info/BezierInfo-2
1
0
Fork 0
mirror of https://github.com/Pomax/BezierInfo-2.git synced 2025-09-08 23:50:43 +02:00
Code Issues Releases Wiki Activity

.

Browse Source
This commit is contained in:
Pomax
2020-08-21 16:08:05 -07:00
parent c90565f493
commit 65173c10a2
17 changed files with 113 additions and 89 deletions
Download Patch File Download Diff File Expand all files Collapse all files

50
docs/zh-CN/index.html
View File

@@ -151,7 +151,7 @@
<li><a href="zh-CN/index.html#inflections">Curve inflections</a></li>
<li>
<a href="zh-CN/index.html#canonical"
>Canonical form (for cubic curves)</a
>The canonical form (for cubic curves)</a
>
</li>
<li><a href="zh-CN/index.html#yforx">Finding Y, given X</a></li>
@@ -3510,7 +3510,7 @@ function getCubicRoots(pa, pb, pc, pd) {
<section id="canonical">
<h1>
<a href="zh-CN/index.html#canonical"
>Canonical form (for cubic curves)</a
>The canonical form (for cubic curves)</a
>
</h1>
<p>
@@ -3694,11 +3694,11 @@ function getCubicRoots(pa, pb, pc, pd) {
<p>
The approach is going to start with a curve that doesn't have
all-colinear points (so we need to make sure the points don't all
fall on a straight line), and then applying four graphics operations
that you will probably have heard of: translation (moving all points
by some fixed x- and y-distance), scaling (multiplying all points by
some x and y scale factor), and shearing (an operation that turns
rectangles into parallelograms).
fall on a straight line), and then applying three graphics
operations that you will probably have heard of: translation (moving
all points by some fixed x- and y-distance), scaling (multiplying
all points by some x and y scale factor), and shearing (an operation
that turns rectangles into parallelograms).
</p>
<p>
Step 1: we translate any curve by -p1.x and -p1.y, so that the curve
@@ -3714,8 +3714,8 @@ function getCubicRoots(pa, pb, pc, pd) {
</p>
<img
class="LaTeX SVG"
src="./images/chapters/canonical/d089cc0687982a3302249bb82af3fc16.svg"
width="499px"
src="./images/chapters/canonical/2a411f175dcc987cdcc12e7df49ca272.svg"
width="464px"
height="55px"
loading="lazy"
/>
@@ -3726,8 +3726,8 @@ function getCubicRoots(pa, pb, pc, pd) {
</p>
<img
class="LaTeX SVG"
src="./images/chapters/canonical/058fa85ac31eb666857a860fdedd79df.svg"
width="477px"
src="./images/chapters/canonical/ba5f418452c3657f3c4dd4b319e59070.svg"
width="447px"
height="57px"
loading="lazy"
/>
@@ -3812,16 +3812,18 @@ function getCubicRoots(pa, pb, pc, pd) {
</p>
<img
class="LaTeX SVG"
src="./images/chapters/canonical/2f85d84f0e3dd14cc25e48583aed3822.svg"
width="460px"
height="80px"
src="./images/chapters/canonical/f039b4e7cf0203df9fac48dad820b2b7.svg"
width="455px"
height="91px"
loading="lazy"
/>
<p>
That looks very complex, but notice that every coordinate value is
being offset by the initial translation, and a lot of terms in there
repeat: it's pretty easy to calculate this fast, since there's so
much we can cache and reuse while we compute this mapped coordinate!
Okay, well, that looks plain ridiculous, but: notice that every
coordinate value is being offset by the initial translation, and
also notice that <em>a lot</em> of terms in that expression are
repeated. Even though the maths looks crazy as a single expression,
we can just pull this apart a little and end up with an
easy-to-calculate bit of code!
</p>
<p>
First, let's just do that translation step as a "preprocessing"
@@ -3830,9 +3832,9 @@ function getCubicRoots(pa, pb, pc, pd) {
</p>
<img
class="LaTeX SVG"
src="./images/chapters/canonical/83262761bb7fa9b832fe483ded436973.svg"
width="359px"
height="56px"
src="./images/chapters/canonical/16fad73cbbbd2202b08ebef05b1579c5.svg"
width="317px"
height="67px"
loading="lazy"
/>
<p>
@@ -3844,9 +3846,9 @@ function getCubicRoots(pa, pb, pc, pd) {
</p>
<img
class="LaTeX SVG"
src="./images/chapters/canonical/f3261ad2802d980ebe6e35b272375700.svg"
width="427px"
height="40px"
src="./images/chapters/canonical/5af3d1772ee07e634d04259a02f7201f.svg"
width="483px"
height="56px"
loading="lazy"
/>
<p>