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