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// Copyright 2014 The Flutter Authors. All rights reserved.
// Use of this source code is governed by a BSD-style license that can be
// found in the LICENSE file.
import 'dart:math' as math;
import 'dart:ui';
import 'package:flutter/foundation.dart';
/// An abstract class providing an interface for evaluating a parametric curve.
///
/// A parametric curve transforms a parameter (hence the name) `t` along a curve
/// to the value of the curve at that value of `t`. The curve can be of
/// arbitrary dimension, but is typically a 1D, 2D, or 3D curve.
///
/// See also:
///
/// * [Curve], a 1D animation easing curve that starts at 0.0 and ends at 1.0.
/// * [Curve2D], a parametric curve that transforms the parameter to a 2D point.
abstract class ParametricCurve<T> {
/// Abstract const constructor to enable subclasses to provide
/// const constructors so that they can be used in const expressions.
const ParametricCurve();
/// Returns the value of the curve at point `t`.
///
/// This method asserts that t is between 0 and 1 before delegating to
/// [transformInternal].
///
/// It is recommended that subclasses override [transformInternal] instead of
/// this function, as the above case is already handled in the default
/// implementation of [transform], which delegates the remaining logic to
/// [transformInternal].
T transform(double t) {
assert(t != null);
assert(t >= 0.0 && t <= 1.0, 'parametric value $t is outside of [0, 1] range.');
return transformInternal(t);
}
/// Returns the value of the curve at point `t`.
///
/// The given parametric value `t` will be between 0.0 and 1.0, inclusive.
@protected
T transformInternal(double t) {
throw UnimplementedError();
}
@override
String toString() => objectRuntimeType(this, 'ParametricCurve');
}
/// An parametric animation easing curve, i.e. a mapping of the unit interval to
/// the unit interval.
///
/// Easing curves are used to adjust the rate of change of an animation over
/// time, allowing them to speed up and slow down, rather than moving at a
/// constant rate.
///
/// A [Curve] must map t=0.0 to 0.0 and t=1.0 to 1.0.
///
/// See also:
///
/// * [Curves], a collection of common animation easing curves.
/// * [CurveTween], which can be used to apply a [Curve] to an [Animation].
/// * [Canvas.drawArc], which draws an arc, and has nothing to do with easing
/// curves.
/// * [Animatable], for a more flexible interface that maps fractions to
/// arbitrary values.
@immutable
abstract class Curve extends ParametricCurve<double> {
/// Abstract const constructor to enable subclasses to provide
/// const constructors so that they can be used in const expressions.
const Curve();
/// Returns the value of the curve at point `t`.
///
/// This function must ensure the following:
/// - The value of `t` must be between 0.0 and 1.0
/// - Values of `t`=0.0 and `t`=1.0 must be mapped to 0.0 and 1.0,
/// respectively.
///
/// It is recommended that subclasses override [transformInternal] instead of
/// this function, as the above cases are already handled in the default
/// implementation of [transform], which delegates the remaining logic to
/// [transformInternal].
@override
double transform(double t) {
if (t == 0.0 || t == 1.0) {
return t;
}
return super.transform(t);
}
/// Returns a new curve that is the reversed inversion of this one.
///
/// This is often useful with [CurvedAnimation.reverseCurve].
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_bounce_in.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_flipped.mp4}
///
/// See also:
///
/// * [FlippedCurve], the class that is used to implement this getter.
/// * [ReverseAnimation], which reverses an [Animation] rather than a [Curve].
/// * [CurvedAnimation], which can take a separate curve and reverse curve.
Curve get flipped => FlippedCurve(this);
}
/// The identity map over the unit interval.
///
/// See [Curves.linear] for an instance of this class.
class _Linear extends Curve {
const _Linear._();
@override
double transformInternal(double t) => t;
}
/// A sawtooth curve that repeats a given number of times over the unit interval.
///
/// The curve rises linearly from 0.0 to 1.0 and then falls discontinuously back
/// to 0.0 each iteration.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_sawtooth.mp4}
class SawTooth extends Curve {
/// Creates a sawtooth curve.
///
/// The [count] argument must not be null.
const SawTooth(this.count) : assert(count != null);
/// The number of repetitions of the sawtooth pattern in the unit interval.
final int count;
@override
double transformInternal(double t) {
t *= count;
return t - t.truncateToDouble();
}
@override
String toString() {
return '${objectRuntimeType(this, 'SawTooth')}($count)';
}
}
/// A curve that is 0.0 until [begin], then curved (according to [curve]) from
/// 0.0 at [begin] to 1.0 at [end], then remains 1.0 past [end].
///
/// An [Interval] can be used to delay an animation. For example, a six second
/// animation that uses an [Interval] with its [begin] set to 0.5 and its [end]
/// set to 1.0 will essentially become a three-second animation that starts
/// three seconds later.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_interval.mp4}
class Interval extends Curve {
/// Creates an interval curve.
///
/// The arguments must not be null.
const Interval(this.begin, this.end, { this.curve = Curves.linear })
: assert(begin != null),
assert(end != null),
assert(curve != null);
/// The largest value for which this interval is 0.0.
///
/// From t=0.0 to t=`begin`, the interval's value is 0.0.
final double begin;
/// The smallest value for which this interval is 1.0.
///
/// From t=`end` to t=1.0, the interval's value is 1.0.
final double end;
/// The curve to apply between [begin] and [end].
final Curve curve;
@override
double transformInternal(double t) {
assert(begin >= 0.0);
assert(begin <= 1.0);
assert(end >= 0.0);
assert(end <= 1.0);
assert(end >= begin);
t = ((t - begin) / (end - begin)).clamp(0.0, 1.0);
if (t == 0.0 || t == 1.0)
return t;
return curve.transform(t);
}
@override
String toString() {
if (curve is! _Linear)
return '${objectRuntimeType(this, 'Interval')}($begin\u22EF$end)\u27A9$curve';
return '${objectRuntimeType(this, 'Interval')}($begin\u22EF$end)';
}
}
/// A curve that is 0.0 until it hits the threshold, then it jumps to 1.0.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_threshold.mp4}
class Threshold extends Curve {
/// Creates a threshold curve.
///
/// The [threshold] argument must not be null.
const Threshold(this.threshold) : assert(threshold != null);
/// The value before which the curve is 0.0 and after which the curve is 1.0.
///
/// When t is exactly [threshold], the curve has the value 1.0.
final double threshold;
@override
double transformInternal(double t) {
assert(threshold >= 0.0);
assert(threshold <= 1.0);
return t < threshold ? 0.0 : 1.0;
}
}
/// A cubic polynomial mapping of the unit interval.
