1 files changed, 484 insertions, 0 deletions
diff --git a/third_party/fontcrunch/quadopt.cc b/third_party/fontcrunch/quadopt.cc
new file mode 100644
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+++ b/third_party/fontcrunch/quadopt.cc
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+/*
+ * Copyright 2014 Google Inc. All rights reserved.
+ *
+ * Licensed under the Apache License, Version 2.0 (the "License");
+ * you may not use this file except in compliance with the License.
+ * You may obtain a copy of the License at
+ *
+ *      http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ *
+ * Contributor: Raph Levien
+ */
+
+#include <iostream>
+#include <fstream>
+#include <cmath>
+#include <vector>
+#include <algorithm>
+
+using std::vector;
+
+#define HALF_STEP 1
+
+class Point {
+public:
+	Point() : x(0), y(0) { }
+	Point(double x, double y) : x(x), y(y) { }
+	double x, y;
+};
+
+bool operator==(const Point& p0, const Point& p1) {
+	return p0.x == p1.x && p0.y == p1.y;
+}
+
+std::ostream& operator<<(std::ostream& os, const Point& p) {
+	os << "(" << p.x << ", " << p.y << ")";
+	return os;
+}
+
+double dist(Point p0, Point p1) {
+	return std::hypot(p0.x - p1.x, p0.y - p1.y);
+}
+
+double dist2(Point p0, Point p1) {
+	double dx = p0.x - p1.x;
+	double dy = p0.y - p1.y;
+	return dx * dx + dy * dy;
+}
+
+Point lerp(double t, Point p0, Point p1) {
+	return Point(p0.x + t * (p1.x - p0.x), p0.y + t * (p1.y - p0.y));
+}
+
+Point unitize(Point p) {
+	double scale = 1/std::hypot(p.x, p.y);
+	return Point(p.x * scale, p.y * scale);
+}
+
+Point round(Point p) {
+	return Point(std::round(p.x), std::round(p.y));
+}
+
+class Quad {
+public:
+	Quad() : p() { }
+	Quad(Point p0, Point p1, Point p2) : p() {
+		p[0] = p0;
+		p[1] = p1;
+		p[2] = p2;
+	}
+	Point p[3];
+	double arclen() const;
+	Point eval(double t) const;
+	bool isLine() const;
+	void print(std::ostream& o) const {
+		o << p[0].x << " " << p[0].y << " " << p[1].x << " " << p[1].y << " "
+			<< p[2].x << " " << p[2].y << std::endl;
+	}
+};
+
+bool Quad::isLine() const {
+	return p[1] == lerp(0.5, p[0], p[2]);
+}
+
+// One step of a 4th-order Runge-Kutta numerical integration
+template <size_t n, typename F>
+void rk4(double y[n], double x, double h, F& derivs) {
+	double dydx[n];
+	double dyt[n];
+	double dym[n];
+	double yt[n];
+	derivs(dydx, x, y);
+	double hh = h * .5;
+	double h6 = h * (1./6);
+	for (size_t i = 0; i < n; i++) {
+		yt[i] = y[i] + hh * dydx[i];
+	}
+	derivs(dyt, x + hh, yt);
+	for (size_t i = 0; i < n; i++) {
+		yt[i] = y[i] + hh * dyt[i];
+	}
+	derivs(dym, x + hh, yt);
+	for (size_t i = 0; i < n; i++) {
+		yt[i] = y[i] + h * dym[i];
+		dym[i] += dyt[i];
+	}
+	derivs(dyt, x + h, yt);
+	for (size_t i = 0; i < n; i++) {
+		y[i] += h6 * (dydx[i] + dyt[i] + 2 * dym[i]);
+	}
+}
+
+class ArclenFunctor {
+public:
+	ArclenFunctor(const Quad& q)
+			: dx0(2 * (q.p[1].x - q.p[0].x))
+			, dx1(2 * (q.p[2].x - q.p[1].x))
+			, dy0(2 * (q.p[1].y - q.p[0].y))
+			, dy1(2 * (q.p[2].y - q.p[1].y)) { }
