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@@ -0,0 +1,5 @@
┌─────────────────┐
│ 2 2
|| B'(t)|| = │B '(t) + B '(t)
⟍│ x y

4
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@@ -0,0 +1,4 @@
╭ B'(t) ╮
N(t) = \bot │ ────────── │
╰ || B'(t)|| ╯

53
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@@ -38,7 +38,7 @@
<meta property="og:locale" content="en-GB" /> <meta property="og:locale" content="en-GB" />
<meta property="og:type" content="article" /> <meta property="og:type" content="article" />
<meta property="og:published_time" content="2013-06-13T12:00:00+00:00" /> <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:author" content="Mike 'Pomax' Kamermans" />
<meta property="og:section" content="Bézier Curves" /> <meta property="og:section" content="Bézier Curves" />
<meta property="og:tag" content="Bézier Curves" /> <meta property="og:tag" content="Bézier Curves" />
@@ -8149,56 +8149,35 @@ O(t) = B(t) + d · N(t)
</p> </p>
<!-- <!--
╭ B'(t) ╮ ╭ B'(t) ╮
N(t) \bot │ ────────── │ N(t) = \bot │ ────────── │
╰ || B'(t)|| ╯ ╰ || B'(t)|| ╯
--> -->
<img <img
class="LaTeX SVG" class="LaTeX SVG"
src="./images/chapters/offsetting/57e62f3f2f7526b2cf7c1b276c17e472.svg" src="./images/chapters/offsetting/e7e8e1f5727387079dbc5770181187c2.svg"
width="120px" width="141px"
height="40px" height="40px"
loading="lazy" loading="lazy"
/> />
<p> <p>The magnitude of <code>B'(t)</code>, usually denoted with double vertical bars, is given by the following formula:</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>
<!-- <!--
┌───────────┐ ┌─────────────────┐
╭ b │ 2 2 │ 2 2
|| f(x,y)|| = | │f ' + f ' || B'(t)|| = │B '(t) + B '(t)
╯ a ⟍│ x y ⟍│ x y
--> -->
<img <img
class="LaTeX SVG" class="LaTeX SVG"
src="./images/chapters/offsetting/cf8e602eb0595cf4d9b851c6bda741af.svg" src="./images/chapters/offsetting/7fd3e895c0eea0965470dd619450b679.svg"
width="169px" width="188px"
height="36px" height="25px"
loading="lazy" loading="lazy"
/> />
<p> <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: And that's where things go wrong: that square root is screwing everything up, because it turns our nice polynomials into things that are
</p> no longer polynomials.
<!--
┌───────────────────┐
╭ 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.
</p> </p>
<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 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

53
docs/ja-JP/index.html generated
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@@ -41,7 +41,7 @@
<meta property="og:locale" content="ja-JP" /> <meta property="og:locale" content="ja-JP" />
<meta property="og:type" content="article" /> <meta property="og:type" content="article" />
<meta property="og:published_time" content="2013-06-13T12:00:00+00:00" /> <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:author" content="Mike 'Pomax' Kamermans" />
<meta property="og:section" content="Bézier Curves" /> <meta property="og:section" content="Bézier Curves" />
<meta property="og:tag" content="Bézier Curves" /> <meta property="og:tag" content="Bézier Curves" />
@@ -8359,56 +8359,35 @@ O(t) = B(t) + d · N(t)
</p> </p>
<!-- <!--
╭ B'(t) ╮ ╭ B'(t) ╮
N(t) \bot │ ────────── │ N(t) = \bot │ ────────── │
╰ || B'(t)|| ╯ ╰ || B'(t)|| ╯
--> -->
<img <img
class="LaTeX SVG" class="LaTeX SVG"
src="./images/chapters/offsetting/57e62f3f2f7526b2cf7c1b276c17e472.svg" src="./images/chapters/offsetting/e7e8e1f5727387079dbc5770181187c2.svg"
width="120px" width="141px"
height="40px" height="40px"
loading="lazy" loading="lazy"
/> />
<p> <p>The magnitude of <code>B'(t)</code>, usually denoted with double vertical bars, is given by the following formula:</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>
<!-- <!--
┌───────────┐ ┌─────────────────┐
╭ b │ 2 2 │ 2 2
|| f(x,y)|| = | │f ' + f ' || B'(t)|| = │B '(t) + B '(t)
╯ a ⟍│ x y ⟍│ x y
--> -->
<img <img
class="LaTeX SVG" class="LaTeX SVG"
src="./images/chapters/offsetting/cf8e602eb0595cf4d9b851c6bda741af.svg" src="./images/chapters/offsetting/7fd3e895c0eea0965470dd619450b679.svg"
width="169px" width="188px"
height="36px" height="25px"
loading="lazy" loading="lazy"
/> />
<p> <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: And that's where things go wrong: that square root is screwing everything up, because it turns our nice polynomials into things that are
</p> no longer polynomials.
<!--
┌───────────────────┐
╭ 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.
</p> </p>
<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 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

