spring_simulation.dart

// Copyright 2016 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.

import 'dart:math' as math;

import 'simulation.dart';
import 'tolerance.dart';
import 'utils.dart';

/// Structure that describes a spring's constants.
///
/// Used to configure a [SpringSimulation].
class SpringDescription {
  /// Creates a spring given the mass, stiffness, and the damping coefficient.
  ///
  /// See [mass], [stiffness], and [damping] for the units of the arguments.
  const SpringDescription({
    this.mass,
    this.stiffness,
    this.damping,
  });

  /// Creates a spring given the mass (m), stiffness (k), and damping ratio (ζ).
  /// The damping ratio is especially useful trying to determining the type of
  /// spring to create. A ratio of 1.0 creates a critically damped spring, > 1.0
  /// creates an overdamped spring and < 1.0 an underdamped one.
  ///
  /// See [mass] and [stiffness] for the units for those arguments. The damping
  /// ratio is unitless.
  SpringDescription.withDampingRatio({
    this.mass,
    this.stiffness,
    double ratio = 1.0,
  }) : damping = ratio * 2.0 * math.sqrt(mass * stiffness);

  /// The mass of the spring (m). The units are arbitrary, but all springs
  /// within a system should use the same mass units.
  final double mass;

  /// The spring constant (k). The units of stiffness are M/T², where M is the
  /// mass unit used for the value of the [mass] property, and T is the time
  /// unit used for driving the [SpringSimulation].
  final double stiffness;

  /// The damping coefficient (c).
  ///
  /// Do not confuse the damping _coefficient_ (c) with the damping _ratio_ (ζ).
  /// To create a [SpringDescription] with a damping ratio, use the [new
  /// SpringDescription.withDampingRatio] constructor.
  ///
  /// The units of the damping coefficient are M/T, where M is the mass unit
  /// used for the value of the [mass] property, and T is the time unit used for
  /// driving the [SpringSimulation].
  final double damping;

  @override
  String toString() => '$runtimeType(mass: ${mass.toStringAsFixed(1)}, stiffness: ${stiffness.toStringAsFixed(1)}, damping: ${damping.toStringAsFixed(1)})';
}

/// The kind of spring solution that the [SpringSimulation] is using to simulate the spring.
///
/// See [SpringSimulation.type].
enum SpringType {
  /// A spring that does not bounce and returns to its rest position in the
  /// shortest possible time.
  criticallyDamped,

  /// A spring that bounces.
  underDamped,

  /// A spring that does not bounce but takes longer to return to its rest
  /// position than a [criticallyDamped] one.
  overDamped,
}

/// A spring simulation.
///
/// Models a particle attached to a spring that follows Hooke's law.
class SpringSimulation extends Simulation {
  /// Creates a spring simulation from the provided spring description, start
  /// distance, end distance, and initial velocity.
  ///
  /// The units for the start and end distance arguments are arbitrary, but must
  /// be consistent with the units used for other lengths in the system.
  ///
  /// The units for the velocity are L/T, where L is the aforementioned
  /// arbitrary unit of length, and T is the time unit used for driving the
  /// [SpringSimulation].
  SpringSimulation(
    SpringDescription spring,
    double start,
    double end,
    double velocity, {
    Tolerance tolerance = Tolerance.defaultTolerance,
  }) : _endPosition = end,
       _solution = _SpringSolution(spring, start - end, velocity),
       super(tolerance: tolerance);

  final double _endPosition;
  final _SpringSolution _solution;

  /// The kind of spring being simulated, for debugging purposes.
  ///
  /// This is derived from the [SpringDescription] provided to the [new
  /// SpringSimulation] constructor.
  SpringType get type => _solution.type;

  @override
  double x(double time) => _endPosition + _solution.x(time);

  @override
  double dx(double time) => _solution.dx(time);

  @override
  bool isDone(double time) {
    return nearZero(_solution.x(time), tolerance.distance) &&
           nearZero(_solution.dx(time), tolerance.velocity);
  }

  @override
  String toString() => '$runtimeType(end: $_endPosition, $type)';
}

/// A SpringSimulation where the value of [x] is guaranteed to have exactly the
/// end value when the simulation isDone().
class ScrollSpringSimulation extends SpringSimulation {
  /// Creates a spring simulation from the provided spring description, start
  /// distance, end distance, and initial velocity.
  ///
  /// See the [new SpringSimulation] constructor on the superclass for a
  /// discussion of the arguments' units.
  ScrollSpringSimulation(
    SpringDescription spring,
    double start,
    double end,
    double velocity, {
    Tolerance tolerance = Tolerance.defaultTolerance,
  }) : super(spring, start, end, velocity, tolerance: tolerance);