///
/// The [Curves] class contains some commonly used cubic curves:
///
/// * [Curves.fastLinearToSlowEaseIn]
/// * [Curves.ease]
/// * [Curves.easeIn]
/// * [Curves.easeInToLinear]
/// * [Curves.easeInSine]
/// * [Curves.easeInQuad]
/// * [Curves.easeInCubic]
/// * [Curves.easeInQuart]
/// * [Curves.easeInQuint]
/// * [Curves.easeInExpo]
/// * [Curves.easeInCirc]
/// * [Curves.easeInBack]
/// * [Curves.easeOut]
/// * [Curves.linearToEaseOut]
/// * [Curves.easeOutSine]
/// * [Curves.easeOutQuad]
/// * [Curves.easeOutCubic]
/// * [Curves.easeOutQuart]
/// * [Curves.easeOutQuint]
/// * [Curves.easeOutExpo]
/// * [Curves.easeOutCirc]
/// * [Curves.easeOutBack]
/// * [Curves.easeInOut]
/// * [Curves.easeInOutSine]
/// * [Curves.easeInOutQuad]
/// * [Curves.easeInOutCubic]
/// * [Curves.easeInOutQuart]
/// * [Curves.easeInOutQuint]
/// * [Curves.easeInOutExpo]
/// * [Curves.easeInOutCirc]
/// * [Curves.easeInOutBack]
/// * [Curves.fastOutSlowIn]
/// * [Curves.slowMiddle]
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_fast_linear_to_slow_ease_in.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_to_linear.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_sine.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_quad.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_cubic.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_quart.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_quint.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_expo.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_circ.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_back.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_out.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_linear_to_ease_out.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_out_sine.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_out_quad.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_out_cubic.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_out_quart.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_out_quint.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_out_expo.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_out_circ.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_out_back.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out_sine.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out_quad.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out_cubic.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out_quart.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out_quint.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out_expo.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out_circ.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out_back.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_fast_out_slow_in.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_slow_middle.mp4}
///
/// The [Cubic] class implements third-order Bézier curves.
///
/// See also:
///
/// * [Curves], where many more predefined curves are available.
/// * [CatmullRomCurve], a curve which passes through specific values.
class Cubic extends Curve {
/// Creates a cubic curve.
///
/// Rather than creating a new instance, consider using one of the common
/// cubic curves in [Curves].
///
/// The [a] (x1), [b](x2), [c](y1), and [d](y2) arguments must not be null.
const Cubic(this.a, this.b, this.c, this.d)
: assert(a != null),
assert(b != null),
assert(c != null),
assert(d != null);
/// The x coordinate of the first control point.
///
/// The line through the point (0, 0) and the first control point is tangent
/// to the curve at the point (0, 0).
final double a;
/// The y coordinate of the first control point.
///
/// The line through the point (0, 0) and the first control point is tangent
/// to the curve at the point (0, 0).
final double b;
/// The x coordinate of the second control point.
///
/// The line through the point (1, 1) and the second control point is tangent
/// to the curve at the point (1, 1).
final double c;
/// The y coordinate of the second control point.
///
/// The line through the point (1, 1) and the second control point is tangent
/// to the curve at the point (1, 1).
final double d;
static const double _cubicErrorBound = 0.001;
double _evaluateCubic(double a, double b, double m) {
return 3 * a * (1 - m) * (1 - m) * m +
3 * b * (1 - m) * m * m +
m * m * m;
}
@override
double transformInternal(double t) {
double start = 0.0;
double end = 1.0;
while (true) {
final double midpoint = (start + end) / 2;
final double estimate = _evaluateCubic(a, c, midpoint);
if ((t - estimate).abs() < _cubicErrorBound)
return _evaluateCubic(b, d, midpoint);
if (estimate < t)
start = midpoint;
else
end = midpoint;
}
}
@override
String toString() {
return '${objectRuntimeType(this, 'Cubic')}(${a.toStringAsFixed(2)}, ${b.toStringAsFixed(2)}, ${c.toStringAsFixed(2)}, ${d.toStringAsFixed(2)})';
}
}
/// A cubic polynomial composed of two curves that share a common center point.
///
/// The curve runs through three points: (0,0), the [midpoint], and (1,1).
///
/// The [Curves] class contains a curve defined with this class:
/// [Curves.easeInOutCubicEmphasized].
///
/// The [ThreePointCubic] class implements third-order Bézier curves, where two
/// curves share an interior [midpoint] that the curve passes through. If the
/// control points surrounding the middle point ([b1], and [a2]) are not
/// colinear with the middle point, then the curve's derivative will have a
/// discontinuity (a cusp) at the shared middle point.
///
/// See also:
///
/// * [Curves], where many more predefined curves are available.
/// * [Cubic], which defines a single cubic polynomial.
/// * [CatmullRomCurve], a curve which passes through specific values.
class ThreePointCubic extends Curve {
/// Creates two cubic curves that share a common control point.
///
/// Rather than creating a new instance, consider using one of the common
/// three-point cubic curves in [Curves].
///
/// The arguments correspond to the control points for the two curves,
/// including the [midpoint], but do not include the two implied end points at
/// (0,0) and (1,1), which are fixed.
const ThreePointCubic(this.a1, this.b1, this.midpoint, this.a2, this.b2);
/// The coordinates of the first control point of the first curve.
///
/// The line through the point (0, 0) and this control point is tangent to the
/// curve at the point (0, 0).
final Offset a1;
/// The coordinates of the second control point of the first curve.
///
/// The line through the [midpoint] and this control point is tangent to the
/// curve approaching the [midpoint].
final Offset b1;
/// The coordinates of the middle shared point.
///
/// The curve will go through this point. If the control points surrounding
/// this middle point ([b1], and [a2]) are not colinear with this point, then
/// the curve's derivative will have a discontinuity (a cusp) at this point.
final Offset midpoint;
/// The coordinates of the first control point of the second curve.
///
/// The line through the [midpoint] and this control point is tangent to the
/// curve approaching the [midpoint].
final Offset a2;
/// The coordinates of the second control point of the second curve.
///
/// The line through the point (1, 1) and this control point is tangent to the
/// curve at (1, 1).
final Offset b2;
@override
double transformInternal(double t) {
final bool firstCurve = t < midpoint.dx;
final double scaleX = firstCurve ? midpoint.dx : 1.0 - midpoint.dx;
final double scaleY = firstCurve ? midpoint.dy : 1.0 - midpoint.dy;
final double scaledT = (t - (firstCurve ? 0.0 : midpoint.dx)) / scaleX;
if (firstCurve) {
return Cubic(
a1.dx / scaleX,
a1.dy / scaleY,
b1.dx / scaleX,
b1.dy / scaleY,
).transform(scaledT) * scaleY;
} else {
return Cubic(
(a2.dx - midpoint.dx) / scaleX,
(a2.dy - midpoint.dy) / scaleY,
(b2.dx - midpoint.dx) / scaleX,
(b2.dy - midpoint.dy) / scaleY,
).transform(scaledT) * scaleY + midpoint.dy;
}
}
@override
String toString() {
return '${objectRuntimeType(this, 'ThreePointCubic($a1, $b1, $midpoint, $a2, $b2)')} ';
}
}
/// Abstract class that defines an API for evaluating 2D parametric curves.
///
/// [Curve2D] differs from [Curve] in that the values interpolated are [Offset]
/// values instead of [double] values, hence the "2D" in the name. They both
/// take a single double `t` that has a range of 0.0 to 1.0, inclusive, as input
/// to the [transform] function . Unlike [Curve], [Curve2D] is not required to
/// map `t=0.0` and `t=1.0` to specific output values.
///
/// The interpolated `t` value given to [transform] represents the progression
/// along the curve, but it doesn't necessarily progress at a constant velocity, so
/// incrementing `t` by, say, 0.1 might move along the curve by quite a lot at one
/// part of the curve, or hardly at all in another part of the curve, depending
/// on the definition of the curve.
///
/// {@tool dartpad}
/// This example shows how to use a [Curve2D] to modify the position of a widget
/// so that it can follow an arbitrary path.
///
/// ** See code in examples/api/lib/animation/curves/curve2_d.0.dart **
/// {@end-tool}
///
abstract class Curve2D extends ParametricCurve<Offset> {
/// Abstract const constructor to enable subclasses to provide const
/// constructors so that they can be used in const expressions.
const Curve2D();
/// Generates a list of samples with a recursive subdivision until a tolerance
/// of `tolerance` is reached.
///
/// Samples are generated in order.