53
docs/ko-KR/index.html generated
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@@ -43,7 +43,7 @@
<meta property="og:locale" content="ko-KR" /> <meta property="og:locale" content="ko-KR" />
<meta property="og:type" content="article" /> <meta property="og:type" content="article" />
<meta property="og:published_time" content="2013-06-13T12:00:00+00:00" /> <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:author" content="Mike 'Pomax' Kamermans" />
<meta property="og:section" content="Bézier Curves" /> <meta property="og:section" content="Bézier Curves" />
<meta property="og:tag" content="Bézier Curves" /> <meta property="og:tag" content="Bézier Curves" />
@@ -8510,56 +8510,35 @@ O(t) = B(t) + d · N(t)
</p> </p>
<!-- <!--
╭ B'(t) ╮ ╭ B'(t) ╮
N(t) \bot │ ────────── │ N(t) = \bot │ ────────── │
╰ || B'(t)|| ╯ ╰ || B'(t)|| ╯
--> -->
<img <img
class="LaTeX SVG" class="LaTeX SVG"
src="./images/chapters/offsetting/57e62f3f2f7526b2cf7c1b276c17e472.svg" src="./images/chapters/offsetting/e7e8e1f5727387079dbc5770181187c2.svg"
width="120px" width="141px"
height="40px" height="40px"
loading="lazy" loading="lazy"
/> />
<p> <p>The magnitude of <code>B'(t)</code>, usually denoted with double vertical bars, is given by the following formula:</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>
<!-- <!--
┌───────────┐ ┌─────────────────┐
╭ b │ 2 2 │ 2 2
|| f(x,y)|| = | │f ' + f ' || B'(t)|| = │B '(t) + B '(t)
╯ a ⟍│ x y ⟍│ x y
--> -->
<img <img
class="LaTeX SVG" class="LaTeX SVG"
src="./images/chapters/offsetting/cf8e602eb0595cf4d9b851c6bda741af.svg" src="./images/chapters/offsetting/7fd3e895c0eea0965470dd619450b679.svg"
width="169px" width="188px"
height="36px" height="25px"
loading="lazy" loading="lazy"
/> />
<p> <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: And that's where things go wrong: that square root is screwing everything up, because it turns our nice polynomials into things that are
</p> no longer polynomials.
<!--
┌───────────────────┐
╭ 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.
</p> </p>
<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 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

2
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@@ -34,7 +34,7 @@
<meta property="og:locale" content="en-GB" /> <meta property="og:locale" content="en-GB" />
<meta property="og:type" content="article" /> <meta property="og:type" content="article" />
<meta property="og:published_time" content="Fri Sep 18 2020 00:00:00 +00:00" /> <meta property="og:published_time" content="Fri Sep 18 2020 00:00:00 +00:00" />
<meta property="og:updated_time" content="Mon Jul 24 2023 16:25:38 +00:00" /> <meta property="og:updated_time" content="Tue Aug 15 2023 15:28:19 +00:00" />
<meta property="og:author" content="Mike 'Pomax' Kamermans" /> <meta property="og:author" content="Mike 'Pomax' Kamermans" />
<meta property="og:section" content="Bézier Curves" /> <meta property="og:section" content="Bézier Curves" />
<meta property="og:tag" content="Bézier Curves" /> <meta property="og:tag" content="Bézier Curves" />

2
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@@ -34,7 +34,7 @@
<meta property="og:locale" content="en-GB" /> <meta property="og:locale" content="en-GB" />
<meta property="og:type" content="article" /> <meta property="og:type" content="article" />
<meta property="og:published_time" content="Sun Nov 22 2020 00:00:00 +00:00" /> <meta property="og:published_time" content="Sun Nov 22 2020 00:00:00 +00:00" />
<meta property="og:updated_time" content="Mon Jul 24 2023 16:25:38 +00:00" /> <meta property="og:updated_time" content="Tue Aug 15 2023 15:28:19 +00:00" />
<meta property="og:author" content="Mike 'Pomax' Kamermans" /> <meta property="og:author" content="Mike 'Pomax' Kamermans" />
<meta property="og:section" content="Bézier Curves" /> <meta property="og:section" content="Bézier Curves" />
<meta property="og:tag" content="Bézier Curves" /> <meta property="og:tag" content="Bézier Curves" />

