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catmull

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Pomax
2016-01-08 16:10:39 -08:00
parent fba01ea536
commit b4910017ac
31 changed files with 3933 additions and 73 deletions
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22
components/Graphic.jsx
View File

@@ -343,6 +343,28 @@ var Graphic = React.createClass({
}
},
drawGrid: function(xcount, ycount, offset) {
var w = this.defaultWidth,
h = this.defaultHeight,
xstep = w/xcount,
ox = xstep/2,
ystep = h/ycount,
oy = ystep/2,
x, xv, y, yv, p1, p2;
for (x=0; x<xcount; x++) {
xv = ox + (x*xstep);
p1 = {x:xv, y:0};
p2 = {x:xv, y:h};
this.drawLine(p1, p2, offset);
}
for (y=0; y<ycount; y++) {
yv = oy + (y*ystep);
p1 = {x:0, y:yv};
p2 = {x:w, y:yv};
this.drawLine(p1, p2, offset);
}
},
drawHull: function(curve, t, offset) {
var hull = typeof curve === "array" ? curve : curve.hull(t);
if(hull.length === 6) {

186
components/sections/abc/index.js
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@@ -7,7 +7,7 @@ var abs = Math.abs;
var ABC = React.createClass({
getDefaultProps: function() {
return {
title: "The 'ABC' curve identity"
title: "The projection identity"
};
},
@@ -69,7 +69,7 @@ var ABC = React.createClass({
api.setFill("black");
api.text("A", {x:10 + A.x, y: A.y});
api.text("B", {x:10 + B.x, y: B.y});
api.text("B (t = " + api.utils.round(api.t,2) + ")", {x:10 + B.x, y: B.y});
api.text("C", {x:10 + C.x, y: C.y});
var d1 = utils.dist(A, B);
@@ -80,6 +80,59 @@ var ABC = React.createClass({
}
},
setCT: function(evt,api) {
api.t = evt.offsetX / api.getPanelWidth();
},
drawCTgraph: function(api) {
api.reset();
api.setColor("black");
var w = api.getPanelWidth();
var h = api.getPanelHeight();
var pad = 20;
var fwh = w - 2*pad;
api.drawAxes(pad, "t",0,1, "u",0,1);
api.setColor("blue");
var uPoint = function(t) {
var value = api.u(t),
res = { x: pad + t*fwh, y: pad + value*fwh };
return res;
}
api.drawFunction(uPoint);
if (api.t) {
var v = api.u(api.t),
v1 = api.utils.round(v,3),
v2 = api.utils.round(1-v,3),
up = uPoint(api.t);
api.drawLine({x:up.x,y:pad}, up);
api.drawLine({x:pad,y:up.y}, up);
api.drawCircle(up,3);
api.setFill("blue");
api.text(" t = " + api.utils.round(api.t,3), {x:up.x+10, y:up.y-7});
api.text("u(t) = " + api.utils.round(v,3), {x:up.x+10, y:up.y+7});
api.setFill("black");
api.text("C = "+v1+" * start + "+v2+" * end", {x:w/2 - pad, y:pad+fwh});
}
},
drawQCT: function(api) {
api.u = api.u || function(t) {
var top = (t-1) * (t-1),
bottom = 2*t*t - 2*t + 1;
return top/bottom;
};
this.drawCTgraph(api);
},
drawCCT: function(api) {
api.u = api.u || function(t) {
var top = (1-t) * (1-t) * (1-t),
bottom = t*t*t + top;
return top/bottom;
};
this.drawCTgraph(api);
},
render: function() {
return (
<section>
@@ -92,16 +145,15 @@ var ABC = React.createClass({
<p>How does that work? Succinctly: we run de Casteljau's algorithm in reverse!</p>
<p>Let's start out with a pre-existing curve, defined by <i>start</i>, two control points, and <i>end</i>. We can
mould this curve by picking a point somewhere on the curve, at some <i>t</i> value, and the moving it to a new
location and reconstructing the curve that goes through <i>start</i>, our new point with the original tangent,
and <i>end</i>. In order to see how and why we can do this, let's look at some identity information for Bézier
curves. There's actually a hidden goldmine of identities that we can exploit when doing Bézier operations, and
this will only scratch the surface. But, in a good way!</p>
<p>In order to run de Casteljau's algorithm in reverse, we need a few basic things: a start and end point, a point
on the curve that want to be moving around, which has an associated <i>t</i> value, and a point we've not explicitly
talked about before, and as far as I know has no explicit name, but lives one iteration higher in the de Casteljau
process then our on-curve point does. I like to call it "A" for reasons that will become obvious.</p>