+	void operator()(double dydx[1], double t, const double y[1]) {
+		Point p(deriv(t));
+		dydx[0] = std::hypot(p.x, p.y);
+	}
+	Point deriv(double t) const {
+		return Point(dx0 + t * (dx1 - dx0), dy0 + t * (dy1 - dy0));
+	}
+private:
+	double dx0, dy0, dx1, dy1;
+};
+
+double Quad::arclen() const {
+	ArclenFunctor derivs(*this);
+	const int n = 10;
+	double dt = 1./n;
+	double t = 0;
+	double y[1] = { 0 };
+	for (int i = 0; i < n; i++) {
+		rk4<1>(y, t, dt, derivs);
+		t += dt;
+	}
+	return y[0];
+}
+
+Point Quad::eval(double t) const {
+	Point p01(lerp(t, p[0], p[1]));
+	Point p12(lerp(t, p[1], p[2]));
+	return lerp(t, p01, p12);
+}
+
+class Thetas {
+public:
+	void init(const vector<Quad>& qs);
+	Point xy(double s) const;
+	Point dir(double s) const;
+	double arclen;
+private:
+	vector<Point> xys;
+	vector<Point> dirs;
+};
+
+void Thetas::init(const vector<Quad>& qs) {
+	xys.clear();
+	dirs.clear();
+	double s = 0;
+	int ix = 0;
+	Point lastxy;
+	Point lastd;
+	double lasts = -1;
+	for (size_t i = 0; i < qs.size(); i++) {
+		const Quad& q = qs[i];
+		ArclenFunctor derivs(q);
+		const int n = 100;
+		double dt = 1./n;
+		double t = 0;
+		double y[1];
+		y[0] = s;
+		for (int j = 0; j < n; j++) {
+			Point thisxy(q.eval(t));
+			Point thisd(derivs.deriv(t));
+			while (ix <= y[0]) {
+				double u = (ix - lasts) / (y[0] - lasts);
+				xys.push_back(lerp(u, lastxy, thisxy));
+				dirs.push_back(unitize(lerp(u, lastd, thisd)));
+				ix++;
+			}
+			lasts = y[0];
+			rk4<1>(y, t, dt, derivs);
+			t += dt;
+			lastxy = thisxy;
+			lastd = thisd;
+		}
+		s = y[0];
+	}
+	const Quad& q = qs[qs.size() - 1];
+	Point thisxy(q.p[2]);
+	Point thisd(ArclenFunctor(q).deriv(1));
+	while (ix <= s + 1) {
+		double u = (ix - lasts) / (s - lasts);
+		xys.push_back(lerp(u, lastxy, thisxy));
+		dirs.push_back(unitize(lerp(u, lastd, thisd)));
+		ix++;
+	}
+	arclen = s;
+}
+
+Point Thetas::xy(double s) const {
+	int bucket = (int)s;
+	double frac = s - bucket;
+	return lerp(frac, xys[bucket], xys[bucket + 1]);
+}
+
+Point Thetas::dir(double s) const {
+	int bucket = (int)s;
+	double frac = s - bucket;
+	return lerp(frac, dirs[bucket], dirs[bucket + 1]);
+}
+
+#define NORM_LEVEL 2
+
+// L1 angle norm, 2, L2 angle norm, 0.05
+// L1 distance norm, 200
+double penalty = 1;
+double dist_factor = .005;
+double angle_factor = 5;
+
+
+class MeasureFunctor {
+public:
+	MeasureFunctor(const Thetas& curve, double s0, double ss, const ArclenFunctor& af,
+			Quad q)
+		: curve(curve), s0(s0), ss(ss), af(af), q(q) { }
+	void operator()(double dydx[2], double t, const double y[2]) {
+		Point dxy(af.deriv(t));
+		dydx[0] = std::hypot(dxy.x, dxy.y);
+
+		// distance error
+		Point curvexy = curve.xy(s0 + y[0] * ss);
+#if NORM_LEVEL == 1
+		double disterr = dist(q.eval(t), curvexy);
+#endif
+#if NORM_LEVEL == 2
+		double disterr = dist2(q.eval(t), curvexy);
+#endif
+		disterr *= dydx[0];
+
+		// angle error
+		Point dir = curve.dir(s0 + y[0] * ss);
