Let's start with the good stuff: when we're talking about Bézier curves, we're talking about the things that you can see in the following graphics. They run from some start point to some end point, with their curvature influenced by one or more "intermediate" control points. Now, because all the graphics on this page are interactive, go manipulate those curves a bit: click-drag the points, and see how their shape changes based on what you do.
+
These curves are used a lot in computer aided design and computer aided manufacturing (CAD/CAM) applications, as well as in graphic design programs like Adobe Illustrator and Photoshop, Inkscape, the Gimp, etc. and in graphic technologies like scalable vector graphics (SVG) and OpenType fonts (ttf/otf). A lot of things use Bézier curves, so if you want to learn more about them... prepare to get your learn on!
; }
@@ -162,22 +162,22 @@ Given \left (
\]
And that's the full description for Bézier curves. Σ in this function indicates that this is a series of additions (using the variable listed below the Σ, starting at ...=<value> and ending at the value listed on top of the Σ).
+
-### How to implement the basis function
-
-We could naively implement the basis function as a mathematical construct, using the function as our guide, like this:
-
-
function Bezier(n,t):
+How to implement the basis function
+We could naively implement the basis function as a mathematical construct, using the function as our guide, like this:
+
+function Bezier(n,t):
sum = 0
- for(k=0; k
-
-I say we could, because we're not going to: the factorial function is incredibly expensive. And, as we can see from the above explanation, we can actually create Pascal's triangle quite easily without it: just start at [1], then [1,1], then [1,2,1], then [1,3,3,1], and so on, with each next row fitting 1 more number than the previous row, starting and ending with "1", with all the numbers in between being the sum of the previous row's elements on either side "above" the one we're computing.
-
-We can generate this as a list of lists lightning fast, and then never have to compute the binomial terms because we have a lookup table:
-
-lut = [ [1], // n=0
+ return sum
+
+
+I say we could, because we're not going to: the factorial function is incredibly expensive. And, as we can see from the above explanation, we can actually create Pascal's triangle quite easily without it: just start at [1], then [1,1], then [1,2,1], then [1,3,3,1], and so on, with each next row fitting 1 more number than the previous row, starting and ending with "1", with all the numbers in between being the sum of the previous row's elements on either side "above" the one we're computing.
+We can generate this as a list of lists lightning fast, and then never have to compute the binomial terms because we have a lookup table:
+
+lut = [ [1], // n=0
[1,1], // n=1
[1,2,1], // n=2
[1,3,3,1], // n=3
@@ -194,42 +194,110 @@ binomial(n,k):
nextRow[i] = lut[prev][i-1] + lut[prev][i]
nextRow[s] = 1
lut.add(nextRow)
- return lut[n][k]
-
-So what's going on here? First, we declare a lookup table with a size that's reasonably large enough to accommodate most lookups. Then, we declare a function to get us the values we need, and we make sure that if an n/k pair is requested that isn't in the LUT yet, we expand it first. Our basis function now looks like this:
-
-function Bezier(n,t):
+ return lut[n][k]
+
+
+So what's going on here? First, we declare a lookup table with a size that's reasonably large enough to accommodate most lookups. Then, we declare a function to get us the values we need, and we make sure that if an n/k pair is requested that isn't in the LUT yet, we expand it first. Our basis function now looks like this:
+
+function Bezier(n,t):
sum = 0
for(k=0; k<=n; k++):
- sum += binomial(n,k) (1-t)^(n-k) t^(k)
- return sum
-
-Perfect. Of course, we can optimize further. For most computer graphics purposes, we don't need arbitrary curves. We need quadratic and cubic curves (this primer actually does do arbitrary curves, so you'll find code similar to shown here), which means we can drastically simplify the code:
-
-function Bezier(2,t):
- t2 = t t
+ sum += binomial(n,k) * (1-t)^(n-k) * t^(k)
+ return sum
+
+
+Perfect. Of course, we can optimize further. For most computer graphics purposes, we don't need arbitrary curves. We need quadratic and cubic curves (this primer actually does do arbitrary curves, so you'll find code similar to shown here), which means we can drastically simplify the code:
+
+function Bezier(2,t):
+ t2 = t * t
mt = 1-t
- mt2 = mt mt
- return mt2 + 2mtt + t2
+ mt2 = mt * mt
+ return mt2 + 2*mt*t + t2
function Bezier(3,t):
- t2 = t t
- t3 = t2 t
+ t2 = t * t
+ t3 = t2 * t
mt = 1-t
- mt2 = mt mt
- mt3 = mt2 mt
- return mt3 + 3mt2t + 3mtt2 + t3
-
-And now we know how to program the basis function. Exellent.
