1/*
2 * Copyright 2020 Google Inc.
3 *
4 * Use of this source code is governed by a BSD-style license that can be
5 * found in the LICENSE file.
6 */
7
8#include "samplecode/Sample.h"
9
10#include "include/core/SkCanvas.h"
11#include "include/core/SkFont.h"
12#include "include/core/SkPaint.h"
13#include "include/core/SkPath.h"
14#include <tuple>
15
16// Math constants are not always defined.
17#ifndef M_PI
18#define M_PI 3.14159265358979323846264338327950288
19#endif
20
21#ifndef M_SQRT2
22#define M_SQRT2 1.41421356237309504880168872420969808
23#endif
24
25constexpr static int kCenterX = 300;
26constexpr static int kCenterY = 325;
27constexpr static int kRadius = 250;
28
29// This sample fits a cubic to the arc between two interactive points on a circle. It also finds the
30// T-coordinate of max error, and outputs it and its value in pixels. (It turns out that max error
31// always occurs at T=0.21132486540519.)
32//
33// Press 'E' to iteratively cut the arc in half and report the improvement in max error after each
34// halving. (It turns out that max error improves by exactly 64x on every halving.)
35class SampleFitCubicToCircle : public Sample {
36    SkString name() override { return SkString("FitCubicToCircle"); }
37    void onOnceBeforeDraw() override { this->fitCubic(); }
38    void fitCubic();
39    void onDrawContent(SkCanvas*) override;
40    Sample::Click* onFindClickHandler(SkScalar x, SkScalar y, skui::ModifierKey) override;
41    bool onClick(Sample::Click*) override;
42    bool onChar(SkUnichar) override;
43
44    // Coordinates of two points on the unit circle. These are the two endpoints of the arc we fit.
45    double fEndptsX[2] = {0, 1};
46    double fEndptsY[2] = {-1, 0};
47
48    // Fitted cubic and info, set by fitCubic().
49    double fControlLength;  // Length of (p1 - p0) and/or (p3 - p2) in unit circle space.
50    double fMaxErrorT;  // T value where the cubic diverges most from the true arc.
51    std::array<double, 4> fCubicX;  // Screen space cubic control points.
52    std::array<double, 4> fCubicY;
53    double fMaxError;  // Max error (in pixels) between the cubic and the screen-space arc.
54    double fTheta;  // Angle of the arc. This is only used for informational purposes.
55    SkTArray<SkString> fInfoStrings;
56
57    class Click;
58};
59
60// Fits a cubic to an arc on the unit circle with endpoints (x0, y0) and (x1, y1). Using the
61// following 3 constraints, we arrive at the formula used in the method:
62//
63//   1) The endpoints and tangent directions at the endpoints must match the arc.
64//   2) The cubic must be symmetric (i.e., length(p1 - p0) == length(p3 - p2)).
65//   3) The height of the cubic must match the height of the arc.
66//
67// Returns the "control length", or length of (p1 - p0) and/or (p3 - p2).
68static float fit_cubic_to_unit_circle(double x0, double y0, double x1, double y1,
69                                      std::array<double, 4>* X, std::array<double, 4>* Y) {
70    constexpr static double kM = -4.0/3;
71    constexpr static double kA = 4*M_SQRT2/3;
72    double d = x0*x1 + y0*y1;
73    double c = (std::sqrt(1 + d) * kM + kA) / std::sqrt(1 - d);
74    *X = {x0, x0 - y0*c, x1 + y1*c, x1};
75    *Y = {y0, y0 + x0*c, y1 - x1*c, y1};
76    return c;
77}
78
79static double lerp(double x, double y, double T) {
80    return x + T*(y - x);
81}
82
83// Evaluates the cubic and 1st and 2nd derivatives at T.
84static std::tuple<double, double, double> eval_cubic(double x[], double T) {
85    // Use De Casteljau's algorithm for better accuracy and stability.