<p>In the following graphic, click anywhere on the curves to see the identity information that we'll
be using to run de Casteljau in reverse (you can manipulate the curve even after picking a point.
Note the "ratio" value when you do so: does it change?):</p>
<p>So let's use graphics instead of text to see where this "A" is, because text only gets us so far: in the
following graphic, click anywhere on the curves to see the identity information that we'll be using to run
de Casteljau in reverse (you can manipulate the curve even after picking a point. Note the "ratio" value
when you do so: does it change?):</p>
<div className="figure">
<Graphic inline={true} preset="abc" title="Projections in a quadratic Bézier curve"
@@ -110,52 +162,81 @@ var ABC = React.createClass({
setup={this.setupCubic} draw={this.draw} onClick={this.onClick} />
</div>
<p>So, what exactly do we see in these graphics? First off, there's the three points <i>A</i>, <i>B</i> and <i>C</i>.</p>
<p>Clicking anywhere on the curves shows us three things:</p>
<p>Point <i>B</i> is our "on curve" point, A is the first "strut" point when running de Casteljau's
algorithm in reverse; for quadratic curves, this happens to also be the curve's control point. For cubic
curves, it's the "top of the triangle" for the struts that lead to point <i>B</i>. Point <i>C</i>, finally,
is the intersection of the line that goes through <i>A</i> and <i>B</i> and the baseline,
between our start and end points.</p>
<ol>
<li>our on-curve point; let's call that <b>B</b>,</li>
<li>a point at the tip of B's "hat", on de Casteljau step up; let's call that <b>A</b>, and</li>
<li>a point that we get by projecting B onto the start--end baseline; let's call that <b>C</b>.</li>
</ol>
<p>There is some important identity information here: as long as we don't pick a new <i>t</i> coordinate,
the location of point <i>C</i> on the line <i>start-end</i> represents a fixed ratio distance. We can drag
around the control points as much as we like, that point won't move at all, and if we can drag around
the start or end point, C will stay at the same ratio-value. For instance, if it was located midway between
start and end, it'll stay midway between start and end, even if the line segment between start and end
becomes longer or shorter.</p>
<p>These three values ABC hide an important identity formula for quadratic and cubic Bézier curves:
for any point on the curve with some <i>t</i> value, the ratio distance of C along baseline is fixed:
if some <i>t</i> value sets up a C that is 20% away from the start and 80% away from the end, then it
doesn't matter where the start, end, or control points are: for that <i>t</i> value, C will <em>always</em> lie
at 20% from the start and 80% from the end point. Go ahead, pick an on-curve point in either graphic
and then move all the other points around: if you only move the control points, start and end won't move,
and so neither will C, and if you move either start or end point, C will move but its relative position
will not change. The following function stays true:</p>
<p>We can also see that the distances for the lines <i>d1 = A-B</i> and <i>d2 = B-C</i> may vary, but the
ratio between them, <i>d1/d2</i>, is a constant value. We can drag any of the start, end, or control points
around as much as we like, but that value also stays the same.</p>
<p>\[
C = u \cdot P_{start} + (1-u) \cdot P_{end}
\]</p>
<p>So that just leaves finding A.</p>
<div className="note">
<p>In fact, because the distance ratio is a fixed value for each point <i>B</i>, which we get by picking
some <i>t</i> value on our curve, the distance ratio is actually an identity function for Bézier curves.
If we were to plot all the ratio values for all possible <i>t</i> values for quadratic and cubic curves,
we'd see two very interesting functions: asymptotic at <i>t=0</i> and <i>t=1</i>, tending towards positive
