1. Chapter 15 A Module of Reactive Animations

2. Motivation  The design of animations in Chapter 13 is elegant, and in fact has the feel of a small domain-specific language (DSL), embedded in Haskell.  However, the language lacks reactivity: the ability to interact with the user or other external stimuli.  In this chapter we add reactivity, and call the resulting DSL functional animation language, or FAL.  In addition, an implementation of FAL is described, using streams.

3. FAL by Example  As before, we use the polymorphic data type “Behavior” to capture time-varying values.  For example: color1 :: Behavior Color color1 = red `untilB` (lbp ->> blue) ball1 :: Behavior Picture ball1 = paint color1 circ circ :: Behavior Region circ = translate (cos time, sin time) (ell 0.2 0.2)  The function “untilB” reflects reactive behavior, and “lbp” corresponds to a left button press.

4. More Reactivity  Recursive reactivity: color1r = red `untilB` lbp ->> blue `untilB` lbp ->> color1r  Choice reactivity: color2 = red `untilB` ((lbp ->> blue).|. (key ->> yellow))  Recursive, choice reactivity: color2r = red `untilB` colorEvent where colorEvent = (lbp ->> blue `untilB` colorEvent).|. (key ->> yellow `untilB` colorEvent)  Pushing recursion into combinator: color2h = red `switch` ((lbp ->> blue).|. (key ->> yellow))

5. Events With Data  Convert button-press events into color events: color1h = red `switch` (lbp `withElem_` cycle [blue, red])  Dispatch on key press: color3 = white `switch` (key =>> \c -> case c of 'R' -> red 'B' -> blue 'Y' -> yellow _ -> white )  Carrying state forward: color4 = white `switch` ((key `snapshot` color4) =>> \(c, old) -> case c of 'R' -> red 'B' -> blue 'Y' -> yellow _ -> lift0 old)

6. Dynamic Events  Not all events are external. For example: while (time >* 42) generates no events until time exceeds 42, and then generates events “infinitely often”. when (time >* 42) generates exactly one event when the time exceeds 42. color5 = red `untilB` (when (time >* 5) ->> blue)

7. Integration  The position of a mass under the influence of an accelerating force f: s, v :: Behavior Float s = s0 + integral v v = v0 + integral f  Combining with reactivity, a bouncing ball: ball2 = paint red (translate (x,y) (ell 0.2 0.2)) whereg = -4 x = -3 + integral 0.5 y = 1.5 + integral v v = integral g `switch` (hit `snapshot_` v =>> \v'-> lift0 (-v') + integral g) hit = when (y <* -1.5) Note similarity to mathematical equations.

8. Implementing FAL  Previously a behavior was conceptually a function: Behavior a ≡ Time -> a  But somehow we must now introduce events. One obvious approach would be: Behavior a ≡ [(UserAction, Time)] -> Time -> a But this would be very inefficient (why?).  Better to do this: Behavior a ≡ [(UserAction, Time)] -> [Time] -> [a]  Or, even more efficient, and now as Haskell code: newtype Behavior a = Behavior ( ([Maybe UserAction], [Time]) -> [a] ) (see text for definition of UserAction)

9. Time and Constants  Recall: newtype Behavior a = Behavior ( ([Maybe UserAction], [Time]) -> [a] )  With this representation, let’s define time: time :: Behavior Time time = Behavior (\(_,ts) -> ts)  Constant behaviors are achieved via lifting: constB :: a -> Behavior a constB x = Behavior (\_ -> repeat x)  For example: red, blue :: Behavior Color red = constB Red blue = constB Blue

10. Curried Liftings  From this “lifted” version of application: ($*) :: Beh (a->b) -> Beh a -> Beh b Beh ff $* Beh fb = Beh (\uts -> zipWith ($) (ff uts) (fb uts))  and the constant lifting operator: lift0 :: a -> Beh a lift0 = constB  all other lifting operators can be defined: lift1 :: (a -> b) -> (Beh a -> Beh b) lift1 f b1 = lift0 f $* b1 lift2 :: (a -> b -> c) -> (Beh a -> Beh b -> Beh c) lift2 f b1 b2 = lift1 f b1 $* b2 lift3 :: (a -> b -> c -> d) -> (Beh a -> Beh b -> Beh c -> Beh d) lift3 f b1 b2 b3 = lift2 f b1 b2 $* b3 (For conciseness, “Beh” is used instead of “Behavior”.)

11. Sample Liftings pairB :: Behavior a -> Behavior b -> Behavior (a,b) pairB = lift2 (,) fstB :: Behavior (a,b) -> Behavior a fstB = lift1 fst paint :: Behavior Color -> Behavior Region -> Behavior Picture paint = lift2 Region red, blue, yellow, green, white, black :: Behavior Color red = lift0 Red blue = lift0 Blue... shape :: Behavior Shape -> Behavior Region shape = lift1 Shape ell, rec :: Behavior Float -> Behavior Float -> Behavior Region ell x y = shape (lift2 Ellipse x y) rec x y = shape (lift2 Rectangle x y) See text for more liftings.