86    double ab = lerp(x[0], x[1], T);
87    double bc = lerp(x[1], x[2], T);
88    double cd = lerp(x[2], x[3], T);
89    double abc = lerp(ab, bc, T);
90    double bcd = lerp(bc, cd, T);
91    double abcd = lerp(abc, bcd, T);
92    return {abcd, 3 * (bcd - abc) /*1st derivative.*/, 6 * (cd - 2*bc + ab) /*2nd derivative.*/};
93}
94
95// Uses newton-raphson convergence to find the point where the provided cubic diverges most from the
96// unit circle. i.e., the point where the derivative of error == 0. For error we use:
97//
98//     error = x^2 + y^2 - 1
99//     error' = 2xx' + 2yy'
100//     error'' = 2xx'' + 2yy'' + 2x'^2 + 2y'^2
101//
102double find_max_error_T(double cubicX[4], double cubicY[4]) {
103    constexpr static double kInitialT = .25;
104    double T = kInitialT;
105    for (int i = 0; i < 64; ++i) {
106        auto [x, dx, ddx] = eval_cubic(cubicX, T);
107        auto [y, dy, ddy] = eval_cubic(cubicY, T);
108        double dError = 2*(x*dx + y*dy);
109        double ddError = 2*(x*ddx + y*ddy + dx*dx + dy*dy);
110        T -= dError / ddError;
111    }
112    return T;
113}
114
115void SampleFitCubicToCircle::fitCubic() {
116    fInfoStrings.reset();
117
118    std::array<double, 4> X, Y;
119    // "Control length" is the length of (p1 - p0) and/or (p3 - p2) in unit circle space.
120    fControlLength = fit_cubic_to_unit_circle(fEndptsX[0], fEndptsY[0], fEndptsX[1], fEndptsY[1],
121                                              &X, &Y);
122    fInfoStrings.push_back().printf("control length=%0.14f", fControlLength);
123
124    fMaxErrorT = find_max_error_T(X.data(), Y.data());
125    fInfoStrings.push_back().printf("max error T=%0.14f", fMaxErrorT);
126
127    for (int i = 0; i < 4; ++i) {
128        fCubicX[i] = X[i] * kRadius + kCenterX;
129        fCubicY[i] = Y[i] * kRadius + kCenterY;
130    }
131    double errX = std::get<0>(eval_cubic(fCubicX.data(), fMaxErrorT)) - kCenterX;
132    double errY = std::get<0>(eval_cubic(fCubicY.data(), fMaxErrorT)) - kCenterY;
133    fMaxError = std::sqrt(errX*errX + errY*errY) - kRadius;
134    fInfoStrings.push_back().printf("max error=%.5gpx", fMaxError);
135
136    fTheta = std::atan2(fEndptsY[1], fEndptsX[1]) - std::atan2(fEndptsY[0], fEndptsX[0]);
137    fTheta = std::abs(fTheta * 180/M_PI);
138    if (fTheta > 180) {
139        fTheta = 360 - fTheta;
140    }
141    fInfoStrings.push_back().printf("(theta=%.2f)", fTheta);
142
143    SkDebugf("\n");
144    for (const SkString& infoString : fInfoStrings) {
145        SkDebugf("%s\n", infoString.c_str());
146    }
147}
148
149void SampleFitCubicToCircle::onDrawContent(SkCanvas* canvas) {
150    canvas->clear(SK_ColorBLACK);
151
152    SkPaint circlePaint;
153    circlePaint.setColor(0x80ffffff);
154    circlePaint.setStyle(SkPaint::kStroke_Style);
155    circlePaint.setStrokeWidth(0);
156    circlePaint.setAntiAlias(true);
157    canvas->drawArc(SkRect::MakeXYWH(kCenterX - kRadius, kCenterY - kRadius, kRadius * 2,
158                                     kRadius * 2), 0, 360, false, circlePaint);
159
160    SkPaint cubicPaint;
161    cubicPaint.setColor(SK_ColorGREEN);
162    cubicPaint.setStyle(SkPaint::kStroke_Style);
163    cubicPaint.setStrokeWidth(10);
164    cubicPaint.setAntiAlias(true);
165    SkPath cubicPath;
166    cubicPath.moveTo(fCubicX[0], fCubicY[0]);
167    cubicPath.cubicTo(fCubicX[1], fCubicY[1], fCubicX[2], fCubicY[2], fCubicX[3], fCubicY[3]);