infinity, with a zero-derivative minimum at <i>t=0.5</i>.</p>
<p>While that relation is fixed, the function <i>u(t)</i> differs depending on whether we're working
with quadratic or cubic curves:</p>
<p>Since these are ratios, we can actually express the ratio values as a function of <i>t</i>. I actually
failed at coming up with the precise functions, but thanks to some help from
<a href="http://mathoverflow.net/questions/122257/finding-the-formula-for-Bézier-curve-ratios-hull-point-point-baseline">Boris
Zbarsky</a> we can see that the ratio functions are actually remarkably simple:</p>
<p>\[\begin{align}
& u(t)_{quadratic} = \frac{(t-1)^2}{2t^2 - 2t + 1} \\
& u(t)_{cubic} = \frac{(1-t)^3}{t^3 + (1-t)^3}
\end{align}\]</p>
<table style={{width:"100%", border:0}}>
<tbody>
<tr>
<td>
<p>Quadratic curves:\[
ratio(t)_2 = \left | \frac{2t^2 - 2t}{2t^2 - 2t + 1} \right |
\]</p>
</td><td>
<p>Cubic curves: \[
ratio(t)_3 = \left | \frac{t^3 + (1-t)^3 - 1}{t^3 + (1-t)^3} \right |
\]</p>
</td>
</tr>
</tbody>
</table>
<p>So, if we know the start and end coordinates, and we know the <i>t</i> value, we know C:</p>
<div className="figure">
<Graphic inline={true} preset="abc" title="Quadratic value of C for t" draw={this.drawQCT} onMouseMove={this.setCT}/>
<Graphic inline={true} preset="abc" title="Cubic value of C for t" draw={this.drawCCT} onMouseMove={this.setCT}/>
</div>
<p>Mouse-over the graphs to see the expression for C, given the <i>t</i> value at the mouse pointer.</p>
</div>
<p>There's also another important bit of information that is inherent to the ABC values: while the distances
between A and B, and B and C, are dynamic (based on where we put B), the <em>ratio</em> between the two
distances is stable: given some <i>t</i> value, the following always holds:</p>
<p>\[
ratio(t) = \frac{distance(B,C)}{distance(A,B)} = Constant
\]</p>
<p>This leads to a pretty powerful bit of knowledge: merely by knowing the <i>t</i> value of some on curve
point, we know where C has to be (as per the above note), and because we know B and C, and thus have the
distance between them, we know where A has to be:</p>
<p>\[
A = B - \frac{C - B}{ratio(t)} = B + \frac{B - C}{ratio(t)}
\]</p>
<p>And that's it, all values found.</p>
<div className="note">
<p>Much like the <i>u(t)</i> function in the above note, the <i>ratio(t)</i> function depends
on whether we're looking at quadratic or cubic curves. Their form is intrinsically related to
the <i>u(t)</i> function in that they both come rolling out of the same function evalution,
explained over on <a href="http://mathoverflow.net/questions/122257/finding-the-formula-for-Bézier-curve-ratios-hull-point-point-baseline">MathOverflow</a> by
Boris Zbarsky and myself. The ratio functions are the "s(t)" functions from the answers there,
while the "u(t)" functions have the same name both here and on MathOverflow.</p>
<p>\[
ratio(t)_{quadratic} = \left | \frac{2t^2 - 2t}{2t^2 - 2t + 1} \right |
\]</p>
<p>\[
ratio(t)_{cubic} = \left | \frac{t^3 + (1-t)^3 - 1}{t^3 + (1-t)^3} \right |
\]</p>
<p>Unfortunately, this trick only works for quadratic and cubic curves. Once we hit higher order curves,
things become a lot less predictable; the "fixed point <i>C</i>" is no longer fixed, moving around as we
@@ -163,6 +244,11 @@ var ABC = React.createClass({
lie on that line before the start, or after the end, and there are no simple ratios that we can exploit.</p>
</div>
<p>So: if we know B and its corresponding <i>t</i> value, then we know all the ABC values, which
—together with a start and end coordinate— gives us the necessary information to reconstruct a curve's
"de Casteljau skeleton", which means that two points and a value between 0 and 1, we can come up with
a curve. And that opens up possibilities: curve manipulation by dragging an on-curve point, curve fitting
of "a bunch of coordinates", these are useful things, and we'll look at both in the next sections.</p>
</section>
);
}