///
/// Samples can be used to render a curve efficiently, since the samples
/// constitute line segments which vary in size with the curvature of the
/// curve. They can also be used to quickly approximate the value of the curve
/// by searching for the desired range in X and linearly interpolating between
/// samples to obtain an approximation of Y at the desired X value. The
/// implementation of [CatmullRomCurve] uses samples for this purpose
/// internally.
///
/// The tolerance is computed as the area of a triangle formed by a new point
/// and the preceding and following point.
///
/// See also:
///
/// * Luiz Henrique de Figueire's Graphics Gem on [the algorithm](http://ariel.chronotext.org/dd/defigueiredo93adaptive.pdf).
Iterable<Curve2DSample> generateSamples({
double start = 0.0,
double end = 1.0,
double tolerance = 1e-10,
}) {
// The sampling algorithm is:
// 1. Evaluate the area of the triangle (a proxy for the "flatness" of the
// curve) formed by two points and a test point.
// 2. If the area of the triangle is small enough (below tolerance), then
// the two points form the final segment.
// 3. If the area is still too large, divide the interval into two parts
// using a random subdivision point to avoid aliasing.
// 4. Recursively sample the two parts.
//
// This algorithm concentrates samples in areas of high curvature.
assert(tolerance != null);
assert(start != null);
assert(end != null);
assert(end > start);
// We want to pick a random seed that will keep the result stable if
// evaluated again, so we use the first non-generated control point.
final math.Random rand = math.Random(samplingSeed);
bool isFlat(Offset p, Offset q, Offset r) {
// Calculates the area of the triangle given by the three points.
final Offset pr = p - r;
final Offset qr = q - r;
final double z = pr.dx * qr.dy - qr.dx * pr.dy;
return (z * z) < tolerance;
}
final Curve2DSample first = Curve2DSample(start, transform(start));
final Curve2DSample last = Curve2DSample(end, transform(end));
final List<Curve2DSample> samples = <Curve2DSample>[first];
void sample(Curve2DSample p, Curve2DSample q, {bool forceSubdivide = false}) {
// Pick a random point somewhat near the center, which avoids aliasing
// problems with periodic curves.
final double t = p.t + (0.45 + 0.1 * rand.nextDouble()) * (q.t - p.t);
final Curve2DSample r = Curve2DSample(t, transform(t));
if (!forceSubdivide && isFlat(p.value, q.value, r.value)) {
samples.add(q);
} else {
sample(p, r);
sample(r, q);
}
}
// If the curve starts and ends on the same point, then we force it to
// subdivide at least once, because otherwise it will terminate immediately.
sample(
first,
last,
forceSubdivide: (first.value.dx - last.value.dx).abs() < tolerance && (first.value.dy - last.value.dy).abs() < tolerance,
);
return samples;
}
/// Returns a seed value used by [generateSamples] to seed a random number
/// generator to avoid sample aliasing.
///
/// Subclasses should override this and provide a custom seed.
///
/// The value returned should be the same each time it is called, unless the
/// curve definition changes.
@protected
int get samplingSeed => 0;
/// Returns the parameter `t` that corresponds to the given x value of the spline.
///
/// This will only work properly for curves which are single-valued in x
/// (where every value of `x` maps to only one value in 'y', i.e. the curve
/// does not loop or curve back over itself). For curves that are not
/// single-valued, it will return the parameter for only one of the values at
/// the given `x` location.
double findInverse(double x) {
assert(x != null);
double start = 0.0;
double end = 1.0;
late double mid;
double offsetToOrigin(double pos) => x - transform(pos).dx;
// Use a binary search to find the inverse point within 1e-6, or 100
// subdivisions, whichever comes first.
const double errorLimit = 1e-6;
int count = 100;
final double startValue = offsetToOrigin(start);
while ((end - start) / 2.0 > errorLimit && count > 0) {
mid = (end + start) / 2.0;
final double value = offsetToOrigin(mid);
if (value.sign == startValue.sign) {
start = mid;
} else {
end = mid;
}
count--;
}
return mid;
}
}
/// A class that holds a sample of a 2D parametric curve, containing the [value]
/// (the X, Y coordinates) of the curve at the parametric value [t].
///
/// See also:
///
/// * [Curve2D.generateSamples], which generates samples of this type.
/// * [Curve2D], a parametric curve that maps a double parameter to a 2D location.
class Curve2DSample {
/// Creates an object that holds a sample; used with [Curve2D] subclasses.
///
/// All arguments must not be null.
const Curve2DSample(this.t, this.value) : assert(t != null), assert(value != null);
/// The parametric location of this sample point along the curve.
final double t;
/// The value (the X, Y coordinates) of the curve at parametric value [t].
final Offset value;
@override
String toString() {
return '[(${value.dx.toStringAsFixed(2)}, ${value.dy.toStringAsFixed(2)}), ${t.toStringAsFixed(2)}]';
}
}
/// A 2D spline that passes smoothly through the given control points using a
/// centripetal Catmull-Rom spline.
///
/// When the curve is evaluated with [transform], the output values will move
/// smoothly from one control point to the next, passing through the control
/// points.
///
/// {@template flutter.animation.CatmullRomSpline}
/// Unlike most cubic splines, Catmull-Rom splines have the advantage that their
/// curves pass through the control points given to them. They are cubic
/// polynomial representations, and, in fact, Catmull-Rom splines can be
/// converted mathematically into cubic splines. This class implements a
/// "centripetal" Catmull-Rom spline. The term centripetal implies that it won't
/// form loops or self-intersections within a single segment.
/// {@endtemplate}
///
/// See also:
/// * [Centripetal Catmull–Rom splines](https://en.wikipedia.org/wiki/Centripetal_Catmull%E2%80%93Rom_spline)
/// on Wikipedia.
/// * [Parameterization and Applications of Catmull-Rom Curves](http://faculty.cs.tamu.edu/schaefer/research/cr_cad.pdf),
/// a paper on using Catmull-Rom splines.
/// * [CatmullRomCurve], an animation curve that uses a [CatmullRomSpline] as its
/// internal representation.
class CatmullRomSpline extends Curve2D {
/// Constructs a centripetal Catmull-Rom spline curve.
///
/// The `controlPoints` argument is a list of four or more points that
/// describe the points that the curve must pass through.
///
/// The optional `tension` argument controls how tightly the curve approaches
/// the given `controlPoints`. It must be in the range 0.0 to 1.0, inclusive. It
/// defaults to 0.0, which provides the smoothest curve. A value of 1.0
/// produces a linear interpolation between points.
///
/// The optional `endHandle` and `startHandle` points are the beginning and
/// ending handle positions. If not specified, they are created automatically
/// by extending the line formed by the first and/or last line segment in the
/// `controlPoints`, respectively. The spline will not go through these handle
/// points, but they will affect the slope of the line at the beginning and
/// end of the spline. The spline will attempt to match the slope of the line
/// formed by the start or end handle and the neighboring first or last
/// control point. The default is chosen so that the slope of the line at the
/// ends matches that of the first or last line segment in the control points.
///
/// The `tension` and `controlPoints` arguments must not be null, and the
/// `controlPoints` list must contain at least four control points to
/// interpolate.
///
/// The internal curve data structures are lazily computed the first time
/// [transform] is called. If you would rather pre-compute the structures,
/// use [CatmullRomSpline.precompute] instead.
CatmullRomSpline(
List<Offset> controlPoints, {
double tension = 0.0,
Offset? startHandle,
Offset? endHandle,
}) : assert(controlPoints != null),
assert(tension != null),
assert(tension <= 1.0, 'tension $tension must not be greater than 1.0.'),
assert(tension >= 0.0, 'tension $tension must not be negative.'),
assert(controlPoints.length > 3, 'There must be at least four control points to create a CatmullRomSpline.'),
_controlPoints = controlPoints,
_startHandle = startHandle,
_endHandle = endHandle,
_tension = tension,
_cubicSegments = <List<Offset>>[];
/// Constructs a centripetal Catmull-Rom spline curve.
