Define accelerated steps time function.

ui/gfx/geometry/cubic_bezier.cc

-// Copyright 2014 The Chromium Authors. All rights reserved.
-// Use of this source code is governed by a BSD-style license that can be
-// found in the LICENSE file.
-
-#include "ui/gfx/geometry/cubic_bezier.h"
-
-#include <algorithm>
-#include <cmath>
-
-#include "base/logging.h"
-
-namespace gfx {
-
-namespace {
-
-static const double kBezierEpsilon = 1e-7;
-static const int MAX_STEPS = 30;
-
-static double eval_bezier(double p1, double p2, double t) {
-  const double p1_times_3 = 3.0 * p1;
-  const double p2_times_3 = 3.0 * p2;
-  const double h3 = p1_times_3;
-  const double h1 = p1_times_3 - p2_times_3 + 1.0;
-  const double h2 = p2_times_3 - 6.0 * p1;
-  return t * (t * (t * h1 + h2) + h3);
-}
-
-static double eval_bezier_derivative(double p1, double p2, double t) {
-  const double h1 = 9.0 * p1 - 9.0 * p2 + 3.0;
-  const double h2 = 6.0 * p2 - 12.0 * p1;
-  const double h3 = 3.0 * p1;
-  return t * (t * h1 + h2) + h3;
-}
-
-// Finds t such that eval_bezier(x1, x2, t) = x.
-// There is a unique solution if x1 and x2 lie within (0, 1).
-static double bezier_interp(double x1,
-                            double x2,
-                            double x) {
-  DCHECK_GE(1.0, x1);
-  DCHECK_LE(0.0, x1);
-  DCHECK_GE(1.0, x2);
-  DCHECK_LE(0.0, x2);
-
-  x1 = std::min(std::max(x1, 0.0), 1.0);
-  x2 = std::min(std::max(x2, 0.0), 1.0);
-  x = std::min(std::max(x, 0.0), 1.0);
-
-  // We're just going to do bisection for now (for simplicity), but we could
-  // easily do some newton steps if this turns out to be a bottleneck.
-  double t = 0.0;
-  double step = 1.0;
-  for (int i = 0; i < MAX_STEPS; ++i, step *= 0.5) {
-    const double error = eval_bezier(x1, x2, t) - x;
-    if (std::abs(error) < kBezierEpsilon)
-      break;
-    t += error > 0.0 ? -step : step;
-  }
-
-  // We should have terminated the above loop because we got close to x, not
-  // because we exceeded MAX_STEPS. Do a DCHECK here to confirm.
-  DCHECK_GT(kBezierEpsilon, std::abs(eval_bezier(x1, x2, t) - x));
-
-  return t;
-}
-
-}  // namespace
-
-CubicBezier::CubicBezier(double x1, double y1, double x2, double y2)
-    : x1_(x1),
-      y1_(y1),
-      x2_(x2),
-      y2_(y2) {
-}
-
-CubicBezier::~CubicBezier() {
-}
-
-double CubicBezier::Solve(double x) const {
-  return eval_bezier(y1_, y2_, bezier_interp(x1_, x2_, x));
-}
-
-double CubicBezier::Slope(double x) const {
-  double t = bezier_interp(x1_, x2_, x);
-  double dx_dt = eval_bezier_derivative(x1_, x2_, t);
-  double dy_dt = eval_bezier_derivative(y1_, y2_, t);
-  return dy_dt / dx_dt;
-}
-
-void CubicBezier::Range(double* min, double* max) const {
-  *min = 0;
-  *max = 1;
-  if (0 <= y1_ && y1_ < 1 && 0 <= y2_ && y2_ <= 1)
-    return;
-
-  // Represent the function's derivative in the form at^2 + bt + c.
-  // (Technically this is (dy/dt)*(1/3), which is suitable for finding zeros
-  // but does not actually give the slope of the curve.)
-  double a = 3 * (y1_ - y2_) + 1;
-  double b = 2 * (y2_ - 2 * y1_);
-  double c = y1_;
-
-  // Check if the derivative is constant.
-  if (std::abs(a) < kBezierEpsilon &&
-      std::abs(b) < kBezierEpsilon)
-    return;
-
-  // Zeros of the function's derivative.
-  double t_1 = 0;
-  double t_2 = 0;
-
-  if (std::abs(a) < kBezierEpsilon) {
-    // The function's derivative is linear.
-    t_1 = -c / b;
-  } else {
-    // The function's derivative is a quadratic. We find the zeros of this
-    // quadratic using the quadratic formula.
-    double discriminant = b * b - 4 * a * c;
-    if (discriminant < 0)
-      return;
-    double discriminant_sqrt = sqrt(discriminant);
-    t_1 = (-b + discriminant_sqrt) / (2 * a);
-    t_2 = (-b - discriminant_sqrt) / (2 * a);
-  }
-
-  double sol_1 = 0;
-  double sol_2 = 0;
-
-  if (0 < t_1 && t_1 < 1)
-    sol_1 = eval_bezier(y1_, y2_, t_1);
-
-  if (0 < t_2 && t_2 < 1)
-    sol_2 = eval_bezier(y1_, y2_, t_2);
-
-  *min = std::min(std::min(*min, sol_1), sol_2);
-  *max = std::max(std::max(*max, sol_1), sol_2);
-}
-
-}  // namespace gfx