417
components/sections/catmullconv/index.js Normal file
View File

@@ -0,0 +1,417 @@
var React = require("react");
var Graphic = require("../../Graphic.jsx");
var SectionHeader = require("../../SectionHeader.jsx");
var CatmullRomConversion = React.createClass({
getDefaultProps: function() {
return {
title: "Bézier curves and Catmull-Rom curves"
};
},
render: function() {
return (
<section>
<SectionHeader {...this.props} />
<p>Taking an excursion to different splines, the other common design curve is
the <a href="https://en.wikipedia.org/wiki/Cubic_Hermite_spline#Catmull.E2.80.93Rom_spline">Catmull-Rom
spline</a>. Now, a Catmull-Rom spline is a form of cubic Hermite spline, and as it so happens
the cubic Bézier curve is also a cubic Hermite spline, so maybe... maybe we can convert one
into the other, and back, with some simple substitutions?</p>
<p>Unlike Bézier curves, Catmull-Rom splines pass through each point used to define the curve,
except the first and last, which makes sense if you read the "natural language" description
for how a Catmull-Rom spline works: a Catmull-Rom spline is a curve that, at each point
P<sub>x</sub>, has a tangent along the line P<sub>x-1</sub> to P<sub>x+1</sub>. The curve
runs from points P<sub>2</sub> to P<sub>n-1</sub>, and has a "tension" that determines
how fast the curve passes through each point. The lower the tension, the faster the curve
goes through each point, and the bigger its local tangent is.</p>
<p>I'll be showing the conversion to and from Catmull-Rom curves for the tension that the
Processing language uses for its Catmull-Rom algorithm.</p>
<p>We start with showing the Catmull-Rom matrix form:</p>
<p>\[
CatmullRom(t) =
\begin{bmatrix}
1 & t & t^2 & t^3
\end{bmatrix}
\cdot
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
-3 & 3 & -2 & -1 \\
2 & -2 & 1 & 1
\end{bmatrix}
\cdot
\begin{bmatrix}
V_1 \\ V_2 \\ V'_1 \\ V'_2
\end{bmatrix}
\]
</p>
<p>However, there's something funny going on here: the coordinate column matrix looks weird.
The reason is that Catmull-Rom curves are actually curve segments that are described by two
points, and two tangents; the curve leaves a point V1 (if we have four coordinates instead,
this is coordinate 2), arriving at a point V2 (coordinate 3), with the curve departing V1
with a tangent vector V'1 (equal to the tangent from coordinate 1 to coordinate 3) and
arriving at V2 with tangent vector V'2 (equal to the tangent from coordinate 2 to coordinate
4). So if we want to express this as a matrix form based on four coordinates, we get this
representation instead:</p>
<p>\[
\begin{bmatrix}
V_1 \\ V_2 \\ V'_1 \\ V'_2
\end{bmatrix}
=
T
\cdot
\begin{bmatrix}
P_1 \\ P_2 \\ P_3 \\ P_4
\end{bmatrix}
=
\begin{bmatrix}
P_2 \\ P_3 \\ \frac{P_3 - P_1}{2} \\ \frac{P_4 - P_2}{2}
\end{bmatrix}