///
/// The same as [new CatmullRomSpline], except that the internal data
/// structures are precomputed instead of being computed lazily.
CatmullRomSpline.precompute(
List<Offset> controlPoints, {
double tension = 0.0,
Offset? startHandle,
Offset? endHandle,
}) : assert(controlPoints != null),
assert(tension != null),
assert(tension <= 1.0, 'tension $tension must not be greater than 1.0.'),
assert(tension >= 0.0, 'tension $tension must not be negative.'),
assert(controlPoints.length > 3, 'There must be at least four control points to create a CatmullRomSpline.'),
_controlPoints = null,
_startHandle = null,
_endHandle = null,
_tension = null,
_cubicSegments = _computeSegments(controlPoints, tension, startHandle: startHandle, endHandle: endHandle);
static List<List<Offset>> _computeSegments(
List<Offset> controlPoints,
double tension, {
Offset? startHandle,
Offset? endHandle,
}) {
// If not specified, select the first and last control points (which are
// handles: they are not intersected by the resulting curve) so that they
// extend the first and last segments, respectively.
startHandle ??= controlPoints[0] * 2.0 - controlPoints[1];
endHandle ??= controlPoints.last * 2.0 - controlPoints[controlPoints.length - 2];
final List<Offset> allPoints = <Offset>[
startHandle,
...controlPoints,
endHandle,
];
// An alpha of 0.5 is what makes it a centripetal Catmull-Rom spline. A
// value of 0.0 would make it a uniform Catmull-Rom spline, and a value of
// 1.0 would make it a chordal Catmull-Rom spline. Non-centripetal values
// for alpha can give self-intersecting behavior or looping within a
// segment.
const double alpha = 0.5;
final double reverseTension = 1.0 - tension;
final List<List<Offset>> result = <List<Offset>>[];
for (int i = 0; i < allPoints.length - 3; ++i) {
final List<Offset> curve = <Offset>[allPoints[i], allPoints[i + 1], allPoints[i + 2], allPoints[i + 3]];
final Offset diffCurve10 = curve[1] - curve[0];
final Offset diffCurve21 = curve[2] - curve[1];
final Offset diffCurve32 = curve[3] - curve[2];
final double t01 = math.pow(diffCurve10.distance, alpha).toDouble();
final double t12 = math.pow(diffCurve21.distance, alpha).toDouble();
final double t23 = math.pow(diffCurve32.distance, alpha).toDouble();
final Offset m1 = (diffCurve21 + (diffCurve10 / t01 - (curve[2] - curve[0]) / (t01 + t12)) * t12) * reverseTension;
final Offset m2 = (diffCurve21 + (diffCurve32 / t23 - (curve[3] - curve[1]) / (t12 + t23)) * t12) * reverseTension;
final Offset sumM12 = m1 + m2;
final List<Offset> segment = <Offset>[
diffCurve21 * -2.0 + sumM12,
diffCurve21 * 3.0 - m1 - sumM12,
m1,
curve[1],
];
result.add(segment);
}
return result;
}
// The list of control point lists for each cubic segment of the spline.
final List<List<Offset>> _cubicSegments;
// This is non-empty only if the _cubicSegments are being computed lazily.
final List<Offset>? _controlPoints;
final Offset? _startHandle;
final Offset? _endHandle;
final double? _tension;
void _initializeIfNeeded() {
if (_cubicSegments.isNotEmpty) {
return;
}
_cubicSegments.addAll(
_computeSegments(_controlPoints!, _tension!, startHandle: _startHandle, endHandle: _endHandle),
);
}
@override
@protected
int get samplingSeed {
_initializeIfNeeded();
final Offset seedPoint = _cubicSegments[0][1];
return ((seedPoint.dx + seedPoint.dy) * 10000).round();
}
@override
Offset transformInternal(double t) {
_initializeIfNeeded();
final double length = _cubicSegments.length.toDouble();
final double position;
final double localT;
final int index;
if (t < 1.0) {
position = t * length;
localT = position % 1.0;
index = position.floor();
} else {
position = length;
localT = 1.0;
index = _cubicSegments.length - 1;
}
final List<Offset> cubicControlPoints = _cubicSegments[index];
final double localT2 = localT * localT;
return cubicControlPoints[0] * localT2 * localT
+ cubicControlPoints[1] * localT2
+ cubicControlPoints[2] * localT
+ cubicControlPoints[3];
}
}
/// An animation easing curve that passes smoothly through the given control
/// points using a centripetal Catmull-Rom spline.
///
/// When this curve is evaluated with [transform], the values will interpolate
/// smoothly from one control point to the next, passing through (0.0, 0.0), the
/// given points, and then (1.0, 1.0).
///
/// {@macro flutter.animation.CatmullRomSpline}
///
/// This class uses a centripetal Catmull-Rom curve (a [CatmullRomSpline]) as
/// its internal representation. The term centripetal implies that it won't form
/// loops or self-intersections within a single segment, and corresponds to a
/// Catmull-Rom α (alpha) value of 0.5.
///
/// See also:
///
/// * [CatmullRomSpline], the 2D spline that this curve uses to generate its values.
/// * A Wikipedia article on [centripetal Catmull-Rom splines](https://en.wikipedia.org/wiki/Centripetal_Catmull%E2%80%93Rom_spline).
/// * [new CatmullRomCurve] for a description of the constraints put on the
/// input control points.
/// * This [paper on using Catmull-Rom splines](http://faculty.cs.tamu.edu/schaefer/research/cr_cad.pdf).
class CatmullRomCurve extends Curve {
/// Constructs a centripetal [CatmullRomCurve].
///
/// It takes a list of two or more points that describe the points that the
/// curve must pass through. See [controlPoints] for a description of the
/// restrictions placed on control points. In addition to the given
/// [controlPoints], the curve will begin with an implicit control point at
/// (0.0, 0.0) and end with an implicit control point at (1.0, 1.0), so that
/// the curve begins and ends at those points.
///
/// The optional [tension] argument controls how tightly the curve approaches
/// the given `controlPoints`. It must be in the range 0.0 to 1.0, inclusive. It
/// defaults to 0.0, which provides the smoothest curve. A value of 1.0
/// is equivalent to a linear interpolation between points.
///
/// The internal curve data structures are lazily computed the first time
/// [transform] is called. If you would rather pre-compute the curve, use
/// [CatmullRomCurve.precompute] instead.
///
/// All of the arguments must not be null.
///
/// See also:
///
/// * This [paper on using Catmull-Rom splines](http://faculty.cs.tamu.edu/schaefer/research/cr_cad.pdf).
CatmullRomCurve(this.controlPoints, {this.tension = 0.0})
: assert(tension != null),
assert(() {
return validateControlPoints(
controlPoints,
tension: tension,
reasons: _debugAssertReasons..clear(),
);
}(), 'control points $controlPoints could not be validated:\n ${_debugAssertReasons.join('\n ')}'),
// Pre-compute samples so that we don't have to evaluate the spline's inverse
// all the time in transformInternal.
_precomputedSamples = <Curve2DSample>[];
/// Constructs a centripetal [CatmullRomCurve].
///
/// Same as [new CatmullRomCurve], but it precomputes the internal curve data
/// structures for a more predictable computation load.