+		double angleerr = dir.x * dxy.y - dir.y * dxy.x;
+#if NORM_LEVEL == 1
+		angleerr = std::abs(angleerr);
+#endif
+#if NORM_LEVEL == 2
+		angleerr = (angleerr * angleerr) / dydx[0];
+#endif
+
+		dydx[1] = dist_factor * disterr + angle_factor * angleerr;
+	}
+private:
+	const Thetas& curve;
+	double s0;
+	double ss;
+	const ArclenFunctor& af;
+	Quad q;
+};
+
+// measure how closely the quad fits the section of curve, using L1 norm
+// of angle mismatch
+double measureQuad(const Thetas& curve, double s0, double s1, const Quad& q) {
+	ArclenFunctor derivs(q);
+	double ss = (s1 - s0) / q.arclen();
+	MeasureFunctor err(curve, s0, ss, derivs, q);
+	const int n = 10;
+	double dt = 1./n;
+	double t = 0;
+	double y[2] = { 0, 0 };
+	for (int i = 0; i < n; i++) {
+		rk4<2>(y, t, dt, err);
+		t += dt;
+	}
+	return y[1];
+}
+
+struct Break {
+	Break(double s, Point xy, Point dir) : s(s), xy(xy), dir(dir) { }
+	double s;
+	Point xy;
+	Point dir;
+};
+
+struct Statelet {
+	void combine(const Statelet* prev, double score, Quad q);
+	const Statelet* prev;
+	double score;
+	Quad q;
+};
+
+void Statelet::combine(const Statelet* newprev, double newscore, Quad newq) {
+	prev = newprev;
+	double pmul = 2;
+	if (newq.isLine()) {
+		pmul = 1;
+	} else if (newprev != 0 && !newprev->q.isLine()
+			&& lerp(0.5, newprev->q.p[1], newq.p[1]) == newq.p[0]) {
+		pmul = 1;
+	}
+	score = (newprev == 0 ? 0 : newprev->score) + penalty * pmul + newscore;
+	q = newq;
+}
+
+struct State {
+	void combine(const State* prev, double score, Quad q);
+	vector<Statelet> sts;
+	bool init;
+};
+
+void State::combine(const State* prev, double score, Quad q) {
+	const Statelet* prevsl = prev->sts.empty() ? 0 : &prev->sts[0];
+	if (prevsl == 0 && !prev->init) {
+		return;
+	}
+	Statelet sl;
+	sl.combine(prevsl, score, q);
+	if (sts.empty()) {
+		sts.push_back(sl);
+	} else {
+		if (sl.score < sts[0].score) {
+			sts[0] = sl;
+		}
+	}
+}
+
+bool isInt(double x) {
+	return x == (int) x;
+}
+
+bool okForHalf(const State* prev, Quad q) {
+	if (isInt(q.p[0].x) && isInt(q.p[0].y)) {
+		return true;
+	}
+	if (q.isLine()) {
+		return false;
+	}
+	const Statelet* prevsl = prev->sts.empty() ? 0 : &prev->sts[0];
+
+	if (prevsl == 0 || prevsl->q.isLine()) {
+		return false;
+	}
+	return lerp(0.5, prevsl->q.p[1], q.p[1]) == q.p[0];
+}
+
+void findBreaks(vector<Break>* breaks, const Thetas& curve) {
+	breaks->clear();
+	double lastd;
+	int n = round(10 * curve.arclen);
+	for (int i = 0; i <= n; i++) {
+		double s = curve.arclen * i / n;
+		Point origp = curve.xy(s);
+#if HALF_STEP
+		Point p(.5 * std::round(2 * origp.x), .5 * std::round(2 * origp.y));
+#else
+		Point p = round(origp);
+#endif
+		double d = dist(p, origp);
+		if (i == 0 || !(p == (*breaks)[breaks->size() - 1].xy)) {
+			Break bk(s, p, curve.dir(s));
+			breaks->push_back(bk);
+			lastd = d;
+		} else if (d < lastd) {
+			(*breaks)[breaks->size() - 1] = Break(s, p, curve.dir(s));
+			lastd = d;
+		}
+	}
+}
+
+bool intersect(Point* result, Point p0, Point dir0, Point p1, Point dir1) {