\ \Rightarrow \
T
=
\frac{1}{2}
\cdot
\begin{bmatrix}
0 & 2 & 0 & 0 \\
0 & 0 & 2 & 0 \\
-1 & 0 & 1 & 0 \\
0 & -1 & 0 & 1
\end{bmatrix}
\]</p>
<div className="note">
<h2>Where did that 2 come from?</h2>
<p>Catmull-Rom splines are based on the concept of tension: the higher the tensions,
the shorter the tangents at the departure and arrival points. The basic Catmull-Rom
curve arrives and departs with tangents equal to half the distance between the two
adjacent points, so that's where that 2 came from.</p>
<p>However, the "real" matrix is this:</p>
<p>\[
T
\cdot
\begin{bmatrix}
P_1 \\ P_2 \\ P_3 \\ P_4
\end{bmatrix}
=
\begin{bmatrix}
P_2 \\ P_3 \\ \frac{P_3 - P_1}{2 \cdot τ} \\ \frac{P_4 - P_2}{2 \cdot τ}
\end{bmatrix}
\Rightarrow \
T
=
\frac{1}{2}
\cdot
\begin{bmatrix}
0 & 2 & 0 & 0 \\
0 & 0 & 2 & 0 \\
-τ & 0 & τ & 0 \\
0 & -τ & 0 & τ
\end{bmatrix}
\]</p>
<p>This bakes in the tension factor τ explicitly.</p>
</div>
<p>Plugging this into the "two coordinates and two tangent vectors" matrix form,
we get:</p>
<p>\[
\begin{bmatrix}
1 & t & t^2 & t^3
\end{bmatrix}
\cdot
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
-3 & 3 & -2 & -1 \\
2 & -2 & 1 & 1
\end{bmatrix}
\cdot
\begin{bmatrix}
V_1 \\ V_2 \\ V'_1 \\ V'_2
\end{bmatrix}
\]</p>
<p>\[
=
\begin{bmatrix}
1 & t & t^2 & t^3
\end{bmatrix}
\cdot
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
-3 & 3 & -2 & -1 \\
2 & -2 & 1 & 1
\end{bmatrix}
\cdot
\left (
\frac{1}{2}
\cdot
\begin{bmatrix}
0 & 2 & 0 & 0 \\
0 & 0 & 2 & 0 \\
-τ & 0 & τ & 0 \\
0 & -τ & 0 & τ
\end{bmatrix}
\right )
\cdot
\begin{bmatrix}
P_1 \\ P_2 \\ P_3 \\ P_4
\end{bmatrix}
\]</p>
<p>\[
=
\begin{bmatrix}
1 & t & t^2 & t^3
\end{bmatrix}
\cdot
\frac{1}{2}
\cdot
\begin{bmatrix}
0 & 2 & 0 & 0 \\
-τ & 0 & τ & 0 \\
2τ & τ-6 & -2(τ-3) & -τ \\
-τ & 4-τ & τ-4 & τ
\end{bmatrix}
\cdot
\begin{bmatrix}
P_1 \\ P_2 \\ P_3 \\ P_4
\end{bmatrix}
\]</p>
<p>So let's find out which transformation matrix we need in order to convert from Catmull-Rom to Bézier:</p>
<p>\[
\begin{bmatrix}
1 & t & t^2 & t^3
\end{bmatrix}
\cdot
\frac{1}{2}
\cdot
\begin{bmatrix}
0 & 2 & 0 & 0 \\
-τ & 0 & τ & 0 \\
2τ & τ-6 & -2(τ-3) & -τ \\
-τ & 4-τ & τ-4 & τ
\end{bmatrix}
\cdot
\begin{bmatrix}
P_1 \\ P_2 \\ P_3 \\ P_4
\end{bmatrix}
=
\begin{bmatrix}
1 & t & t^2 & t^3
\end{bmatrix}
\cdot
\begin{bmatrix}
1 & 0 & 0 & 0 \\
-3 & 3 & 0 & 0 \\
3 & -6 & 3 & 0 \\
-1 & 3 & -3 & 1
\end{bmatrix}
\cdot
A
\cdot
\begin{bmatrix}
P_1 \\ P_2 \\ P_3 \\ P_4
\end{bmatrix}
\]</p>
<p>The difference is somewhere in the actual hermite matrix, since the <em>t</em> and coordinate
values are identical, so let's solve that matrix equasion:</p>
<p>\[
\frac{1}{2}
\cdot
\begin{bmatrix}
0 & 2 & 0 & 0 \\
-τ & 0 & τ & 0 \\
2τ & τ-6 & -2(τ-3) & -τ \\
-τ & 4-τ & τ-4 & τ
\end{bmatrix}
=
\begin{bmatrix}
1 & 0 & 0 & 0 \\
-3 & 3 & 0 & 0 \\
