Use fastOutSlowIn curve for transitions on iOS when not controlled by user gesture. (#5666)

BUG=https://github.com/flutter/flutter/issues/5599

packages/flutter/lib/src/material/page.dart

@@ -48,13 +48,14 @@ class _CupertinoPageTransition extends AnimatedWidget {
 
   _CupertinoPageTransition({
     Key key,
+    Curve curve,
     Animation<double> animation,
     this.child
   }) : super(
     key: key,
     animation: _kTween.animate(new CurvedAnimation(
       parent: animation,
-      curve: new _CupertinoTransitionCurve()
+      curve: new _CupertinoTransitionCurve(curve)
     )
   ));
 
@@ -84,16 +85,33 @@ class AnimationMean extends CompoundAnimation<double> {
   double get value => (first.value + next.value) / 2.0;
 }
 
-// Custom curve for iOS page transitions. The halfway point is when the page
-// is fully on-screen. 0.0 is fully off-screen to the right. 1.0 is off-screen
-// to the left.
+// Custom curve for iOS page transitions.
 class _CupertinoTransitionCurve extends Curve {
-  _CupertinoTransitionCurve();
+  _CupertinoTransitionCurve(this.curve);
+
+  Curve curve;
 
   @override
   double transform(double t) {
-    if (t > 0.5)
-      return (t - 0.5) / 3.0 + 0.5;
+    // The input [t] is the average of the current and next route's animation.
+    // This means t=0.5 represents when the route is fully onscreen. At
+    // t > 0.5, it is partially offscreen to the left (which happens when there
+    // is another route on top). At t < 0.5, the route is to the right.
+    // We divide the range into two halves, each with a different transition,
+    // and scale each half to the range [0.0, 1.0] before applying curves so that
+    // each half goes through the full range of the curve.
+    if (t > 0.5) {
+      // Route is to the left of center.
+      t = (t - 0.5) * 2.0;
+      if (curve != null)
+        t = curve.transform(t);
+      t = t / 3.0;
+      t = t / 2.0 + 0.5;
+    } else {
+      // Route is to the right of center.
+      if (curve != null)
+        t = curve.transform(t * 2.0) / 2.0;
+    }
     return t;
   }
 }
@@ -220,7 +238,8 @@ class MaterialPageRoute<T> extends PageRoute<T> {
     // TODO(mpcomplete): This hack prevents the previousRoute from animating
     // when we pop(). Remove once we fix this bug:
     // https://github.com/flutter/flutter/issues/5577
-    if (!Navigator.of(context).userGestureInProgress)
+    bool userGesture = Navigator.of(context).userGestureInProgress;
+    if (!userGesture)
       forwardAnimation = kAlwaysDismissedAnimation;
 
     ThemeData theme = Theme.of(context);
@@ -233,6 +252,10 @@ class MaterialPageRoute<T> extends PageRoute<T> {
         );
       case TargetPlatform.iOS:
         return new _CupertinoPageTransition(
+          // Use a linear curve when controlled by a user gesture. This ensures
+          // the animation tracks the user's finger 1:1.
+          // See https://github.com/flutter/flutter/issues/5664
+          curve: userGesture ? null : Curves.fastOutSlowIn,
           animation: new AnimationMean(left: animation, right: forwardAnimation),
           child: child
         );