is, that square root is
screwing everything up, because it turns our nice polynomials into things that are no longer polynomials.
And that's where things go wrong: that square root is screwing everything up, because it turns our nice polynomials into things that are
no longer polynomials.
</p>
<p>
There is a small class of polynomials where the square root is also a polynomial, but they're utterly useless to us: any polynomial with