CatmullRomCurve.precompute(this.controlPoints, {this.tension = 0.0})
: assert(tension != null),
assert(() {
return validateControlPoints(
controlPoints,
tension: tension,
reasons: _debugAssertReasons..clear(),
);
}(), 'control points $controlPoints could not be validated:\n ${_debugAssertReasons.join('\n ')}'),
// Pre-compute samples so that we don't have to evaluate the spline's inverse
// all the time in transformInternal.
_precomputedSamples = _computeSamples(controlPoints, tension);
static List<Curve2DSample> _computeSamples(List<Offset> controlPoints, double tension) {
return CatmullRomSpline.precompute(
// Force the first and last control points for the spline to be (0, 0)
// and (1, 1), respectively.
<Offset>[Offset.zero, ...controlPoints, const Offset(1.0, 1.0)],
tension: tension,
).generateSamples(tolerance: 1e-12).toList();
}
/// A static accumulator for assertion failures. Not used in release mode.
static final List<String> _debugAssertReasons = <String>[];
// The precomputed approximation curve, so that evaluation of the curve is
// efficient.
//
// If the curve is constructed lazily, then this will be empty, and will be filled
// the first time transform is called.
final List<Curve2DSample> _precomputedSamples;
/// The control points used to create this curve.
///
/// The `dx` value of each [Offset] in [controlPoints] represents the
/// animation value at which the curve should pass through the `dy` value of
/// the same control point.
///
/// The [controlPoints] list must meet the following criteria:
///
/// * The list must contain at least two points.
/// * The X value of each point must be greater than 0.0 and less then 1.0.
/// * The X values of each point must be greater than the
/// previous point's X value (i.e. monotonically increasing). The Y values
/// are not constrained.
/// * The resulting spline must be single-valued in X. That is, for each X
/// value, there must be exactly one Y value. This means that the control
/// points must not generated a spline that loops or overlaps itself.
///
/// The static function [validateControlPoints] can be used to check that
/// these conditions are met, and will return true if they are. In debug mode,
/// it will also optionally return a list of reasons in text form. In debug
/// mode, the constructor will assert that these conditions are met and print
/// the reasons if the assert fires.
///
/// When the curve is evaluated with [transform], the values will interpolate
/// smoothly from one control point to the next, passing through (0.0, 0.0), the
/// given control points, and (1.0, 1.0).
final List<Offset> controlPoints;
/// The "tension" of the curve.
///
/// The `tension` attribute controls how tightly the curve approaches the
/// given [controlPoints]. It must be in the range 0.0 to 1.0, inclusive. It
/// is optional, and defaults to 0.0, which provides the smoothest curve. A
/// value of 1.0 is equivalent to a linear interpolation between control
/// points.
final double tension;
/// Validates that a given set of control points for a [CatmullRomCurve] is
/// well-formed and will not produce a spline that self-intersects.
///
/// This method is also used in debug mode to validate a curve to make sure
/// that it won't violate the contract for the [new CatmullRomCurve]
/// constructor.
///
/// If in debug mode, and `reasons` is non-null, this function will fill in
/// `reasons` with descriptions of the problems encountered. The `reasons`
/// argument is ignored in release mode.
///
/// In release mode, this function can be used to decide if a proposed
/// modification to the curve will result in a valid curve.
static bool validateControlPoints(
List<Offset>? controlPoints, {
double tension = 0.0,
List<String>? reasons,
}) {
assert(tension != null);
if (controlPoints == null) {
assert(() {
reasons?.add('Supplied control points cannot be null');
return true;
}());
return false;
}
if (controlPoints.length < 2) {
assert(() {
reasons?.add('There must be at least two points supplied to create a valid curve.');
return true;
}());
return false;
}
controlPoints = <Offset>[Offset.zero, ...controlPoints, const Offset(1.0, 1.0)];
final Offset startHandle = controlPoints[0] * 2.0 - controlPoints[1];
final Offset endHandle = controlPoints.last * 2.0 - controlPoints[controlPoints.length - 2];
controlPoints = <Offset>[startHandle, ...controlPoints, endHandle];
double lastX = -double.infinity;
for (int i = 0; i < controlPoints.length; ++i) {
if (i > 1 &&
i < controlPoints.length - 2 &&
(controlPoints[i].dx <= 0.0 || controlPoints[i].dx >= 1.0)) {
assert(() {
reasons?.add(
'Control points must have X values between 0.0 and 1.0, exclusive. '
'Point $i has an x value (${controlPoints![i].dx}) which is outside the range.',
);
return true;
}());
return false;
}
if (controlPoints[i].dx <= lastX) {
assert(() {
reasons?.add(
'Each X coordinate must be greater than the preceding X coordinate '
'(i.e. must be monotonically increasing in X). Point $i has an x value of '
'${controlPoints![i].dx}, which is not greater than $lastX',
);
return true;
}());
return false;
}
lastX = controlPoints[i].dx;
}
bool success = true;
// An empirical test to make sure things are single-valued in X.
lastX = -double.infinity;
const double tolerance = 1e-3;
final CatmullRomSpline testSpline = CatmullRomSpline(controlPoints, tension: tension);
final double start = testSpline.findInverse(0.0);
final double end = testSpline.findInverse(1.0);
final Iterable<Curve2DSample> samplePoints = testSpline.generateSamples(start: start, end: end);
/// If the first and last points in the samples aren't at (0,0) or (1,1)
/// respectively, then the curve is multi-valued at the ends.
if (samplePoints.first.value.dy.abs() > tolerance || (1.0 - samplePoints.last.value.dy).abs() > tolerance) {
bool bail = true;
success = false;
assert(() {
reasons?.add(
'The curve has more than one Y value at X = ${samplePoints.first.value.dx}. '
'Try moving some control points further away from this value of X, or increasing '
'the tension.',
);
// No need to keep going if we're not giving reasons.
bail = reasons == null;
return true;
}());
if (bail) {
// If we're not in debug mode, then we want to bail immediately
// instead of checking everything else.
return false;
}
}
for (final Curve2DSample sample in samplePoints) {
final Offset point = sample.value;
final double t = sample.t;
final double x = point.dx;
if (t >= start && t <= end && (x < -1e-3 || x > 1.0 + 1e-3)) {
bool bail = true;
success = false;
assert(() {
reasons?.add(
'The resulting curve has an X value ($x) which is outside '
'the range [0.0, 1.0], inclusive.',
);
// No need to keep going if we're not giving reasons.
bail = reasons == null;
return true;
}());
if (bail) {
// If we're not in debug mode, then we want to bail immediately
// instead of checking all the segments.
return false;
}
}
if (x < lastX) {
bool bail = true;
success = false;
assert(() {
reasons?.add(
'The curve has more than one Y value at x = $x. Try moving '
'some control points further apart in X, or increasing the tension.',
);
// No need to keep going if we're not giving reasons.
bail = reasons == null;
return true;
}());
if (bail) {
// If we're not in debug mode, then we want to bail immediately
// instead of checking all the segments.
return false;
}
}
lastX = x;
}
return success;
}
@override
double transformInternal(double t) {
// Linearly interpolate between the two closest samples generated when the
// curve was created.
if (_precomputedSamples.isEmpty) {
// Compute the samples now if we were constructed lazily.
_precomputedSamples.addAll(_computeSamples(controlPoints, tension));
}
int start = 0;
int end = _precomputedSamples.length - 1;
int mid;
Offset value;
Offset startValue = _precomputedSamples[start].value;
Offset endValue = _precomputedSamples[end].value;
// Use a binary search to find the index of the sample point that is just
// before t.
while (end - start > 1) {
mid = (end + start) ~/ 2;
value = _precomputedSamples[mid].value;
if (t >= value.dx) {
start = mid;
startValue = value;
} else {
end = mid;
endValue = value;
}
}
// Now interpolate between the found sample and the next one.
final double t2 = (t - startValue.dx) / (endValue.dx - startValue.dx);
return lerpDouble(startValue.dy, endValue.dy, t2)!;
}
}
/// A curve that is the reversed inversion of its given curve.
