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2021-08-30 15:01:03 +00:00
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@@ -41,7 +41,7 @@
<meta property="og:locale" content="ja-JP" />
<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="2021-08-30T14:51:35+00:00" />
<meta property="og:updated_time" content="2021-08-30T15:00:31+00:00" />
<meta property="og:author" content="Mike 'Pomax' Kamermans" />
<meta property="og:section" content="Bézier Curves" />
<meta property="og:tag" content="Bézier Curves" />
@@ -6234,30 +6234,30 @@ lli = function(line1, line2):
<!--
\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
╭ A' - e
│ 1
│ v = A' - ───────
╡ 1 1 - t
│ A' - e
│ 2
│ v = A' - ───────
╰ 2 t
╭ A - e
│ 1
│ v = A - ──────
╡ 1 1 - t
│ A - e
│ 2
│ v = A - ──────
╰ 2 t
-->
<img class="LaTeX SVG" src="./images/chapters/abc/eccc1bdb9423bbfe2d42418fc8a7dd24.svg" width="132px" height="75px" loading="lazy" />
<img class="LaTeX SVG" src="./images/chapters/abc/68a25507037f1a9420c60a5cd3d10f47.svg" width="121px" height="73px" loading="lazy" />
<p>And then reverse engineer the curve's control points:</p>
<!--
\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
╭ v - start
│ 1
│ C '= start + ───────────
╡ 1 t
│ v - end
│ 2
│ C '= end + ─────────
╰ 2 1 - t
╭ v - start
│ 1
│ C = start + ───────────
╡ 1 t
│ v - end
│ 2
│ C = end + ─────────
╰ 2 1 - t
-->
<img class="LaTeX SVG" src="./images/chapters/abc/634d373310711268cc188f45e5699d8d.svg" width="163px" height="73px" loading="lazy" />
<img class="LaTeX SVG" src="./images/chapters/abc/2184aa2a897df864b3a67984be18ef27.svg" width="163px" height="73px" loading="lazy" />
<p>
So: if we have a curve's start and end points, then for any <code>t</code> value we implicitly know all the ABC values, which (combined
with an educated guess on appropriate <code>e1</code> and <code>e2</code> coordinates for cubic curves) gives us the necessary information