diff --git a/Shared/Player.cs b/Shared/Player.cs
index 93931d8..bbca5cf 100644
--- a/Shared/Player.cs
+++ b/Shared/Player.cs
@@ -55,11 +55,14 @@ namespace SemiColinGames {
 
       // Broad test: remove all collision targets nowhere near the player.
       var candidates = new List<Aabb>();
-      // TODO: This is strictly larger than it needs to be. We could expand only in the actual
-      // direction of movement.
+      // Expand the box in the direction of movement. The center is the midpoint of the line
+      // between the player's current position and their desired movement. The width increases by
+      // the magnitude of the movement in each direction. We add 1 to each dimension just to be
+      // sure (the only downside is a small number of false-positive AABBs, which should be
+      // discarded by later tests anyhow.)
       Aabb largeBox = new Aabb(
-        new Vector2(position.X, position.Y),  // current player position
-        new Vector2(halfSize.X + Math.Abs(movement.X), halfSize.Y + Math.Abs(movement.Y)));
+        new Vector2(position.X + movement.X / 2, position.Y + movement.Y / 2),
+        new Vector2(halfSize.X + Math.Abs(movement.X) + 1, halfSize.Y + Math.Abs(movement.Y) + 1));
       foreach (var box in collisionTargets) {
         if (box.Intersect(largeBox) != null) {
           Debug.AddRect(box, Color.Green);
diff --git a/Shared/World.cs b/Shared/World.cs
index 80b59ed..ce948e7 100644
--- a/Shared/World.cs
+++ b/Shared/World.cs
@@ -153,14 +153,26 @@ namespace SemiColinGames {
         }
       }
       tiles = tilesList.ToArray();
-      collisionTargets = new Aabb[tiles.Length];
 
+      // Because we added tiles from left to right, the collisionTargets are sorted by x-position.
+      // We maintain this invariant so that it's possible to efficiently find collisionTargets that
+      // are nearby a given x-position.
+      collisionTargets = new Aabb[tiles.Length + 2];
+
+      // Add a synthetic collisionTarget on the left side of the world.
+      collisionTargets[0] = new Aabb(new Vector2(-1, 0), new Vector2(1, float.MaxValue));
+
+      // Now add all the normal collisionTargets for every static terrain tile.
       Vector2 halfSize = new Vector2(TileSize / 2, TileSize / 2);
       for (int i = 0; i < tiles.Length; i++) {
         Vector2 center = new Vector2(
           tiles[i].Position.Left + halfSize.X, tiles[i].Position.Top + halfSize.Y);
-        collisionTargets[i] = new Aabb(center, halfSize);
+        collisionTargets[i + 1] = new Aabb(center, halfSize);
       }
+
+      // Add a final synthetic collisionTarget on the right side of the world.
+      collisionTargets[tiles.Length + 1] = new Aabb(
+          new Vector2(Width * TileSize + 1, 0), new Vector2(1, float.MaxValue));
     }
 
     public void Draw(SpriteBatch spriteBatch, Camera camera) {