# Class BrachistoSim

The Brachistochrone is a classic problem about finding the path between two points where a ball will travel fastest (without friction). From greek: brachistos (shortest) and chronos (time).

The equations of motion are the same as those of RollerSingleSim. For derivation equations of motion see https://www.myphysicslab.com/RollerSimple.html and https://www.myphysicslab.com/RollerSpring.html.

## Variables Array

The variables are stored in the VarsList as follows

vars[0] time
vars[1] position ball 0
vars[2] velocity ball 0
vars[3] x-position ball 0 (derived from vars[1] and path)
vars[4] y-position ball 0 (derived from vars[1] and path)
vars[5] position ball 1
vars[6] velocity ball 1
vars[7] x-position ball 1 (derived from vars[5] and path)
vars[8] y-position ball 1 (derived from vars[5] and path)
etc.

TO DO Graph is not very useful now. More useful would be to show each ball with a different color, and show y position vs. time. Perhaps name variables after the associated path. Or number the various paths and draw numbers on the balls.

TO DO Puzzle: if you turn off the repeat time and just let it run, the circle ball is ahead of the brachistochrone. Because it is a shorter path? But all the balls reach the same height at the same moment on the first repetition. After that they are no longer in sync. Why?

TO DO Don't put the x-position and y-position in the vars[] array; instead calculate these in getVariable(). See RollerFlightSim for an example.

TO DO could make an array of PathPoint, to avoid creating them so often.

## Constructors

• #### Parameters

• ##### paths: ParametricPath[]

the set of paths to show

• ##### Optionalopt_name: string

name of this as a Subject

• ##### Optionalopt_simList: SimList

optional SimList where SimObjects should be stored

## Properties

choice_: number = -1

Which path was chosen by user, or -1 if no path chosen.

initialState_: null | number[] = null

Initial values.

simObjects_: SimObject[] = []

keep track of our SimObjects so that we can erase them

PATH_CHOSEN: "PATH_CHOSEN" = 'PATH_CHOSEN'

Event broadcast when a path is chosen.

## Methods

• Adds the given Observer to this Subject's list of Observers, so that the Observer will be notified of changes in this Subject. An Observer may call Subject.addObserver during its observe method.

#### Parameters

• ##### observer: Observer

the Observer to add

#### Returns void

• Adds the Parameter to the list of this Subject's available Parameters.

#### Parameters

• ##### parameter: Parameter

the Parameter to add

#### Throws

if a Parameter with the same name already exists.

• Notifies all Observers that the Parameter with the given name has changed by calling observe on each Observer.

#### Parameters

• ##### name: string

the language-independent or English name of the Parameter that has changed

#### Throws

if there is no Parameter with the given name

• Defines the differential equations of this ODESim; for an input set of variables, returns the current rate of change for each variable (the first derivative of each variable with respect to time).

The timeStep is the time since the state variables were last fully calculated, which can be and often is zero. The current time can be regarded as getTime() + timeStep. The input variables correspond to the Simulation state at that time. Note that timeStep is different from the time step used to advance the Simulation (as in AdvanceStrategy.advance). The timeStep is typically used when finding collisions in CollisionSim.findCollisions.

#### Parameters

• ##### vars: number[]

the current array of state variables (input), corresponding to the state at getTime() + timeStep

• ##### change: number[]

array of change rates for each variable (output), all values are zero on entry.

• ##### _timeStep: number

the current time step (might be zero)

#### Returns null | object

null if the evaluation succeeds, otherwise an object relating to the error that occurred. The change array contains the output results.

• Called at the end of a mouse drag operation, performs whatever action is appropriate. Only called if startDrag returned true.

#### Parameters

• ##### _simObject: null | SimObject

the SimObject being dragged, or null if no SimObject was found

• ##### _location: Vector

the location of the mouse in simulation coordinates of the SimView where simObject was found, or in the focus view if simObject is null.

• ##### _offset: Vector

distance from the initial object position to the mouse location at start of drag.

#### Returns void

• Returns the Parameter with the given name.

#### Parameters

• ##### name: string

the language-independent or English name of the Parameter

#### Returns Parameter

the Parameter with the given name

#### Throws

if there is no Parameter with the given name

• Returns the ParameterBoolean with the given name.

#### Parameters

• ##### name: string

the language-independent or English name of the ParameterBoolean

#### Returns ParameterBoolean

the ParameterBoolean with the given name

#### Throws

if there is no ParameterBoolean with the given name

• Returns the ParameterNumber with the given name.

