diff --git a/Lib/ufo2ft/filters/__init__.py b/Lib/ufo2ft/filters/__init__.py
index e43223c98..4f9afedc4 100644
--- a/Lib/ufo2ft/filters/__init__.py
+++ b/Lib/ufo2ft/filters/__init__.py
@@ -6,6 +6,7 @@
 from ufo2ft.util import _loadPluginFromString
 
 from .base import BaseFilter
+from .addExtrema import AddExtremaFilter
 from .cubicToQuadratic import CubicToQuadraticFilter
 from .decomposeComponents import DecomposeComponentsFilter
 from .decomposeTransformedComponents import DecomposeTransformedComponentsFilter
@@ -18,6 +19,7 @@
 
 __all__ = [
     "BaseFilter",
+    "AddExtremaFilter",
     "CubicToQuadraticFilter",
     "DecomposeComponentsFilter",
     "DecomposeTransformedComponentsFilter",
diff --git a/Lib/ufo2ft/filters/addExtrema.py b/Lib/ufo2ft/filters/addExtrema.py
new file mode 100644
index 000000000..d70230798
--- /dev/null
+++ b/Lib/ufo2ft/filters/addExtrema.py
@@ -0,0 +1,104 @@
+import logging
+from math import sqrt
+
+from fontTools.misc.bezierTools import splitCubicAtT
+from fontTools.pens.filterPen import FilterPen
+from fontTools.pens.recordingPen import RecordingPen
+
+from ufo2ft.filters import BaseFilter
+
+logger = logging.getLogger(__name__)
+
+
+def quadraticRoots(a, b, c):
+    """Returns real roots of at^2 + bt + c = 0 if 0 < root < 1"""
+    roots = []
+    if a != 0.0 and b * b - 4 * a * c > 0.0:
+        x = -b / (2 * a)
+        y = sqrt(b * b - 4 * a * c) / (2 * a)
+        t1 = x - y
+        if 0.0 <= t1 <= 1.0:
+            roots.append(t1)
+        t2 = x + y
+        if 0.0 <= t2 <= 1.0:
+            roots.append(t2)
+    return roots
+
+
+class AddExtremaPen(FilterPen):
+    def __init__(self, outPen):
+        self._outPen = outPen
+        self.affected = False
+
+    def moveTo(self, pt):
+        self.lastPt = pt
+        self._outPen.moveTo(pt)
+
+    def lineTo(self, pt):
+        self.lastPt = pt
+        self._outPen.lineTo(pt)
+
+    def qCurveTo(self, *points):
+        self.lastPt = points[-1]
+        self._outPen.qCurveTo(*points)
+
+    def curveTo(self, *points):
+        # Make it a full segment
+        points = [self.lastPt] + list(points)
+
+        # Does this curve have extremes?
+        derivative = [
+            ((points[1][0] - points[0][0]) * 3, (points[1][1] - points[0][1]) * 3),
+            ((points[2][0] - points[1][0]) * 3, (points[2][1] - points[1][1]) * 3),
+            ((points[3][0] - points[2][0]) * 3, (points[3][1] - points[2][1]) * 3),
+        ]
+        roots = []
+        roots.extend(
+            quadraticRoots(
+                derivative[0][0] - 2 * derivative[1][0] + derivative[2][0],
+                2 * (derivative[1][0] - derivative[0][0]),
+                derivative[0][0],
+            )
+        )
+        roots.extend(
+            quadraticRoots(
+                derivative[0][1] - 2 * derivative[1][1] + derivative[2][1],
+                2 * (derivative[1][1] - derivative[0][1]),
+                derivative[0][1],
+            )
+        )
+
+        # To find inflections, we need to take the second derivative
+        d1 = derivative[0][0] - 2 * derivative[1][0] + derivative[2][0]
+        d2 = derivative[0][1] - 2 * derivative[1][1] + derivative[2][1]
+        if d1 != 0:
+            r1 = (derivative[0][0] - derivative[1][0]) / d1
+            roots.append(r1)
+        if d2 != 0:
+            r2 = (derivative[0][1] - derivative[1][1]) / d2
+            roots.append(r2)
+
+        roots = [x for x in roots if 0.01 <= x <= 0.99]
+        if roots:
+            for subcurve in splitCubicAtT(*points, *roots):
+                self._outPen.curveTo(*subcurve[1:])
+            self.affected = True
+        else:
+            self._outPen.curveTo(*points[1:])
+        self.lastPt = points[-1]
+
+
+class AddExtremaFilter(BaseFilter):
+    def filter(self, glyph):
+        if not len(glyph):
+            return False
+
+        contours = list(glyph)
+        outpen = RecordingPen()
+        p = AddExtremaPen(outpen)
+        for contour in contours:
+            contour.draw(p)
+        if p.affected:
+            glyph.clearContours()
+            outpen.replay(glyph.getPen())
+        return p.affected
diff --git a/tests/filters/addExtrema_test.py b/tests/filters/addExtrema_test.py
new file mode 100644
index 000000000..52f05a504
--- /dev/null
+++ b/tests/filters/addExtrema_test.py
@@ -0,0 +1,23 @@
+from ufo2ft.filters.addExtrema import AddExtremaFilter
+
+
+class AddExtremaFilterTest:
+    def test_add_extrema_O(self, FontClass):
+        ufo = FontClass()
+        a = ufo.newGlyph("O")
+        a.width = 300
+        pen = a.getPen()
+        pen.moveTo((150, 0))
+        pen.curveTo((249, 0), (249, 150), (150, 150))
+        pen.curveTo((52, 150), (52, 0), (150, 0))
+        pen.closePath()
+
+        contour = ufo["O"][0]
+        assert len(contour) == 6
+
+        filter_ = AddExtremaFilter()
+
+        assert filter_(ufo)
+
+        contour = ufo["O"][0]
+        assert len(contour) == 12