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258 lines
9.6 KiB
258 lines
9.6 KiB
/*
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* Copyright 2020 Google Inc.
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*
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* Use of this source code is governed by a BSD-style license that can be
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* found in the LICENSE file.
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*/
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#include "samplecode/Sample.h"
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#include "include/core/SkCanvas.h"
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#include "include/core/SkFont.h"
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#include "include/core/SkPaint.h"
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#include "include/core/SkPath.h"
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#include <tuple>
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// Math constants are not always defined.
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#ifndef M_PI
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#define M_PI 3.14159265358979323846264338327950288
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#endif
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#ifndef M_SQRT2
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#define M_SQRT2 1.41421356237309504880168872420969808
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#endif
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constexpr static int kCenterX = 300;
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constexpr static int kCenterY = 325;
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constexpr static int kRadius = 250;
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// This sample fits a cubic to the arc between two interactive points on a circle. It also finds the
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// T-coordinate of max error, and outputs it and its value in pixels. (It turns out that max error
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// always occurs at T=0.21132486540519.)
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//
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// Press 'E' to iteratively cut the arc in half and report the improvement in max error after each
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// halving. (It turns out that max error improves by exactly 64x on every halving.)
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class SampleFitCubicToCircle : public Sample {
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SkString name() override { return SkString("FitCubicToCircle"); }
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void onOnceBeforeDraw() override { this->fitCubic(); }
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void fitCubic();
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void onDrawContent(SkCanvas*) override;
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Sample::Click* onFindClickHandler(SkScalar x, SkScalar y, skui::ModifierKey) override;
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bool onClick(Sample::Click*) override;
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bool onChar(SkUnichar) override;
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// Coordinates of two points on the unit circle. These are the two endpoints of the arc we fit.
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double fEndptsX[2] = {0, 1};
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double fEndptsY[2] = {-1, 0};
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// Fitted cubic and info, set by fitCubic().
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double fControlLength; // Length of (p1 - p0) and/or (p3 - p2) in unit circle space.
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double fMaxErrorT; // T value where the cubic diverges most from the true arc.
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std::array<double, 4> fCubicX; // Screen space cubic control points.
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std::array<double, 4> fCubicY;
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double fMaxError; // Max error (in pixels) between the cubic and the screen-space arc.
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double fTheta; // Angle of the arc. This is only used for informational purposes.
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SkTArray<SkString> fInfoStrings;
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class Click;
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};
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// Fits a cubic to an arc on the unit circle with endpoints (x0, y0) and (x1, y1). Using the
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// following 3 constraints, we arrive at the formula used in the method:
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//
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// 1) The endpoints and tangent directions at the endpoints must match the arc.
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// 2) The cubic must be symmetric (i.e., length(p1 - p0) == length(p3 - p2)).
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// 3) The height of the cubic must match the height of the arc.
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//
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// Returns the "control length", or length of (p1 - p0) and/or (p3 - p2).
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static float fit_cubic_to_unit_circle(double x0, double y0, double x1, double y1,
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std::array<double, 4>* X, std::array<double, 4>* Y) {
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constexpr static double kM = -4.0/3;
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constexpr static double kA = 4*M_SQRT2/3;
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double d = x0*x1 + y0*y1;
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double c = (std::sqrt(1 + d) * kM + kA) / std::sqrt(1 - d);
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*X = {x0, x0 - y0*c, x1 + y1*c, x1};
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*Y = {y0, y0 + x0*c, y1 - x1*c, y1};
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return c;
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}
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static double lerp(double x, double y, double T) {
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return x + T*(y - x);
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}
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// Evaluates the cubic and 1st and 2nd derivatives at T.
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static std::tuple<double, double, double> eval_cubic(double x[], double T) {
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// Use De Casteljau's algorithm for better accuracy and stability.