///
/// This curve evaluates the given curve in reverse (i.e., from 1.0 to 0.0 as t
/// increases from 0.0 to 1.0) and returns the inverse of the given curve's
/// value (i.e., 1.0 minus the given curve's value).
///
/// This is the class used to implement the [flipped] getter on curves.
///
/// This is often useful with [CurvedAnimation.reverseCurve].
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_bounce_in.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_flipped.mp4}
///
/// See also:
///
/// * [Curve.flipped], which provides the [FlippedCurve] of a [Curve].
/// * [ReverseAnimation], which reverses an [Animation] rather than a [Curve].
/// * [CurvedAnimation], which can take a separate curve and reverse curve.
class FlippedCurve extends Curve {
/// Creates a flipped curve.
///
/// The [curve] argument must not be null.
const FlippedCurve(this.curve) : assert(curve != null);
/// The curve that is being flipped.
final Curve curve;
@override
double transformInternal(double t) => 1.0 - curve.transform(1.0 - t);
@override
String toString() {
return '${objectRuntimeType(this, 'FlippedCurve')}($curve)';
}
}
/// A curve where the rate of change starts out quickly and then decelerates; an
/// upside-down `f(t) = t²` parabola.
///
/// This is equivalent to the Android `DecelerateInterpolator` class with a unit
/// factor (the default factor).
///
/// See [Curves.decelerate] for an instance of this class.
class _DecelerateCurve extends Curve {
const _DecelerateCurve._();
@override
double transformInternal(double t) {
// Intended to match the behavior of:
// https://android.googlesource.com/platform/frameworks/base/+/master/core/java/android/view/animation/DecelerateInterpolator.java
// ...as of December 2016.
t = 1.0 - t;
return 1.0 - t * t;
}
}
// BOUNCE CURVES
double _bounce(double t) {
if (t < 1.0 / 2.75) {
return 7.5625 * t * t;
} else if (t < 2 / 2.75) {
t -= 1.5 / 2.75;
return 7.5625 * t * t + 0.75;
} else if (t < 2.5 / 2.75) {
t -= 2.25 / 2.75;
return 7.5625 * t * t + 0.9375;
}
t -= 2.625 / 2.75;
return 7.5625 * t * t + 0.984375;
}
/// An oscillating curve that grows in magnitude.
///
/// See [Curves.bounceIn] for an instance of this class.
class _BounceInCurve extends Curve {
const _BounceInCurve._();
@override
double transformInternal(double t) {
return 1.0 - _bounce(1.0 - t);
}
}
/// An oscillating curve that shrink in magnitude.
///
/// See [Curves.bounceOut] for an instance of this class.
class _BounceOutCurve extends Curve {
const _BounceOutCurve._();
@override
double transformInternal(double t) {
return _bounce(t);
}
}
/// An oscillating curve that first grows and then shrink in magnitude.
///
/// See [Curves.bounceInOut] for an instance of this class.
class _BounceInOutCurve extends Curve {
const _BounceInOutCurve._();
@override
double transformInternal(double t) {
if (t < 0.5)
return (1.0 - _bounce(1.0 - t * 2.0)) * 0.5;
else
return _bounce(t * 2.0 - 1.0) * 0.5 + 0.5;
}
}
// ELASTIC CURVES
/// An oscillating curve that grows in magnitude while overshooting its bounds.
///
/// An instance of this class using the default period of 0.4 is available as
/// [Curves.elasticIn].
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_elastic_in.mp4}
class ElasticInCurve extends Curve {
/// Creates an elastic-in curve.
///
/// Rather than creating a new instance, consider using [Curves.elasticIn].
const ElasticInCurve([this.period = 0.4]);
/// The duration of the oscillation.
final double period;
@override
double transformInternal(double t) {
final double s = period / 4.0;
t = t - 1.0;
return -math.pow(2.0, 10.0 * t) * math.sin((t - s) * (math.pi * 2.0) / period);
}
@override
String toString() {
return '${objectRuntimeType(this, 'ElasticInCurve')}($period)';
}
}
/// An oscillating curve that shrinks in magnitude while overshooting its bounds.
///
/// An instance of this class using the default period of 0.4 is available as
/// [Curves.elasticOut].
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_elastic_out.mp4}
class ElasticOutCurve extends Curve {
/// Creates an elastic-out curve.
///
/// Rather than creating a new instance, consider using [Curves.elasticOut].
const ElasticOutCurve([this.period = 0.4]);
/// The duration of the oscillation.
final double period;
@override
double transformInternal(double t) {
final double s = period / 4.0;
return math.pow(2.0, -10 * t) * math.sin((t - s) * (math.pi * 2.0) / period) + 1.0;
}
@override
String toString() {
return '${objectRuntimeType(this, 'ElasticOutCurve')}($period)';
}
}
/// An oscillating curve that grows and then shrinks in magnitude while
/// overshooting its bounds.
///
/// An instance of this class using the default period of 0.4 is available as
/// [Curves.elasticInOut].
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_elastic_in_out.mp4}
class ElasticInOutCurve extends Curve {
/// Creates an elastic-in-out curve.
///
/// Rather than creating a new instance, consider using [Curves.elasticInOut].
const ElasticInOutCurve([this.period = 0.4]);
/// The duration of the oscillation.
final double period;
@override
double transformInternal(double t) {
final double s = period / 4.0;
t = 2.0 * t - 1.0;
if (t < 0.0)
return -0.5 * math.pow(2.0, 10.0 * t) * math.sin((t - s) * (math.pi * 2.0) / period);
else
return math.pow(2.0, -10.0 * t) * math.sin((t - s) * (math.pi * 2.0) / period) * 0.5 + 1.0;
}
@override
String toString() {
return '${objectRuntimeType(this, 'ElasticInOutCurve')}($period)';
}
}
// PREDEFINED CURVES
/// A collection of common animation curves.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_bounce_in.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_bounce_in_out.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_bounce_out.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_decelerate.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_sine.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_quad.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_cubic.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_quart.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_quint.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_expo.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_circ.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_back.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out_sine.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out_quad.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out_cubic.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out_quart.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out_quint.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out_expo.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out_circ.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out_back.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_out.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_out_sine.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_out_quad.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_out_cubic.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_out_quart.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_out_quint.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_out_expo.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_out_circ.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_out_back.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_elastic_in.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_elastic_in_out.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_elastic_out.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_fast_out_slow_in.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_slow_middle.mp4}
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_linear.mp4}
///
/// See also:
///
/// * [Curve], the interface implemented by the constants available from the
/// [Curves] class.
class Curves {
// This class is not meant to be instantiated or extended; this constructor
// prevents instantiation and extension.
Curves._();
/// A linear animation curve.
///
/// This is the identity map over the unit interval: its [Curve.transform]
/// method returns its input unmodified. This is useful as a default curve for
/// cases where a [Curve] is required but no actual curve is desired.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_linear.mp4}
static const Curve linear = _Linear._();
/// A curve where the rate of change starts out quickly and then decelerates; an
/// upside-down `f(t) = t²` parabola.
///
/// This is equivalent to the Android `DecelerateInterpolator` class with a unit
/// factor (the default factor).
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_decelerate.mp4}
static const Curve decelerate = _DecelerateCurve._();
/// A curve that is very steep and linear at the beginning, but quickly flattens out
/// and very slowly eases in.