12. Events and Reactivity  Abstractly, we can think of events as: type Event a = Behavior (Maybe a)  But for type safety, this is better: newtype Event a = Event ( ([Maybe UserAction], [Time]) -> [Maybe a] )  Core of FAL’s reactivity: untilB:: Behavior a -> Event (Behavior a) -> Behavior a switch :: Behavior a -> Event (Behavior a) -> Behavior a (->>) :: Event a -> b -> Event b (=>>) :: Event a -> (a->b) -> Event b plus primitive events such as: lbp :: Event ( )

13. Primitive Events  “lbp” must look for a “left button press” in the stream of UserActions: lbp :: Event ( ) lbp = Event (\(uas,_) -> map getlbp uas) where getlbp (Just (Button _ True True)) = Just ( ) getlbp _ = Nothing  Similarly for “key”: key :: Event Char key = Event (\(uas,_) -> map getkey uas) where getkey (Just (Key ch True)) = Just ch getkey _= Nothing

14. Implementing UntilB untilB switches into a new behavior carried by the event. untilB:: Behavior a -> Event (Behavior a) -> Behavior a Behavior fb `untilB` Event fe = memoB $ Behavior (\uts@(us,ts) -> loop us ts (fe uts) (fb uts)) where loop (_:us) (_:ts) ~(e:es) (b:bs) = b : case e of Nothing -> loop us ts es bs Just (Behavior fb') -> fb' (us,ts) memoB :: Behavior a -> Behavior a memoB (Behavior fb) = Behavior (memo1 fb) Stare at this code until you understand it completely! The definition of “switch” is very similar (see text).

15. Event Map  Recall: color1 :: Behavior Color color1 = red `untilB` (lbp ->> blue) What does “->>” do?  Consider types: red, blue :: Behavior Color untilB :: Behavior Color -> Event (Behavior Color) -> Behavior Color lbp :: Event ( ) (->>) :: Event ( ) -> Behavior Color -> Event (Behavior Color)  So (->>) somehow “tags” an event with a Behavior. Polymorphically speaking: (->>) :: Event a -> b -> Event b  It is actually a special case of the more general: (=>>) :: Event a -> (a->b) -> Event b

16. Implementing Event Map  (=>>) is defined as: Event fe =>> f = Event (\uts -> map aux (fe uts)) whereaux (Just a)= Just (f a) aux Nothing= Nothing  Which can be defined more succinctly using fmap from the Functor class (discussed in Chapter 18!): Event fe =>> f = Event (map (fmap f). fe)  (->>) is then defined in terms of (=>>): e ->> v = e =>> \_ -> v

17. Implementing Predicate Events  “while” is defined as: while :: Behavior Bool -> Event () while (Behavior fb) = Event (\uts -> map aux (fb uts)) where aux True= Just () aux False = Nothing  “when” is defined similarly (see text).

18. Implementing Integration  “integral” is defined by: integral :: Behavior Float -> Behavior Float integral (Behavior fb) = Behavior (\uts@(us,t:ts) -> 0 : loop t 0 ts (fb uts)) where loop t0 acc (t1:ts) (a:as) = let acc' = acc + (t1-t0)*a in acc' : loop t1 acc' ts as  This corresponds to the standard definition of integration as a limit in calculus (see text).

19. “Steppers”  “Steppers” are convenient variations of switch: step :: a -> Event a -> Behavior a a `step` e = constB a `switch` e =>> constB stepAccum :: a -> Event (a->a) -> Behavior a a `stepAccum` e = b where b = a `step` (e `snapshot` b =>> uncurry ($))  For example, a counter: counter = 0 `stepAccum` lbp ->> (+1) an example involving `step` is on the next slide.

20. Mouse Movement  It’s convenient to treat mouse position as a pair of Behaviors: mouse :: (Behavior Float, Behavior Float) mouse = (fstB m, sndB m) where m = (0,0) `step` mm  where “mm” is defined as: mm :: Event Coordinate mm = Event (\(uas,_) -> map getmm uas) where getmm (Just (MouseMove pt)) = Just (gPtToPt pt) getmm _ = Nothing

21. Final Example: Paddleball!  A paddleball game consists of three parts: paddleball vel = walls `over` paddle `over` pball vel  Where ”walls” and ”paddle” are defined by: walls = let upper = paint blue (translate ( 0,1.7) (rec 4.4 0.05)) left = paint blue (translate (-2.2,0) (rec 0.05 3.4)) right = paint blue (translate ( 2.2,0) (rec 0.05 3.4)) in upper `over` left `over` right paddle = paint red (translate (fst mouse, -1.7) (rec 0.5 0.05))  The core of the game is in “pball”.

22. Putting it All Together pball vel = letxvel = vel `stepAccum` xbounce ->> negate xpos = integral xvel xbounce = when (xpos >* 2 ||* xpos > negate ypos = integral yvel ybounce = when (ypos >* 1.5 ||* ypos `between` (-2.0,-1.5) &&* fst mouse `between` (xpos-0.25,xpos+0.25)) in paint yellow (translate (xpos, ypos) (ell 0.2 0.2)) x `between` (a,b) = x >* a &&* x <* b