#### Parameters

• ##### name: string

the language-independent or English name of the ParameterNumber

#### Returns ParameterNumber

the ParameterNumber with the given name

#### Throws

if there is no ParameterNumber with the given name

• Returns the ParameterString with the given name.

#### Parameters

• ##### name: string

the language-independent or English name of the ParameterString

#### Returns ParameterString

the ParameterString with the given name

#### Throws

if there is no ParameterString with the given name

• Which path was chosen by user, or -1 if no path chosen.

#### Returns number

Which path was chosen by user, or -1 if no path chosen.

• Returns the current Simulation time.

#### Returns number

the current Simulation time.

#### Throws

if there is no time variable for the simulation

• Called when a key is pressed or released, performs whatever action is appropriate for that event.

#### Parameters

• ##### _evt: KeyboardEvent

the KeyboardEvent that happened

• ##### _pressed: boolean

true means this is a key-down event; false means a key-up event

• ##### _modifiers: ModifierKeys

the modifier keys down during event

#### Returns void

• Called at each movement during a mouse drag, performs whatever action is appropriate. Only called if startDrag returned true. The SimObject being moved is passed in, along with the current mouse position, in simulation coordinates, and an offset calculated at the start of the drag.

Setting the SimObject position to (x - offsetX, y - offsetY) will move the SimObject smoothly along with the mouse movement.

#### Parameters

• ##### _simObject: null | SimObject

the SimObject being dragged, or null if no SimObject was found

• ##### _location: Vector

the location of the mouse in simulation coordinates of the SimView where simObject was found, or in the focus view if simObject is null.

• ##### _offset: Vector

distance from the initial object position (from DisplayObject.getPosition) to the mouse location at start of drag.

#### Returns void

• Removes the Observer from this Subject's list of Observers. An Observer may call removeObserver during its observe method.

#### Parameters

• ##### observer: Observer

the Observer to detach from list of Observers

#### Returns void

• Removes the Parameter from the list of this Subject's available Parameters.

#### Parameters

• ##### parameter: Parameter

the Parameter to remove

#### Returns void

• Sets whether this Subject will broadcast events, typically used to temporarily disable broadcasting. Intended to be used in situations where a subclass overrides a method that broadcasts an event. This allows the subclass to prevent the superclass broadcasting that event, so that the subclass can broadcast the event when the method is completed.

#### Parameters

• ##### value: boolean

whether this Subject should broadcast events

#### Returns boolean

the previous value

#### Returns void

• Set which path the user has chosen for the 'fastest path'. Resets the variables to initial conditions. Creates a new set of SimObjects to represent the balls on the path. Creates the text message to display instructions to the user.

Note: the simulation will still run without choosing a path; the path choice is only to allow decorating one of the paths with a different colored ball, and to present text instructions. *

#### Parameters

• ##### choice: number

Which path was chosen by user, or -1 if no path chosen.

#### Returns void

• Sets the Terminal object that this simulation can print data into.

#### Parameters

• ##### terminal: null | Terminal

the Terminal object that this simulation can print data into.

#### Returns void

• Sets the VarsList for this simulation.

#### Parameters

• ##### varsList: VarsList

the VarsList to use in this simulation

#### Returns void

• Called at the start of a mouse drag. The nearest dragable SimObject is passed in, along with mouse position in simulation coordinates. If no dragable SimObject was found, null is passed for the first argument. If the EventHandler does not recognize the SimObject then it should return false.

#### Parameters

• ##### _simObject: null | SimObject

the SimObject that is nearest to the mouse drag coordinates, or null if no SimObject was found

• ##### location: Vector

the location of the mouse in simulation coordinates of the SimView where simObject was found, or in the focus view if simObject is null.

• ##### _offset: Vector

distance from the initial object position (from DisplayObject.getPosition) to the mouse location at start of drag

• ##### _dragBody: null | Vector

location of 'drag point' on the SimObject in body coordinates of the SimObject; this is where for example a spring will be attached on the SimObject when dragging; or null when no SimObject was found

• ##### _modifiers: ModifierKeys

the modifier keys down during event

#### Returns boolean

true if the EventHandler will handle dragging the SimObject

• Returns a minimal string representation of this object, usually giving just identity information like the class name and name of the object.

For an object whose main purpose is to represent another Printable object, it is recommended to include the result of calling toStringShort on that other object. For example, calling toStringShort() on a DisplayShape might return something like this:

DisplayShape{polygon:Polygon{'chain3'}}

#### Returns string

a minimal string representation of this object.

#### Returns string[]

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