///
/// By default is the curve used to animate pages on iOS back to their original
/// position if a swipe gesture is ended midway through a swipe.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_fast_linear_to_slow_ease_in.mp4}
static const Cubic fastLinearToSlowEaseIn = Cubic(0.18, 1.0, 0.04, 1.0);
/// A cubic animation curve that speeds up quickly and ends slowly.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease.mp4}
static const Cubic ease = Cubic(0.25, 0.1, 0.25, 1.0);
/// A cubic animation curve that starts slowly and ends quickly.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in.mp4}
static const Cubic easeIn = Cubic(0.42, 0.0, 1.0, 1.0);
/// A cubic animation curve that starts slowly and ends linearly.
///
/// The symmetric animation to [linearToEaseOut].
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_to_linear.mp4}
static const Cubic easeInToLinear = Cubic(0.67, 0.03, 0.65, 0.09);
/// A cubic animation curve that starts slowly and ends quickly. This is
/// similar to [Curves.easeIn], but with sinusoidal easing for a slightly less
/// abrupt beginning and end. Nonetheless, the result is quite gentle and is
/// hard to distinguish from [Curves.linear] at a glance.
///
/// Derived from Robert Penner’s easing functions.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_sine.mp4}
static const Cubic easeInSine = Cubic(0.47, 0.0, 0.745, 0.715);
/// A cubic animation curve that starts slowly and ends quickly. Based on a
/// quadratic equation where `f(t) = t²`, this is effectively the inverse of
/// [Curves.decelerate].
///
/// Compared to [Curves.easeInSine], this curve is slightly steeper.
///
/// Derived from Robert Penner’s easing functions.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_quad.mp4}
static const Cubic easeInQuad = Cubic(0.55, 0.085, 0.68, 0.53);
/// A cubic animation curve that starts slowly and ends quickly. This curve is
/// based on a cubic equation where `f(t) = t³`. The result is a safe sweet
/// spot when choosing a curve for widgets animating off the viewport.
///
/// Compared to [Curves.easeInQuad], this curve is slightly steeper.
///
/// Derived from Robert Penner’s easing functions.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_cubic.mp4}
static const Cubic easeInCubic = Cubic(0.55, 0.055, 0.675, 0.19);
/// A cubic animation curve that starts slowly and ends quickly. This curve is
/// based on a quartic equation where `f(t) = t⁴`.
///
/// Animations using this curve or steeper curves will benefit from a longer
/// duration to avoid motion feeling unnatural.
///
/// Compared to [Curves.easeInCubic], this curve is slightly steeper.
///
/// Derived from Robert Penner’s easing functions.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_quart.mp4}
static const Cubic easeInQuart = Cubic(0.895, 0.03, 0.685, 0.22);
/// A cubic animation curve that starts slowly and ends quickly. This curve is
/// based on a quintic equation where `f(t) = t⁵`.
///
/// Compared to [Curves.easeInQuart], this curve is slightly steeper.
///
/// Derived from Robert Penner’s easing functions.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_quint.mp4}
static const Cubic easeInQuint = Cubic(0.755, 0.05, 0.855, 0.06);
/// A cubic animation curve that starts slowly and ends quickly. This curve is
/// based on an exponential equation where `f(t) = 2¹⁰⁽ᵗ⁻¹⁾`.
///
/// Using this curve can give your animations extra flare, but a longer
/// duration may need to be used to compensate for the steepness of the curve.
///
/// Compared to [Curves.easeInQuint], this curve is slightly steeper.
///
/// Derived from Robert Penner’s easing functions.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_expo.mp4}
static const Cubic easeInExpo = Cubic(0.95, 0.05, 0.795, 0.035);
/// A cubic animation curve that starts slowly and ends quickly. This curve is
/// effectively the bottom-right quarter of a circle.
///
/// Like [Curves.easeInExpo], this curve is fairly dramatic and will reduce
/// the clarity of an animation if not given a longer duration.
///
/// Derived from Robert Penner’s easing functions.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_circ.mp4}
static const Cubic easeInCirc = Cubic(0.6, 0.04, 0.98, 0.335);
/// A cubic animation curve that starts slowly and ends quickly. This curve
/// is similar to [Curves.elasticIn] in that it overshoots its bounds before
/// reaching its end. Instead of repeated swinging motions before ascending,
/// though, this curve overshoots once, then continues to ascend.
///
/// Derived from Robert Penner’s easing functions.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_back.mp4}
static const Cubic easeInBack = Cubic(0.6, -0.28, 0.735, 0.045);
/// A cubic animation curve that starts quickly and ends slowly.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_out.mp4}
static const Cubic easeOut = Cubic(0.0, 0.0, 0.58, 1.0);
/// A cubic animation curve that starts linearly and ends slowly.
///
/// A symmetric animation to [easeInToLinear].
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_linear_to_ease_out.mp4}
static const Cubic linearToEaseOut = Cubic(0.35, 0.91, 0.33, 0.97);
/// A cubic animation curve that starts quickly and ends slowly. This is
/// similar to [Curves.easeOut], but with sinusoidal easing for a slightly
/// less abrupt beginning and end. Nonetheless, the result is quite gentle and
/// is hard to distinguish from [Curves.linear] at a glance.
///
/// Derived from Robert Penner’s easing functions.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_out_sine.mp4}
static const Cubic easeOutSine = Cubic(0.39, 0.575, 0.565, 1.0);
/// A cubic animation curve that starts quickly and ends slowly. This is
/// effectively the same as [Curves.decelerate], only simulated using a cubic
/// bezier function.
///
/// Compared to [Curves.easeOutSine], this curve is slightly steeper.
///
/// Derived from Robert Penner’s easing functions.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_out_quad.mp4}
static const Cubic easeOutQuad = Cubic(0.25, 0.46, 0.45, 0.94);
/// A cubic animation curve that starts quickly and ends slowly. This curve is
/// a flipped version of [Curves.easeInCubic].
///
/// The result is a safe sweet spot when choosing a curve for animating a
/// widget's position entering or already inside the viewport.
///
/// Compared to [Curves.easeOutQuad], this curve is slightly steeper.
///
/// Derived from Robert Penner’s easing functions.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_out_cubic.mp4}
static const Cubic easeOutCubic = Cubic(0.215, 0.61, 0.355, 1.0);
/// A cubic animation curve that starts quickly and ends slowly. This curve is
/// a flipped version of [Curves.easeInQuart].
///
/// Animations using this curve or steeper curves will benefit from a longer
/// duration to avoid motion feeling unnatural.
///
/// Compared to [Curves.easeOutCubic], this curve is slightly steeper.
///
/// Derived from Robert Penner’s easing functions.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_out_quart.mp4}
static const Cubic easeOutQuart = Cubic(0.165, 0.84, 0.44, 1.0);
/// A cubic animation curve that starts quickly and ends slowly. This curve is
/// a flipped version of [Curves.easeInQuint].
///
/// Compared to [Curves.easeOutQuart], this curve is slightly steeper.
///
/// Derived from Robert Penner’s easing functions.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_out_quint.mp4}
static const Cubic easeOutQuint = Cubic(0.23, 1.0, 0.32, 1.0);
/// A cubic animation curve that starts quickly and ends slowly. This curve is
/// a flipped version of [Curves.easeInExpo]. Using this curve can give your
/// animations extra flare, but a longer duration may need to be used to
/// compensate for the steepness of the curve.
///
/// Derived from Robert Penner’s easing functions.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_out_expo.mp4}
static const Cubic easeOutExpo = Cubic(0.19, 1.0, 0.22, 1.0);
/// A cubic animation curve that starts quickly and ends slowly. This curve is
/// effectively the top-left quarter of a circle.