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double ab = lerp(x[0], x[1], T);
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double bc = lerp(x[1], x[2], T);
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double cd = lerp(x[2], x[3], T);
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double abc = lerp(ab, bc, T);
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double bcd = lerp(bc, cd, T);
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double abcd = lerp(abc, bcd, T);
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return {abcd, 3 * (bcd - abc) /*1st derivative.*/, 6 * (cd - 2*bc + ab) /*2nd derivative.*/};
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}
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// Uses newton-raphson convergence to find the point where the provided cubic diverges most from the
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// unit circle. i.e., the point where the derivative of error == 0. For error we use:
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//
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// error = x^2 + y^2 - 1
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// error' = 2xx' + 2yy'
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// error'' = 2xx'' + 2yy'' + 2x'^2 + 2y'^2
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//
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double find_max_error_T(double cubicX[4], double cubicY[4]) {
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constexpr static double kInitialT = .25;
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double T = kInitialT;
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for (int i = 0; i < 64; ++i) {
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auto [x, dx, ddx] = eval_cubic(cubicX, T);
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auto [y, dy, ddy] = eval_cubic(cubicY, T);
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double dError = 2*(x*dx + y*dy);
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double ddError = 2*(x*ddx + y*ddy + dx*dx + dy*dy);
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T -= dError / ddError;
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}
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return T;
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}
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void SampleFitCubicToCircle::fitCubic() {
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fInfoStrings.reset();
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std::array<double, 4> X, Y;
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// "Control length" is the length of (p1 - p0) and/or (p3 - p2) in unit circle space.
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fControlLength = fit_cubic_to_unit_circle(fEndptsX[0], fEndptsY[0], fEndptsX[1], fEndptsY[1],
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&X, &Y);
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fInfoStrings.push_back().printf("control length=%0.14f", fControlLength);
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fMaxErrorT = find_max_error_T(X.data(), Y.data());
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fInfoStrings.push_back().printf("max error T=%0.14f", fMaxErrorT);
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for (int i = 0; i < 4; ++i) {
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fCubicX[i] = X[i] * kRadius + kCenterX;
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fCubicY[i] = Y[i] * kRadius + kCenterY;
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}
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double errX = std::get<0>(eval_cubic(fCubicX.data(), fMaxErrorT)) - kCenterX;
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double errY = std::get<0>(eval_cubic(fCubicY.data(), fMaxErrorT)) - kCenterY;
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fMaxError = std::sqrt(errX*errX + errY*errY) - kRadius;
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fInfoStrings.push_back().printf("max error=%.5gpx", fMaxError);
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fTheta = std::atan2(fEndptsY[1], fEndptsX[1]) - std::atan2(fEndptsY[0], fEndptsX[0]);
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fTheta = std::abs(fTheta * 180/M_PI);
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if (fTheta > 180) {
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fTheta = 360 - fTheta;
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}
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fInfoStrings.push_back().printf("(theta=%.2f)", fTheta);
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SkDebugf("\n");
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for (const SkString& infoString : fInfoStrings) {
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SkDebugf("%s\n", infoString.c_str());
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}
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}
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void SampleFitCubicToCircle::onDrawContent(SkCanvas* canvas) {
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canvas->clear(SK_ColorBLACK);
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SkPaint circlePaint;
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circlePaint.setColor(0x80ffffff);
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circlePaint.setStyle(SkPaint::kStroke_Style);
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circlePaint.setStrokeWidth(0);
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circlePaint.setAntiAlias(true);
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canvas->drawArc(SkRect::MakeXYWH(kCenterX - kRadius, kCenterY - kRadius, kRadius * 2,
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kRadius * 2), 0, 360, false, circlePaint);
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SkPaint cubicPaint;
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cubicPaint.setColor(SK_ColorGREEN);
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cubicPaint.setStyle(SkPaint::kStroke_Style);
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cubicPaint.setStrokeWidth(10);
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cubicPaint.setAntiAlias(true);
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SkPath cubicPath;
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cubicPath.moveTo(fCubicX[0], fCubicY[0]);
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cubicPath.cubicTo(fCubicX[1], fCubicY[1], fCubicX[2], fCubicY[2], fCubicX[3], fCubicY[3]);
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canvas->drawPath(cubicPath, cubicPaint);
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SkPaint endpointsPaint;