+ mt2 = mt * mt
+ mt3 = mt2 * mt
+ return mt3 + 3*mt2*t + 3*mt*t2 + t3
+
+
+
And now we know how to program the basis function. Exellent.
So, now we know what the base function(s) look(s) like, time to add in the magic that makes Bézier curves so special: control points.
; }
},
"control": {
- "title": "Unknown title (control)",
+ "title": "Controlling Bézier curvatures",
"getContent": function(handler) { return
+
+Bézier curves are (like all "splines") interpolation functions, meaning they take a set of points, and generate values somewhere "between" those points. (One of the consequences of this is that you'll never be able to generate a point that lies outside the outline for the control points, commonly called the "hull" for the curve. Useful information!). In fact, we can visualize how each point contributes to the value generated by the function, so we can see which points are important, where, in the curve.
+The following graphs show the interpolation functions for quadratic and cubic curves, with "S" being the strength of a point's contribution to the total sum of the Bézier function. Click or click-drag to see the interpolation percentages for each curve-defining point at a specific t value.
+
+
+
+
+
+
+Also shown is the interpolation function for a 15th order Bézier function. As you can see, the start and end point contribute considerably more to the curve's shape than any other point in the control point set.
+If we want to change the curve, we need to change the weights of each point, effectively changing the interpolations. The way to do this is about as straight forward as possible: just multiply each point with a value that changes its strength. These values are conventionally called "Weights", and we can add them to our original Bézier function:
+
+\[
+ Bézier(n,t) = \sum_{i=0}^{n}
+ \underset{binomial\ term}{\underbrace{\binom{n}{i}}}
+ \cdot\
+ \underset{polynomial\ term}{\underbrace{(1-t)^{n-i} \cdot t^{i}}}
+ \cdot\
+ \underset{weight}{\underbrace{w_i}}
+\]
+
+That looks complicated, but as it so happens, the "weights" are actually just the coordinate values we want our curve to have: for an nth
+ order curve, w0 is our start coordinate, wn is our last coordinate, and everything in between is a controlling coordinate. Say we want a cubic curve that starts at (120,160), is controlled by (35,200) and (220,260) and ends at (220,40), we use this Bézier curve:
+
+\[
+\left \{ \begin{matrix}
+ x = BLUE[120] \cdot (1-t)^3 + BLUE[35] \cdot 3 \cdot (1-t)^2 \cdot t + BLUE[220] \cdot 3 \cdot (1-t) \cdot t^2 + BLUE[220] \cdot t^3 \\
+ y = BLUE[160] \cdot (1-t)^3 + BLUE[200] \cdot 3 \cdot (1-t)^2 \cdot t + BLUE[260] \cdot 3 \cdot (1-t) \cdot t^2 + BLUE[40] \cdot t^3
+\end{matrix} \right.
+\]
+
+Which gives us the curve we saw at the top of the article:
+
+
+What else can we do with Bézier curves? Quite a lot, actually. The rest of this article covers a multitude of possible operations and algorithms that we can apply, and the tasks they achieve.
+
+
+
How to implement the weighted basis function
+
Given that we already know how to implement basis function, adding in the control points is remarkably easy:
+
+function Bezier(n,t,w[]):
+ sum = 0
+ for(k=0; k<n; k++):
+ sum += w[k] * binomial(n,k) * (1-t)^(n-k) * t^(k)
+ return sum
+
+
+
And for the extremely optimized versions:
+
+function Bezier(2,t,w[]):
+ t2 = t * t
+ mt = 1-t
+ mt2 = mt * mt
+ return w[0]*mt2 + w[1]*2*mt*t + w[2]*t2
+
+function Bezier(3,t,w[]):
+ t2 = t * t
+ t3 = t2 * t
+ mt = 1-t
+ mt2 = mt * mt
+ mt3 = mt2 * mt
+ return w[0]*mt3 + 3*w[1]*mt2*t + 3*w[2]*mt*t2 + w[3]*t3
+
+
+
And now we know how to program the weighted basis function.