  @override
  double x(double time) => isDone(time) ? _endPosition : super.x(time);
}


// SPRING IMPLEMENTATIONS

abstract class _SpringSolution {
  factory _SpringSolution(
    SpringDescription spring,
    double initialPosition,
    double initialVelocity,
  ) {
    assert(spring != null);
    assert(spring.mass != null);
    assert(spring.stiffness != null);
    assert(spring.damping != null);
    assert(initialPosition != null);
    assert(initialVelocity != null);
    final double cmk = spring.damping * spring.damping - 4 * spring.mass * spring.stiffness;
    if (cmk == 0.0)
      return _CriticalSolution(spring, initialPosition, initialVelocity);
    if (cmk > 0.0)
      return _OverdampedSolution(spring, initialPosition, initialVelocity);
    return _UnderdampedSolution(spring, initialPosition, initialVelocity);
  }

  double x(double time);
  double dx(double time);
  SpringType get type;
}

class _CriticalSolution implements _SpringSolution {
  factory _CriticalSolution(
    SpringDescription spring,
    double distance,
    double velocity,
  ) {
    final double r = -spring.damping / (2.0 * spring.mass);
    final double c1 = distance;
    final double c2 = velocity / (r * distance);
    return _CriticalSolution.withArgs(r, c1, c2);
  }

  _CriticalSolution.withArgs(double r, double c1, double c2)
    : _r = r,
      _c1 = c1,
      _c2 = c2;

  final double _r, _c1, _c2;

  @override
  double x(double time) {
    return (_c1 + _c2 * time) * math.pow(math.e, _r * time);
  }

  @override
  double dx(double time) {
    final double power = math.pow(math.e, _r * time);
    return _r * (_c1 + _c2 * time) * power + _c2 * power;
  }

  @override
  SpringType get type => SpringType.criticallyDamped;
}

class _OverdampedSolution implements _SpringSolution {
  factory _OverdampedSolution(
    SpringDescription spring,
    double distance,
    double velocity,
  ) {
    final double cmk = spring.damping * spring.damping - 4 * spring.mass * spring.stiffness;
    final double r1 = (-spring.damping - math.sqrt(cmk)) / (2.0 * spring.mass);
    final double r2 = (-spring.damping + math.sqrt(cmk)) / (2.0 * spring.mass);
    final double c2 = (velocity - r1 * distance) / (r2 - r1);
    final double c1 = distance - c2;
    return _OverdampedSolution.withArgs(r1, r2, c1, c2);
  }

  _OverdampedSolution.withArgs(double r1, double r2, double c1, double c2)
    : _r1 = r1,
      _r2 = r2,
      _c1 = c1,
      _c2 = c2;

  final double _r1, _r2, _c1, _c2;

  @override
  double x(double time) {
    return _c1 * math.pow(math.e, _r1 * time) +
           _c2 * math.pow(math.e, _r2 * time);
  }

  @override
  double dx(double time) {
    return _c1 * _r1 * math.pow(math.e, _r1 * time) +
           _c2 * _r2 * math.pow(math.e, _r2 * time);
  }

  @override
  SpringType get type => SpringType.overDamped;
}

class _UnderdampedSolution implements _SpringSolution {
  factory _UnderdampedSolution(
    SpringDescription spring,
    double distance,
    double velocity,
  ) {
    final double w = math.sqrt(4.0 * spring.mass * spring.stiffness -
                     spring.damping * spring.damping) / (2.0 * spring.mass);
    final double r = -(spring.damping / 2.0 * spring.mass);
    final double c1 = distance;
    final double c2 = (velocity - r * distance) / w;
    return _UnderdampedSolution.withArgs(w, r, c1, c2);
  }

  _UnderdampedSolution.withArgs(double w, double r, double c1, double c2)
    : _w = w,
      _r = r,
      _c1 = c1,
      _c2 = c2;

  final double _w, _r, _c1, _c2;

  @override
  double x(double time) {
    return math.pow(math.e, _r * time) *
           (_c1 * math.cos(_w * time) + _c2 * math.sin(_w * time));
  }

  @override
  double dx(double time) {
    final double power = math.pow(math.e, _r * time);
    final double cosine = math.cos(_w * time);
    final double sine = math.sin(_w * time);
    return      power * (_c2 * _w * cosine - _c1 * _w * sine) +
           _r * power * (_c2 *      sine   + _c1 *      cosine);
  }

  @override
  SpringType get type => SpringType.underDamped;
}