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endpointsPaint.setColor(SK_ColorBLUE);
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endpointsPaint.setStrokeWidth(8);
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endpointsPaint.setAntiAlias(true);
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SkPoint points[2] = {{(float)fCubicX[0], (float)fCubicY[0]},
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{(float)fCubicX[3], (float)fCubicY[3]}};
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canvas->drawPoints(SkCanvas::kPoints_PointMode, 2, points, endpointsPaint);
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SkPaint textPaint;
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textPaint.setColor(SK_ColorWHITE);
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constexpr static float kInfoTextSize = 16;
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SkFont font(nullptr, kInfoTextSize);
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int infoY = 10 + kInfoTextSize;
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for (const SkString& infoString : fInfoStrings) {
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canvas->drawString(infoString.c_str(), 10, infoY, font, textPaint);
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infoY += kInfoTextSize * 3/2;
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}
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}
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class SampleFitCubicToCircle::Click : public Sample::Click {
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public:
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Click(int ptIdx) : fPtIdx(ptIdx) {}
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void doClick(SampleFitCubicToCircle* that) {
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double dx = fCurr.fX - kCenterX;
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double dy = fCurr.fY - kCenterY;
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double l = std::sqrt(dx*dx + dy*dy);
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that->fEndptsX[fPtIdx] = dx/l;
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that->fEndptsY[fPtIdx] = dy/l;
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if (that->fEndptsX[0] * that->fEndptsY[1] - that->fEndptsY[0] * that->fEndptsX[1] < 0) {
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std::swap(that->fEndptsX[0], that->fEndptsX[1]);
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std::swap(that->fEndptsY[0], that->fEndptsY[1]);
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fPtIdx = 1 - fPtIdx;
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}
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that->fitCubic();
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}
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private:
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int fPtIdx;
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};
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Sample::Click* SampleFitCubicToCircle::onFindClickHandler(SkScalar x, SkScalar y,
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skui::ModifierKey) {
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double dx0 = x - fCubicX[0];
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double dy0 = y - fCubicY[0];
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double dx3 = x - fCubicX[3];
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double dy3 = y - fCubicY[3];
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if (dx0*dx0 + dy0*dy0 < dx3*dx3 + dy3*dy3) {
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return new Click(0);
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} else {
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return new Click(1);
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}
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}
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bool SampleFitCubicToCircle::onClick(Sample::Click* click) {
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Click* myClick = (Click*)click;
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myClick->doClick(this);
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return true;
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}
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bool SampleFitCubicToCircle::onChar(SkUnichar unichar) {
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if (unichar == 'E') {
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constexpr static double kMaxErrorT = 0.21132486540519; // Always the same.
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// Split the arc in half until error =~0, and report the improvement after each halving.
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double lastError = -1;
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for (double theta = fTheta; lastError != 0; theta /= 2) {
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double rads = theta * M_PI/180;
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std::array<double, 4> X, Y;
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fit_cubic_to_unit_circle(1, 0, std::cos(rads), std::sin(rads), &X, &Y);
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auto [x, dx, ddx] = eval_cubic(X.data(), kMaxErrorT);
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auto [y, dy, ddy] = eval_cubic(Y.data(), kMaxErrorT);
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double error = std::sqrt(x*x + y*y) * kRadius - kRadius;
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if ((float)error <= 0) {
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error = 0;
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}
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SkDebugf("%6.2f degrees: error= %10.5gpx", theta, error);
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if (lastError > 0) {
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SkDebugf(" (%17.14fx improvement)", lastError / error);
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}
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SkDebugf("\n");
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lastError = error;
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}
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return true;
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}
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return false;
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}
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DEF_SAMPLE(return new SampleFitCubicToCircle;)
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