+
; }
},
diff --git a/make-locales.js b/make-locales.js
index cc7148d4..da3e4556 100644
--- a/make-locales.js
+++ b/make-locales.js
@@ -1,4 +1,3 @@
-const glue = '';
var marked = require("marked");
var fs = require("fs-extra");
@@ -17,34 +16,99 @@ Module.prototype.require = function() {
};
-/**
- * Split data up into "markdown" and "latex"
- */
-function chunk(data) {
- var p = 0, e = 0, chunks = [];
+function chunkDivs(data, chunks, chunkMore) {
+ var p = 0,
+ next = chunkMore ? chunkMore[0] : false,
+ otherChunkers = chunkMore ? chunkMore.slice(1) : false,
+ divMatch = '\n
\n',
+ divClosingTag = '
\n';
+
while (p !== -1) {
- let s = data.indexOf('\n\\[', p);
- if (s === -1) {
- chunks.push({
- latex: false,
- data: data.substring(p)
- });
+ // Let's check for a
tag
+ let div = data.indexOf(divMatch, p);
+ if (div === -1) {
+ // No div tags found: we're done here. Parse the remaining
+ // data for whatever else might be in there.
+ performChunking(data.substring(p), chunks, next, otherChunkers);
break;
}
- chunks.push({
- latex: false,
- data: data.substring(p, s)
- });
+ // First parse the non-div data for whatever else might be in there.
+ performChunking(data.substring(p, div), chunks, next, otherChunkers);
- e = data.indexOf('\\]\n\n', s) + 4;
- chunks.push({
- latex: true,
- data: data.substring(s, e)
- });
+ // Now, if we have a div, there's a few options:
+ //
+ // - "figure" contains HTML content, not to be converted
+ // - "note" contains markdown content, to be converted
+ // - "howtocode" contains markdown content, to be converted
+ let className = data.substring(div).match(/className="([^"]+)"/);
+ if (className !== null) { className = className[1]; }
- p = e;
+ let eod, type="div";
+ if (className === "figure") {
+ eod = data.indexOf(divClosingTag, div) + divClosingTag.length;
+ type += ".figure";
+ } else {
+ eod = data.indexOf(divEnd, div) + divEnd.length;
+ }
+
+ chunks.push({ convert: false, type: type, s:div, e:eod, data: data.substring(div, eod) });
+ p = eod;
}
+}
+
+/**
+ * Split data up into "latex" and "not latex".
+ * Anything that is not latex might still be "not markdown"
+ * though, so we hand that data off to additional chunkers
+ */
+function chunkLatex(data, chunks, chunkMore) {
+ var p = 0,
+ next = chunkMore ? chunkMore[0] : false,
+ otherChunkers = chunkMore ? chunkMore.slice(1) : false,
+ latexEnd = '\\]\n\n';
+
+ while (p !== -1) {
+ // Let's check a LaTeX block
+ let latex = data.indexOf('\n\\[', p);
+ if (latex === -1) {
+ // No LaTeX block found: we're done here. Parse the remaining
+ // data for whatever else might be in there.
+ performChunking(data.substring(p), chunks, next, otherChunkers);
+ break;
+ }
+
+ // First parse the non-LaTeX data for whatever else might be in there.
+ performChunking(data.substring(p, latex), chunks, next, otherChunkers);
+
+ // Then capture the LaTeX block and mark it as "don't convert"
+ let eol = data.indexOf(latexEnd, latex) + latexEnd.length;
+ chunks.push({ convert: false, type: "latex", s:latex, e:eol, data: data.substring(latex, eol) });
+ p = eol;
+ }
+}
+
+// in-place chunking
+function performChunking(data, chunks, chunker, moreChunkers) {
+ // If there's no further chunking function to run, just
+ // record this data as a chunk of convertible data.
+ if (!chunker) {
+ chunks.push({ convert: true, data: data });
+ return "early";
+ }
+
+ // otherwise, perform more chunking.
+ chunker(data, chunks, moreChunkers);
+}
+
+/**
+ * Split data up into "markdown" and "not markdown" parts.
+ * We'll only run markdown conversion on the markdown parts.
+ */
+function chunk(data) {
+ var chunks = [];
+ performChunking(data, chunks, chunkLatex, [chunkDivs]);
return chunks;
}
@@ -59,26 +123,23 @@ sections.forEach((cname, number) => {
var data, title;
try {
data = fs.readFileSync(loc).toString();
-
- // convert all non-latex data
- data = data.split(glue).map(content => {
- return chunk(content).map(chunk => {
- if (chunk.latex) return chunk.data;
- let d = marked(chunk.data)
- // And then some post-processing...
- d = d.replace(/
]+>([^<]+)<\/h1>/,function(_,t) {
- title = t;
- return ``;
- });
- d = d.replace('', '