///
/// Like [Curves.easeOutExpo], this curve is fairly dramatic and will reduce
/// the clarity of an animation if not given a longer duration.
///
/// Derived from Robert Penner’s easing functions.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_out_circ.mp4}
static const Cubic easeOutCirc = Cubic(0.075, 0.82, 0.165, 1.0);
/// A cubic animation curve that starts quickly and ends slowly. This curve is
/// similar to [Curves.elasticOut] in that it overshoots its bounds before
/// reaching its end. Instead of repeated swinging motions after ascending,
/// though, this curve only overshoots once.
///
/// Derived from Robert Penner’s easing functions.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_out_back.mp4}
static const Cubic easeOutBack = Cubic(0.175, 0.885, 0.32, 1.275);
/// A cubic animation curve that starts slowly, speeds up, and then ends
/// slowly.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out.mp4}
static const Cubic easeInOut = Cubic(0.42, 0.0, 0.58, 1.0);
/// A cubic animation curve that starts slowly, speeds up, and then ends
/// slowly. This is similar to [Curves.easeInOut], but with sinusoidal easing
/// for a slightly less abrupt beginning and end.
///
/// Derived from Robert Penner’s easing functions.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out_sine.mp4}
static const Cubic easeInOutSine = Cubic(0.445, 0.05, 0.55, 0.95);
/// A cubic animation curve that starts slowly, speeds up, and then ends
/// slowly. This curve can be imagined as [Curves.easeInQuad] as the first
/// half, and [Curves.easeOutQuad] as the second.
///
/// Compared to [Curves.easeInOutSine], this curve is slightly steeper.
///
/// Derived from Robert Penner’s easing functions.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out_quad.mp4}
static const Cubic easeInOutQuad = Cubic(0.455, 0.03, 0.515, 0.955);
/// A cubic animation curve that starts slowly, speeds up, and then ends
/// slowly. This curve can be imagined as [Curves.easeInCubic] as the first
/// half, and [Curves.easeOutCubic] as the second.
///
/// The result is a safe sweet spot when choosing a curve for a widget whose
/// initial and final positions are both within the viewport.
///
/// Compared to [Curves.easeInOutQuad], this curve is slightly steeper.
///
/// Derived from Robert Penner’s easing functions.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out_cubic.mp4}
static const Cubic easeInOutCubic = Cubic(0.645, 0.045, 0.355, 1.0);
/// A cubic animation curve that starts slowly, speeds up shortly thereafter,
/// and then ends slowly. This curve can be imagined as a steeper version of
/// [easeInOutCubic].
///
/// The result is a more emphasized eased curve when choosing a curve for a
/// widget whose initial and final positions are both within the viewport.
///
/// Compared to [Curves.easeInOutCubic], this curve is slightly steeper.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out_cubic_emphasized.mp4}
static const ThreePointCubic easeInOutCubicEmphasized = ThreePointCubic(
Offset(0.05, 0), Offset(0.133333, 0.06),
Offset(0.166666, 0.4),
Offset(0.208333, 0.82), Offset(0.25, 1),
);
/// A cubic animation curve that starts slowly, speeds up, and then ends
/// slowly. This curve can be imagined as [Curves.easeInQuart] as the first
/// half, and [Curves.easeOutQuart] as the second.
///
/// Animations using this curve or steeper curves will benefit from a longer
/// duration to avoid motion feeling unnatural.
///
/// Compared to [Curves.easeInOutCubic], this curve is slightly steeper.
///
/// Derived from Robert Penner’s easing functions.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out_quart.mp4}
static const Cubic easeInOutQuart = Cubic(0.77, 0.0, 0.175, 1.0);
/// A cubic animation curve that starts slowly, speeds up, and then ends
/// slowly. This curve can be imagined as [Curves.easeInQuint] as the first
/// half, and [Curves.easeOutQuint] as the second.
///
/// Compared to [Curves.easeInOutQuart], this curve is slightly steeper.
///
/// Derived from Robert Penner’s easing functions.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out_quint.mp4}
static const Cubic easeInOutQuint = Cubic(0.86, 0.0, 0.07, 1.0);
/// A cubic animation curve that starts slowly, speeds up, and then ends
/// slowly.
///
/// Since this curve is arrived at with an exponential function, the midpoint
/// is exceptionally steep. Extra consideration should be taken when designing
/// an animation using this.
///
/// Compared to [Curves.easeInOutQuint], this curve is slightly steeper.
///
/// Derived from Robert Penner’s easing functions.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out_expo.mp4}
static const Cubic easeInOutExpo = Cubic(1.0, 0.0, 0.0, 1.0);
/// A cubic animation curve that starts slowly, speeds up, and then ends
/// slowly. This curve can be imagined as [Curves.easeInCirc] as the first
/// half, and [Curves.easeOutCirc] as the second.
///
/// Like [Curves.easeInOutExpo], this curve is fairly dramatic and will reduce
/// the clarity of an animation if not given a longer duration.
///
/// Compared to [Curves.easeInOutExpo], this curve is slightly steeper.
///
/// Derived from Robert Penner’s easing functions.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out_circ.mp4}
static const Cubic easeInOutCirc = Cubic(0.785, 0.135, 0.15, 0.86);
/// A cubic animation curve that starts slowly, speeds up, and then ends
/// slowly. This curve can be imagined as [Curves.easeInBack] as the first
/// half, and [Curves.easeOutBack] as the second.
///
/// Since two curves are used as a basis for this curve, the resulting
/// animation will overshoot its bounds twice before reaching its end - first
/// by exceeding its lower bound, then exceeding its upper bound and finally
/// descending to its final position.
///
/// Derived from Robert Penner’s easing functions.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_ease_in_out_back.mp4}
static const Cubic easeInOutBack = Cubic(0.68, -0.55, 0.265, 1.55);
/// A curve that starts quickly and eases into its final position.
///
/// Over the course of the animation, the object spends more time near its
/// final destination. As a result, the user isn’t left waiting for the
/// animation to finish, and the negative effects of motion are minimized.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_fast_out_slow_in.mp4}
///
/// See also:
///
/// * [standardEasing], the name for this curve in the Material specification.
static const Cubic fastOutSlowIn = Cubic(0.4, 0.0, 0.2, 1.0);
/// A cubic animation curve that starts quickly, slows down, and then ends
/// quickly.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_slow_middle.mp4}
static const Cubic slowMiddle = Cubic(0.15, 0.85, 0.85, 0.15);
/// An oscillating curve that grows in magnitude.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_bounce_in.mp4}
static const Curve bounceIn = _BounceInCurve._();
/// An oscillating curve that first grows and then shrink in magnitude.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_bounce_out.mp4}
static const Curve bounceOut = _BounceOutCurve._();
/// An oscillating curve that first grows and then shrink in magnitude.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_bounce_in_out.mp4}
static const Curve bounceInOut = _BounceInOutCurve._();
/// An oscillating curve that grows in magnitude while overshooting its bounds.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_elastic_in.mp4}
static const ElasticInCurve elasticIn = ElasticInCurve();
/// An oscillating curve that shrinks in magnitude while overshooting its bounds.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_elastic_out.mp4}
static const ElasticOutCurve elasticOut = ElasticOutCurve();
/// An oscillating curve that grows and then shrinks in magnitude while overshooting its bounds.
///
/// {@animation 464 192 https://flutter.github.io/assets-for-api-docs/assets/animation/curve_elastic_in_out.mp4}
static const ElasticInOutCurve elasticInOut = ElasticInOutCurve();
}