diff --git a/Lib/fontTools/misc/arrayTools.py b/Lib/fontTools/misc/arrayTools.py
index ced8d87a61..0c99395303 100644
--- a/Lib/fontTools/misc/arrayTools.py
+++ b/Lib/fontTools/misc/arrayTools.py
@@ -208,7 +208,7 @@ def sectRect(rect1, rect2):
         min(xMax1, xMax2),
         min(yMax1, yMax2),
     )
-    if xMin >= xMax or yMin >= yMax:
+    if xMin > xMax or yMin > yMax:
         return False, (0, 0, 0, 0)
     return True, (xMin, yMin, xMax, yMax)
 
diff --git a/Lib/fontTools/misc/bezierTools.py b/Lib/fontTools/misc/bezierTools.py
index a1a707b098..213e2544f4 100644
--- a/Lib/fontTools/misc/bezierTools.py
+++ b/Lib/fontTools/misc/bezierTools.py
@@ -1323,7 +1323,8 @@ def midpoint(r):
         return 0.5 * (r[0] + r[1])
 
     # If they do overlap but they're tiny, approximate
-    if rectArea(bounds1) < precision and rectArea(bounds2) < precision:
+    precision2 = precision * precision
+    if rectArea(bounds1) < precision2 and rectArea(bounds2) < precision2:
         return [(midpoint(range1), midpoint(range2))]
 
     c11, c12 = _split_segment_at_t(curve1, 0.5)
@@ -1356,7 +1357,7 @@ def midpoint(r):
         )
     )
 
-    unique_key = lambda ts: (int(ts[0] / precision), int(ts[1] / precision))
+    unique_key = lambda ts: (round(ts[0] / precision), round(ts[1] / precision))
     seen = set()
     unique_values = []
 
@@ -1394,7 +1395,7 @@ def curveCurveIntersections(curve1, curve2):
         >>> len(intersections)
         3
         >>> intersections[0].pt
-        (81.7831487395506, 109.88904552375288)
+        (81.7904225872997, 109.89939633675462)
     """
     if _is_linelike(curve1):
         line1 = curve1[0], curve1[-1]
@@ -1433,7 +1434,7 @@ def segmentSegmentIntersections(seg1, seg2):
         >>> len(intersections)
         3
         >>> intersections[0].pt
-        (81.7831487395506, 109.88904552375288)
+        (81.7904225872997, 109.89939633675462)
         >>> curve3 = [ (100, 240), (30, 60), (210, 230), (160, 30) ]
         >>> line  = [ (25, 260), (230, 20) ]
         >>> intersections = segmentSegmentIntersections(curve3, line)
diff --git a/Tests/misc/arrayTools_test.py b/Tests/misc/arrayTools_test.py
index c8de7bda0e..8ed7a52f5c 100644
--- a/Tests/misc/arrayTools_test.py
+++ b/Tests/misc/arrayTools_test.py
@@ -89,6 +89,13 @@ def test_sectRect():
     assert rect == (5, 20, 20, 30)
 
 
+def test_sectRect_perpendicular():
+    # https://github.com/fonttools/fonttools/issues/3352
+    intersects, rect = sectRect((0.0, 0.0, 0.0, 1.0), (0.0, 0.5, 1.0, 0.5))
+    assert intersects
+    assert rect == (0.0, 0.5, 0.0, 0.5)
+
+
 def test_unionRect():
     assert unionRect((0, 10, 20, 30), (0, 40, 20, 50)) == (0, 10, 20, 50)
 
diff --git a/Tests/misc/bezierTools_test.py b/Tests/misc/bezierTools_test.py
index ce8a9e17e4..a88a32a04c 100644
--- a/Tests/misc/bezierTools_test.py
+++ b/Tests/misc/bezierTools_test.py
@@ -197,3 +197,12 @@ def test_intersections_linelike():
     seg2 = [(0.0, 0.5), (0.25, 0.5), (0.75, 0.5), (1.0, 0.5)]
     pt = curveCurveIntersections(seg1, seg2)[0][0]
     assert pt == (0.0, 0.5)
+
+
+def test_curve_curve_touching_each_other():
+    seg1 = [(0, 0), (0, 100), (100, 100), (100, 0)]
+    seg2 = [(0, 150), (0, 50), (100, 50), (100, 150)]
+    intersections = curveCurveIntersections(seg1, seg2)
+    assert len(intersections) == 1
+    pt = intersections[0][0]
+    assert pt == pytest.approx((50.0, 75.0), rel=1e-3)