168    canvas->drawPath(cubicPath, cubicPaint);
169
170    SkPaint endpointsPaint;
171    endpointsPaint.setColor(SK_ColorBLUE);
172    endpointsPaint.setStrokeWidth(8);
173    endpointsPaint.setAntiAlias(true);
174    SkPoint points[2] = {{(float)fCubicX[0], (float)fCubicY[0]},
175                         {(float)fCubicX[3], (float)fCubicY[3]}};
176    canvas->drawPoints(SkCanvas::kPoints_PointMode, 2, points, endpointsPaint);
177
178    SkPaint textPaint;
179    textPaint.setColor(SK_ColorWHITE);
180    constexpr static float kInfoTextSize = 16;
181    SkFont font(nullptr, kInfoTextSize);
182    int infoY = 10 + kInfoTextSize;
183    for (const SkString& infoString : fInfoStrings) {
184        canvas->drawString(infoString.c_str(), 10, infoY, font, textPaint);
185        infoY += kInfoTextSize * 3/2;
186    }
187}
188
189class SampleFitCubicToCircle::Click : public Sample::Click {
190public:
191    Click(int ptIdx) : fPtIdx(ptIdx) {}
192
193    void doClick(SampleFitCubicToCircle* that) {
194        double dx = fCurr.fX - kCenterX;
195        double dy = fCurr.fY - kCenterY;
196        double l = std::sqrt(dx*dx + dy*dy);
197        that->fEndptsX[fPtIdx] = dx/l;
198        that->fEndptsY[fPtIdx] = dy/l;
199        if (that->fEndptsX[0] * that->fEndptsY[1] - that->fEndptsY[0] * that->fEndptsX[1] < 0) {
200            std::swap(that->fEndptsX[0], that->fEndptsX[1]);
201            std::swap(that->fEndptsY[0], that->fEndptsY[1]);
202            fPtIdx = 1 - fPtIdx;
203        }
204        that->fitCubic();
205    }
206
207private:
208    int fPtIdx;
209};
210
211Sample::Click* SampleFitCubicToCircle::onFindClickHandler(SkScalar x, SkScalar y,
212                                                          skui::ModifierKey) {
213    double dx0 = x - fCubicX[0];
214    double dy0 = y - fCubicY[0];
215    double dx3 = x - fCubicX[3];
216    double dy3 = y - fCubicY[3];
217    if (dx0*dx0 + dy0*dy0 < dx3*dx3 + dy3*dy3) {
218        return new Click(0);
219    } else {
220        return new Click(1);
221    }
222}
223
224bool SampleFitCubicToCircle::onClick(Sample::Click* click) {
225    Click* myClick = (Click*)click;
226    myClick->doClick(this);
227    return true;
228}
229
230bool SampleFitCubicToCircle::onChar(SkUnichar unichar) {
231    if (unichar == 'E') {
232        constexpr static double kMaxErrorT = 0.21132486540519;  // Always the same.
233        // Split the arc in half until error =~0, and report the improvement after each halving.
234        double lastError = -1;
235        for (double theta = fTheta; lastError != 0; theta /= 2) {
236            double rads = theta * M_PI/180;
237            std::array<double, 4> X, Y;
238            fit_cubic_to_unit_circle(1, 0, std::cos(rads), std::sin(rads), &X, &Y);
239            auto [x, dx, ddx] = eval_cubic(X.data(), kMaxErrorT);
240            auto [y, dy, ddy] = eval_cubic(Y.data(), kMaxErrorT);
241            double error = std::sqrt(x*x + y*y) * kRadius - kRadius;
242            if ((float)error <= 0) {
243                error = 0;
244            }
245            SkDebugf("%6.2f degrees:   error= %10.5gpx", theta, error);
246            if (lastError > 0) {
247                SkDebugf(" (%17.14fx improvement)", lastError / error);
248            }
249            SkDebugf("\n");
250            lastError = error;
251        }
252        return true;
253    }
254    return false;
255}
256
257DEF_SAMPLE(return new SampleFitCubicToCircle;)
258