3 & -6 & 3 & 0 \\
-1 & 3 & -3 & 1
\end{bmatrix}
\cdot
A
\]</p>
<p>We left-multiply both sides by the inverse of the Bézier matrix, to get rid of the
Bézier matrix on the right side of the equals sign:</p>
<p>\[
{
\begin{bmatrix}
1 & 0 & 0 & 0 \\
-3 & 3 & 0 & 0 \\
3 & -6 & 3 & 0 \\
-1 & 3 & -3 & 1
\end{bmatrix}
}^{-1}
\cdot
\frac{1}{2}
\cdot
\begin{bmatrix}
0 & 2 & 0 & 0 \\
-τ & 0 & τ & 0 \\
2τ & τ-6 & -2(τ-3) & -τ \\
-τ & 4-τ & τ-4 & τ
\end{bmatrix}
=
{
\begin{bmatrix}
1 & 0 & 0 & 0 \\
-3 & 3 & 0 & 0 \\
3 & -6 & 3 & 0 \\
-1 & 3 & -3 & 1
\end{bmatrix}
}^{-1}
\cdot
\begin{bmatrix}
1 & 0 & 0 & 0 \\
-3 & 3 & 0 & 0 \\
3 & -6 & 3 & 0 \\
-1 & 3 & -3 & 1
\end{bmatrix}
\cdot
A
\ =\
I \cdot A
\ =\
A
\]
</p>
<p>Which gives us:</p>
<p>\[
\frac{1}{6}
\cdot
\begin{bmatrix}
0 & 6 & 0 & 0 \\
-τ & 6 & τ & 0 \\
0 & τ & 0 & -τ \\
0 & 0 & 6 & 0
\end{bmatrix}
=
A
\]</p>
<p>Multiplying this <strong><em>A</em></strong> with our coordinates
will give us a proper Bézier matrix expression again:</p>
<p>\[
\begin{bmatrix}
1 & t & t^2 & t^3
\end{bmatrix}
\cdot
\begin{bmatrix}
1 & 0 & 0 & 0 \\
-3 & 3 & 0 & 0 \\
3 & -6 & 3 & 0 \\
-1 & 3 & -3 & 1
\end{bmatrix}
\cdot
\frac{1}{6}
\cdot
\begin{bmatrix}
0 & 6 & 0 & 0 \\
-τ & 6 & τ & 0 \\
0 & τ & 0 & -τ \\
0 & 0 & 6 & 0
\end{bmatrix}
\cdot
\begin{bmatrix}
P_1 \\ P_2 \\ P_3 \\ P_4
\end{bmatrix}
\]</p>
<p>\[
=
\begin{bmatrix}
1 & t & t^2 & t^3
\end{bmatrix}
\cdot
\begin{bmatrix}
1 & 0 & 0 & 0 \\
-3 & 3 & 0 & 0 \\
3 & -6 & 3 & 0 \\
-1 & 3 & -3 & 1
\end{bmatrix}
\cdot
\begin{bmatrix}
P_2 \\
P_2 + \frac{P_3-P_1}{6 \cdot τ} \\
P_3 - \frac{P_4-P_2}{6 \cdot τ} \\
P_3
\end{bmatrix}
\]</p>
<p>So a Catmull-Rom to Bézier conversion, based on coordinates, requires turning
the Catmull-Rom coordinates on the left into the Bézier coordinates on the right
(with τ being our tension factor):</p>
<p>\[
\begin{bmatrix}
P_1 \\
P_2 \\
P_3 \\
P_4
\end{bmatrix}_{CatmullRom}
\Rightarrow
\begin{bmatrix}
P_2 \\
P_2 + \frac{P_3-P_1}{6 \cdot τ} \\
P_3 - \frac{P_4-P_2}{6 \cdot τ} \\
P_3
\end{bmatrix}_{Bézier}
\]</p>
<p>And the other way around, a Bézier to Catmull-Rom conversion requires turning
the Bézier coordinates on the left this time into the Catmull-Rom coordinates
on the right. Note that there is no tension this time, because Bézier curves
don't have any. Converting from Bézier to Catmull-Rom is simply a default-tension
Catmull-Rom curve:</p>
<p>\[
\begin{bmatrix}
P_1 \\
P_2 \\
P_3 \\
P_4
\end{bmatrix}_{Bézier}
\Rightarrow
\begin{bmatrix}
P_4 + 6 \cdot (P_1 - P_2) \\
P_1 \\
P_4 \\
P_1 + 6 \cdot (P_4 - P_3)
\end{bmatrix}_{CatmullRom}
\]</p>
<p>Done. We can now draw the curves we want using either Bézier curves or Catmull-Rom
splines, the choice mostly being which drawing algorithms we have natively available.</p>
</section>
);
}
});
module.exports = CatmullRomConversion;