2
docs/news/index.html generated
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@@ -33,7 +33,7 @@
<meta property="og:description" content="" /> <meta property="og:description" content="" />
<meta property="og:locale" content="en-GB" /> <meta property="og:locale" content="en-GB" />
<meta property="og:type" content="article" /> <meta property="og:type" content="article" />
<meta property="og:published_time" content="Mon Jul 24 2023 16:25:38 GMT+0000 (Coordinated Universal Time)" /> <meta property="og:published_time" content="Tue Aug 15 2023 15:28:19 GMT+0000 (Coordinated Universal Time)" />
<meta property="og:updated_time" content="" /> <meta property="og:updated_time" content="" />
<meta property="og:author" content="Mike 'Pomax' Kamermans" /> <meta property="og:author" content="Mike 'Pomax' Kamermans" />
<meta property="og:section" content="Bézier Curves" /> <meta property="og:section" content="Bézier Curves" />

2
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@@ -6,7 +6,7 @@
<atom:link href="https://pomax.github.io/bezierinfo" rel="self"></atom:link> <atom:link href="https://pomax.github.io/bezierinfo" rel="self"></atom:link>
<description>News updates for the <a href="https://pomax.github.io/bezierinfo">primer on Bézier Curves</a> by Pomax</description> <description>News updates for the <a href="https://pomax.github.io/bezierinfo">primer on Bézier Curves</a> by Pomax</description>
<language>en-GB</language> <language>en-GB</language>
<lastBuildDate>Mon Jul 24 2023 16:25:38 +00:00</lastBuildDate> <lastBuildDate>Tue Aug 15 2023 15:28:19 +00:00</lastBuildDate>
<image> <image>
<url>https://pomax.github.io/bezierinfo/images/og-image.png</url> <url>https://pomax.github.io/bezierinfo/images/og-image.png</url>
<title>A Primer on Bézier Curves</title> <title>A Primer on Bézier Curves</title>

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docs/ru-RU/index.html generated
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@@ -34,7 +34,7 @@
<meta property="og:locale" content="ru-RU" /> <meta property="og:locale" content="ru-RU" />
<meta property="og:type" content="article" /> <meta property="og:type" content="article" />
<meta property="og:published_time" content="2013-06-13T12:00:00+00:00" /> <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:author" content="Mike 'Pomax' Kamermans" />
<meta property="og:section" content="Bézier Curves" /> <meta property="og:section" content="Bézier Curves" />
<meta property="og:tag" content="Bézier Curves" /> <meta property="og:tag" content="Bézier Curves" />
@@ -8593,56 +8593,35 @@ O(t) = B(t) + d · N(t)
</p> </p>
<!-- <!--
╭ B'(t) ╮ ╭ B'(t) ╮
N(t) \bot │ ────────── │ N(t) = \bot │ ────────── │
╰ || B'(t)|| ╯ ╰ || B'(t)|| ╯
--> -->
<img <img
class="LaTeX SVG" class="LaTeX SVG"
src="./images/chapters/offsetting/57e62f3f2f7526b2cf7c1b276c17e472.svg" src="./images/chapters/offsetting/e7e8e1f5727387079dbc5770181187c2.svg"
width="120px" width="141px"
height="40px" height="40px"
loading="lazy" loading="lazy"
/> />
<p> <p>The magnitude of <code>B'(t)</code>, usually denoted with double vertical bars, is given by the following formula:</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>
<!-- <!--
┌───────────┐ ┌─────────────────┐
╭ b │ 2 2 │ 2 2
|| f(x,y)|| = | │f ' + f ' || B'(t)|| = │B '(t) + B '(t)
╯ a ⟍│ x y ⟍│ x y
--> -->
<img <img
class="LaTeX SVG" class="LaTeX SVG"
src="./images/chapters/offsetting/cf8e602eb0595cf4d9b851c6bda741af.svg" src="./images/chapters/offsetting/7fd3e895c0eea0965470dd619450b679.svg"
width="169px" width="188px"
height="36px" height="25px"
loading="lazy" loading="lazy"
/> />
<p> <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: And that's where things go wrong: that square root is screwing everything up, because it turns our nice polynomials into things that are
</p> no longer polynomials.
<!--
┌───────────────────┐
╭ 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.
</p> </p>
<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 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

