Doing some basic calculus using Reactive Banana

问题:

Setup:

I am using Reactive Banana along with OpenGL and I have a gear that I want to spin. I have the following signals:

bTime :: Behavior t Int -- the time in ms from start of rendering
bAngularVelosity :: Behavior t Double -- the angular velocity
                                      -- which can be increase or
                                      -- decreased by the user
eDisplay :: Event t ()     -- need to redraw the screen
eKey :: Event t KeyState   -- user input

Ultimately, I need to calculate bAngle which is then past to the drawing function:

reactimate $ (draw gears) <$> (bAngle <@ eDisp)

The angle is easy to calculate: a = ∫v(t) dt

Question:

I think what I want to do is to approximate this integral as a = ∑ v Δt for each eDisplay event (or more often if I need to). Is this the correct way to go about this? If so, how do I get Δt from bTime?

See Also: I suspect that answer uses the mapAccum function. If so, please also see my other question as well.

回答1:

Edit: to answer the question, yes, you're right to use the approximation you're using, it's Euler's method of solving a first order differential equation, and is accurate enough for your purposes, particularly since the user doesn't have an absolute value for the angular velocity lying around to judge you against. Decreasing your time interval would make it more accurate, but that's not important.

You can do this in fewer, larger steps (see below), but this way seems clearest to me, I hope it is to you.

Why bother with this longer solution? This works even when eDisplay happens at irregular intervals, because it calculates eDeltaT.

Let's give ourselves a time event:

eTime :: Event t Int
eTime = bTime <@ eDisplay

To get DeltaT, we'll need to keep track of the time interval passing:

type TimeInterval = (Int,Int) -- (previous time, current time)

so we can convert them to deltas:

delta :: TimeInterval -> Int
delta (t0,t1) = t1 - t0

How should we update a time interval when we get a new one t2?

tick :: Int -> TimeInterval -> TimeInterval
tick t2 (t0,t1) = (t1,t2)

So let's partially apply that to the time, to give us an interval updater:

eTicker :: Event t (TimeInterval->TimeInterval)
eTicker = tick <$> eTime

and then we can accumE-accumulate that function on an initial time interval:

eTimeInterval :: Event t TimeInterval
eTimeInterval = accumE (0,0) eTicker

Since eTime is measured since the start of rendering, an initial (0,0) is appropriate.

Finally we can have our DeltaT event, by just applying (fmapping) delta on the time interval.

eDeltaT :: Event t Int
eDeltaT = delta <$> eTimeInterval

Now we need to update the angle, using similar ideas.

I'll make an angle updater, by just turning the bAngularVelocity into a multiplier:

bAngleMultiplier :: Behaviour t (Double->Double)
bAngleMultiplier = (*) <$> bAngularVelocity

then we can use that to make eDeltaAngle: (edit: changed to (+) and converted to Double)

eDeltaAngle :: Event t (Double -> Double)
eDeltaAngle = (+) <$> (bAngleMultiplier <@> ((fromInteger.toInteger) <$> eDeltaT)

and accumulate to get the angle:

eAngle :: Event t Double
eAngle = accumE 0.0 eDeltaAngle

If you like one-liners, you can write

eDeltaT = delta <$> (accumE (0,0) $ tick <$> (bTime <@ eDisplay)) where
    delta (t0,t1) = t1 - t0
    tick t2 (t0,t1) = (t1,t2)

eAngle = accumE 0.0 $ (+) <$> ((*) <$> bAngularVelocity <@> eDeltaT) = 

but I don't think that's terribly illuminating, and to be honest, I'm not sure I've got my fixities right since I've not tested this in ghci.

Of course, since I made eAngle instead of bAngle, you need

reactimate $ (draw gears) <$> eAngle

instead of your original

reactimate $ (draw gears) <$> (bAngle <@ eDisp)

回答2:

A simple approach is to assume that eDisplay happens at regular time intervals, and consider bAngularVelocity to be a relative rather than absolute measure, whch would give you the really rather short solution below. [Note that this is no good if eDisplay is out of your control, or if it fires visibly irregularly, or varies in regularity because it will cause your gear to rotate at different speeds as the interval of your eDisplay changes. You'd need my other (longer) approach if that's the case.]

eDeltaAngle :: Event t (Double -> Double)
eDeltaAngle = (+) <$> bAngularVelocity <@ eDisplay

i.e. turn the bAngularVelocity into an adder Event that fires when you eDisplay, so then

eAngle :: Event t Double
eAngle = accumE 0.0 eDeltaAngle

and finally

reactimate $ (draw gears) <$> eAngle

Yes, approximating the integral as a sum is appropriate, and here I'm further approximating by making possibly slightly inaccurate assumtions about the step width, but it's clear and should be smooth as long as your eDisplay is more-or-less regular.