184
components/sections/catmullmoulding/index.js Normal file
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@@ -0,0 +1,184 @@
var React = require("react");
var Graphic = require("../../Graphic.jsx");
var SectionHeader = require("../../SectionHeader.jsx");
var CatmullRomMoulding = React.createClass({
getDefaultProps: function() {
return {
title: "Creating a Catmull-Rom curve from three points"
};
},
setup: function(api) {
api.setPanelCount(3);
api.lpts = [
{x:56, y:153},
{x:144,y:83},
{x:188,y:185}
];
api.distance = 0;
},
onKeyDown: function(evt, api) {
if (evt.key === "ArrowUp") {
evt.preventDefault();
api.distance += 1;
} else if (evt.key === "ArrowDown") {
evt.preventDefault();
api.distance -= 1;
}
},
convert: function(p1, p2, p3, p4) {
var t = 0.5;
return [
p2,
{
x: p2.x + (p3.x-p1.x)/(6*t),
y: p2.y + (p3.y-p1.y)/(6*t),
},
{
x: p3.x - (p4.x-p2.x)/(6*t),
y: p3.y - (p4.y-p2.y)/(6*t),
},
p3
];
},
draw: function(api) {
api.reset();
api.setColor("lightblue");
api.drawGrid(10,10);
var pts = api.lpts;
api.setColor("black");
api.setFill("black");
pts.forEach((p,pos) => {
api.drawCircle(p, 3);
api.text("point "+(pos+1), p, {x:10, y:7});
});
var w = api.getPanelWidth();
var h = api.getPanelHeight();
var offset = {x:w, y:0};
api.setColor("lightblue");
api.drawGrid(10,10,offset);
api.setColor("black");
api.drawLine({x:0,y:0}, {x:0,y:h}, offset);
pts.forEach((p,pos) => {
api.drawCircle(p, 3, offset);
});
var p1 = pts[0], p2 = pts[1], p3 = pts[2];
var dx = p3.x - p1.x,
dy = p3.y - p1.y,
m = Math.sqrt(dx*dx + dy*dy);
dx /= m;
dy /= m;
api.drawLine(p1, p3, offset);
var p0 = {
x: p1.x + (p3.x - p2.x) - api.distance * dx,
y: p1.y + (p3.y - p2.y) - api.distance * dy
};
var p4 = {
x: p1.x + (p3.x - p2.x) + api.distance * dx,
y: p1.y + (p3.y - p2.y) + api.distance * dy
};
var center = api.utils.lli4(p1,p3,p2,{
x: (p0.x + p4.x)/2,
y: (p0.y + p4.y)/2
});
api.setColor("blue");
api.drawCircle(center, 3, offset);
api.drawLine(pts[1],center, offset);
api.setColor("#666");
api.drawLine(center, p0, offset);
api.drawLine(center, p4, offset);
api.setFill("blue");
api.text("p0", p0, {x:-20 + offset.x, y:offset.y + 2});
api.text("p4", p4, {x:+10 + offset.x, y:offset.y + 2});
// virtual point p0
api.setColor("red");
api.drawCircle(p0, 3, offset);
api.drawLine(p2, p0, offset);
api.drawLine(p1, {
x: p1.x + (p2.x - p0.x)/5,
y: p1.y + (p2.y - p0.y)/5
}, offset);
// virtual point p4
api.setColor("#00FF00");
api.drawCircle(p4, 3, offset);
api.drawLine(p2, p4, offset);
api.drawLine(p3, {
x: p3.x + (p4.x - p2.x)/5,
y: p3.y + (p4.y - p2.y)/5
}, offset);
// Catmull-Rom curve for p0-p1-p2-p3-p4
var c1 = new api.Bezier(this.convert(p0,p1,p2,p3)),
c2 = new api.Bezier(this.convert(p1,p2,p3,p4));
api.setColor("lightgrey");
api.drawCurve(c1, offset);
api.drawCurve(c2, offset);
offset.x += w;
api.setColor("lightblue");
api.drawGrid(10,10,offset);
api.setColor("black");
api.drawLine({x:0,y:0}, {x:0,y:h}, offset);
api.drawCurve(c1, offset);
api.drawCurve(c2, offset);
api.drawPoints(c1.points, offset);
api.drawPoints(c2.points, offset);
api.setColor("lightgrey");
api.drawLine(c1.points[0], c1.points[1], offset);
api.drawLine(c1.points[2], c2.points[1], offset);
api.drawLine(c2.points[2], c2.points[3], offset);
},
render: function() {
return (
<section>
<SectionHeader {...this.props} />
<p>Now, we saw how to fit a Bézier curve to three points, but if Catmull-Rom curves go through points,
why can't we just use those to do curve fitting, instead?</p>
<p>As a matter of fact, we can, but there's a difference between the kind of curve fitting we did in the
previous section, and the kind of curve fitting that we can do with Catmull-Rom curves. In the previous
section we came up with a single curve that goes through three points. There was a decent amount of
maths and computation involved, and the end result was three or four coordinates that described a
single curve, depending on whether we were fitting a quadratic or cubic curve.</p>
<p>Using Catmull-Rom curves, we need virtually no computation, but even though we end up with one Catmull-Rom
curve of <i>n</i> points, in order to draw the equivalent curve using cubic Bézier curves we need a
massive <i>3n-1</i> points (and that's without double-counting points that are shared by consecutive
cubic curves).</p>
<p>In the following graphic, on the left we see three points that we want to draw a Catmull-Rom curve through
(which we can move around freely, by the way), with in the second panel some of the "interesting" Catmull-Rom
information: in black there's the baseline start--end, which will act as tangent orientation for the curve at
point p2. We also see a virtual point p0 and p4, which are initially just point p2 reflected over the baseline.
However, by using the up and down cursor key we can offset these points parallel to the baseline. Why would
we want to do this? Because the line p0--p2 acts as departure tangent at p1, and the line p2--p4 acts as
arrival tangent at p3. Play around with the graphic a bit to get an idea of what all of that meant:</p>
<Graphic preset="threepanel" title="Catmull-Rom curve fitting" setup={this.setup} draw={this.draw} onKeyDown={this.onKeyDown}/>
<p>As should be obvious by now, Catmull-Rom curves are great for "fitting a curvature to some points", but if we want
to convert that curve to Bézier form we're going to end up with a lot of separate (but visually joined) Bézier curves.
Depending on what we want to do, that'll be either unnecessary work, or exactly what we want: which it is depends
entirely on you.</p>
</section>
);
}
});
module.exports = CatmullRomMoulding;