53
docs/uk-UA/index.html generated
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@@ -39,7 +39,7 @@
<meta property="og:locale" content="uk-UA" /> <meta property="og:locale" content="uk-UA" />
<meta property="og:type" content="article" /> <meta property="og:type" content="article" />
<meta property="og:published_time" content="2013-06-13T12:00:00+00:00" /> <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:author" content="Mike 'Pomax' Kamermans" />
<meta property="og:section" content="Bézier Curves" /> <meta property="og:section" content="Bézier Curves" />
<meta property="og:tag" content="Bézier Curves" /> <meta property="og:tag" content="Bézier Curves" />
@@ -8553,56 +8553,35 @@ O(t) = B(t) + d · N(t)
</p> </p>
<!-- <!--
╭ B'(t) ╮ ╭ B'(t) ╮
N(t) \bot │ ────────── │ N(t) = \bot │ ────────── │
╰ || B'(t)|| ╯ ╰ || B'(t)|| ╯
--> -->
<img <img
class="LaTeX SVG" class="LaTeX SVG"
src="./images/chapters/offsetting/57e62f3f2f7526b2cf7c1b276c17e472.svg" src="./images/chapters/offsetting/e7e8e1f5727387079dbc5770181187c2.svg"
width="120px" width="141px"
height="40px" height="40px"
loading="lazy" loading="lazy"
/> />
<p> <p>The magnitude of <code>B'(t)</code>, usually denoted with double vertical bars, is given by the following formula:</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>
<!-- <!--
┌───────────┐ ┌─────────────────┐
╭ b │ 2 2 │ 2 2
|| f(x,y)|| = | │f ' + f ' || B'(t)|| = │B '(t) + B '(t)
╯ a ⟍│ x y ⟍│ x y
--> -->
<img <img
class="LaTeX SVG" class="LaTeX SVG"
src="./images/chapters/offsetting/cf8e602eb0595cf4d9b851c6bda741af.svg" src="./images/chapters/offsetting/7fd3e895c0eea0965470dd619450b679.svg"
width="169px" width="188px"
height="36px" height="25px"
loading="lazy" loading="lazy"
/> />
<p> <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: And that's where things go wrong: that square root is screwing everything up, because it turns our nice polynomials into things that are
</p> no longer polynomials.
<!--
┌───────────────────┐
╭ 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.
</p> </p>
<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 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

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@@ -35,7 +35,7 @@
<meta property="og:locale" content="zh-CN" /> <meta property="og:locale" content="zh-CN" />
<meta property="og:type" content="article" /> <meta property="og:type" content="article" />
<meta property="og:published_time" content="2013-06-13T12:00:00+00:00" /> <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:author" content="Mike 'Pomax' Kamermans" />
<meta property="og:section" content="Bézier Curves" /> <meta property="og:section" content="Bézier Curves" />
<meta property="og:tag" content="Bézier Curves" /> <meta property="og:tag" content="Bézier Curves" />
@@ -7872,56 +7872,35 @@ O(t) = B(t) + d · N(t)
</p> </p>
<!-- <!--
╭ B'(t) ╮ ╭ B'(t) ╮
N(t) \bot │ ────────── │ N(t) = \bot │ ────────── │
╰ || B'(t)|| ╯ ╰ || B'(t)|| ╯
--> -->
<img <img
class="LaTeX SVG" class="LaTeX SVG"
src="./images/chapters/offsetting/57e62f3f2f7526b2cf7c1b276c17e472.svg" src="./images/chapters/offsetting/e7e8e1f5727387079dbc5770181187c2.svg"
width="120px" width="141px"
height="40px" height="40px"
loading="lazy" loading="lazy"
/> />
<p> <p>The magnitude of <code>B'(t)</code>, usually denoted with double vertical bars, is given by the following formula:</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>
<!-- <!--
┌───────────┐ ┌─────────────────┐
╭ b │ 2 2 │ 2 2
|| f(x,y)|| = | │f ' + f ' || B'(t)|| = │B '(t) + B '(t)
╯ a ⟍│ x y ⟍│ x y
--> -->
<img <img
class="LaTeX SVG" class="LaTeX SVG"
src="./images/chapters/offsetting/cf8e602eb0595cf4d9b851c6bda741af.svg" src="./images/chapters/offsetting/7fd3e895c0eea0965470dd619450b679.svg"
width="169px" width="188px"
height="36px" height="25px"
loading="lazy" loading="lazy"
/> />
<p> <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: And that's where things go wrong: that square root is screwing everything up, because it turns our nice polynomials into things that are
</p> no longer polynomials.
<!--
┌───────────────────┐
╭ 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.
</p> </p>
<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 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