+	double det = dir0.x * dir1.y - dir0.y * dir1.x;
+	if (std::abs(det) < 1e-6) return false;
+	det = 1 / det;
+	double a = p0.y * dir0.x - p0.x * dir0.y;
+	double b = p1.y * dir1.x - p1.x * dir1.y;
+	result->x = (a * dir1.x - b * dir0.x) * det;
+	result->y = (a * dir1.y - b * dir0.y) * det;
+	return true;
+}
+
+void tryQuad(const State* prev, State* st, const Thetas& curve,
+	const Break& bk0, const Break& bk1, const Quad& q) {
+	double score = measureQuad(curve, bk0.s, bk1.s, q);
+	st->combine(prev, score, q);
+}
+
+void tryLineQuad(const State* prev, State* st, const Thetas& curve,
+	const Break& bk0, const Break& bk1) {
+	if (isInt(bk0.xy.x) && isInt(bk0.xy.y)) {
+		Quad line(bk0.xy, lerp(0.5, bk0.xy, bk1.xy), bk1.xy);
+		tryQuad(prev, st, curve, bk0, bk1, line);
+	}
+	Point pmid;
+	if (intersect(&pmid, bk0.xy, bk0.dir, bk1.xy, bk1.dir)) {
+		Quad q(bk0.xy, round(pmid), bk1.xy);
+		if (okForHalf(prev, q)) {
+			tryQuad(prev, st, curve, bk0, bk1, q);
+		}
+	}
+}
+
+vector<Quad> optimize(const Thetas& curve) {
+	vector<Break> breaks;
+	findBreaks(&breaks, curve);
+	int n = breaks.size() - 1;
+	vector<State> states;
+	states.resize(n + 1);
+	states[0].init = true;
+	tryLineQuad(&states[0], &states[n], curve, breaks[0], breaks[n]);
+	if (states[n].sts[0].score <= 3 * penalty) {
+		goto done;
+	}
+	for (int i = 1; i < n; i++) {
+		tryLineQuad(&states[0], &states[i], curve, breaks[0], breaks[i]);
+		tryLineQuad(&states[i], &states[n], curve, breaks[i], breaks[n]);
+	}
+	if (states[n].sts[0].score <= 4 * penalty) {
+		goto done;
+	}
+	for (int i = 1; i <= n; i++) {
+		for (int j = i - 1; j >= 0; j--) {
+			tryLineQuad(&states[j], &states[i], curve, breaks[j], breaks[i]);
+		}
+	}
+done:
+	vector<Quad> result;
+	for (const Statelet* sl = &states[n].sts[0]; sl != 0; sl = sl->prev) {
+		result.push_back(sl->q);
+	}
+	std::reverse(result.begin(), result.end());
+	return result;
+}
+
+void readBzs(vector<Quad>* result, std::istream& is) {
+	double x0, y0, x1, y1, x2, y2;
+	while (is >> x0 >> y0 >> x1 >> y1 >> x2 >> y2) {
+		result->push_back(Quad(Point(x0, y0), Point(x1, y1), Point(x2, y2)));
+	}
+	// Round the endpoints, they must be on integers
+	(*result)[0].p[0] = round((*result)[0].p[0]);
+	Quad* lastq = &(*result)[(*result).size()];
+	lastq->p[2] = round(lastq->p[2]);
+}
+
+int main(int argc, char** argv) {
+	if (argc != 3) {
+		std::cerr << "usage: quadopt in out\n";
+		return 1;
+	}
+#if 0
+	Quad q(Point(100, 0), Point(0, 0), Point(0, 100));
+	std::cout.precision(8);
+	std::cout << q.arclen() << "\n";
+#endif
+	vector<Quad> bzs;
+	std::ifstream is;
+	is.open(argv[1]);
+	readBzs(&bzs, is);
+	Thetas thetas;
+	thetas.init(bzs);
+#if 0
+	for (int i = 0; i < thetas.arclen; i++) {
+		Point xy = thetas.dir(i);
+		std::cout << xy.x << " " << xy.y << std::endl;
+	}
+#endif
+	vector<Quad> optbzs = optimize(thetas);
+	std::ofstream os;
+	os.open(argv[2]);
+	for (size_t i = 0; i < optbzs.size(); i++) {
+		optbzs[i].print(os);
+	}
+	return 0;
+}