11
components/sections/index.js
View File

@@ -34,14 +34,14 @@ module.exports = {
abc: require("./abc"),
moulding: require("./moulding"),
pointcurves: require("./pointcurves")
pointcurves: require("./pointcurves"),
catmullconv: require("./catmullconv"),
catmullmoulding: require("./catmullmoulding")
};
/*
catmullconv: require("./catmullconv"),
catmullmoulding: require("./catmullmoulding"),
polybezier: require("./polybezier"),
shapes: require("./shapes"),
@@ -56,9 +56,6 @@ module.exports = {
*/
/*
Bézier curves and Catmull-Rom curves
Creating a Catmull-Rom curve from three points
Forming poly-Bézier curves
Boolean shape operations
Projecting a point onto a Bézier curve

8
components/sections/moulding/index.js
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@@ -234,7 +234,7 @@ var Moulding = React.createClass({
we can compute the ABC ratio for this configuration, and we know that our new point A' should like at a distance:</p>
<p>\[
A' = B' + \frac{B' - C}{ratio} = B' - \frac{C - B'}{ratio}
A' = B' - \frac{C - B'}{ratio} = B' + \frac{B - C}{ratio}
\]</p>
<p>For quadratic curves, this means we're done, since the new point A' is equivalent to the new quadratic control point.
@@ -249,7 +249,9 @@ var Moulding = React.createClass({
curve. Given A', B', and the endpoints e1 and e2 of the strut line relative to B', we can now compute where the new control points
should be. Remember that B' lies on line e1--e2 at a distance <i>t</i>, because that's how Bézier curves work. In the same manner,
we know the distance A--e1 is only line-interval [0,t] of the full segment, and A--e2 is only line-interval [t,1], so constructing
the new control points is fairly easy:</p>
the new control points is fairly easy.</p>
<p>First, we construct the one-level-of-de-Casteljau-up points:</p>
<p>\[
\left \{ \begin{align}
@@ -258,6 +260,8 @@ var Moulding = React.createClass({
\end{align} \right .
\]</p>
<p>And then we can compute the new control points:</p>
<p>\[
\left \{ \begin{align}
C1' &= v1 + \frac{v1 - start}{t} \\

10
components/sections/pointcurves/index.js
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@@ -12,7 +12,11 @@ var PointCurves = React.createClass({
},
setup: function(api) {
api.lpts = [];
api.lpts = [
{x:56, y:153},
{x:144,y:83},
{x:188,y:185}
];
},
onClick: function(evt, api) {
@@ -209,8 +213,8 @@ var PointCurves = React.createClass({
<p>In each graphic, the blue parts are the values that we "just have" simply by setting up
our three points, combined with our decision on which <i>t</i> value to use (and construction line
orientation and length for cubic curves). There are of course many ways to determine a combination
of <i>t</i> and tangent values that lead to a more "aesthetic" curve, but this will be left as an
exercise to the reader, since there are many, and aesthetics are often quite personal.</p>
of <i>t</i> and tangent values that lead to a more "æsthetic" curve, but this will be left as an
exercise to the reader, since there are many, and æsthetics are often quite personal.</p>
</section>
);
}

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Side by Side

After

Width:  |  Height:  |  Size: 17 KiB

1
stylesheets/figure.less
View File

@@ -18,6 +18,7 @@ figure {
&:focus {
border: 1px solid grey;
outline: none;
}
}

11
stylesheets/note.less Normal file
View File

@@ -0,0 +1,11 @@
div.note {
font-size: 90%;
margin: 1em 2em;
padding: 1em;
border: 1px solid grey;
background: rgba(150,150,50,0.05);
h1,h2,h3,h4,h5,h6,p { margin: 0; padding: 0; }
p { margin: 1em 0; }
div.MathJax_Display { margin: 1em 0; }
}

12
stylesheets/section.less
View File

@@ -22,14 +22,4 @@ section {
}
}
div.note {
font-size: 90%;
margin: 1em 2em;
padding: 1em;
border: 1px solid grey;
background: rgba(150,150,50,0.05);
* { margin: 0; padding: 0; }
p { margin: 1em 0; }
div.MathJax_Display { margin: 1em 0; }
}
@import "note.less";