diff --git a/masque/shapes/arc.py b/masque/shapes/arc.py
index a068700..ab2e47b 100644
--- a/masque/shapes/arc.py
+++ b/masque/shapes/arc.py
@@ -268,7 +268,7 @@ class Arc(PositionableImpl, Shape):
             """ Figure out the parameter values at which we should place vertices to meet the arclength constraint"""
             dr = -wh if inner else wh
 
-            n_pts = numpy.ceil(2 * pi * max(self.radii + dr) / max_arclen).astype(int)
+            n_pts = max(2, int(numpy.ceil(2 * pi * max(self.radii + dr) / max_arclen)))
             arc_lengths, thetas = get_arclens(n_pts, *a_ranges[0 if inner else 1], dr=dr)
 
             keep = [0]
diff --git a/masque/shapes/circle.py b/masque/shapes/circle.py
index 8dad165..a6f7af1 100644
--- a/masque/shapes/circle.py
+++ b/masque/shapes/circle.py
@@ -108,7 +108,7 @@ class Circle(PositionableImpl, Shape):
             n += [num_vertices]
         if max_arclen is not None:
             n += [2 * pi * self.radius / max_arclen]
-        num_vertices = int(round(max(n)))
+        num_vertices = max(3, int(round(max(n))))
         thetas = numpy.linspace(2 * pi, 0, num_vertices, endpoint=False)
         xs = numpy.cos(thetas) * self.radius
         ys = numpy.sin(thetas) * self.radius
diff --git a/masque/shapes/ellipse.py b/masque/shapes/ellipse.py
index 8a5c287..55ce9fd 100644
--- a/masque/shapes/ellipse.py
+++ b/masque/shapes/ellipse.py
@@ -168,7 +168,7 @@ class Ellipse(PositionableImpl, Shape):
             n += [num_vertices]
         if max_arclen is not None:
             n += [perimeter / max_arclen]
-        num_vertices = int(round(max(n)))
+        num_vertices = max(3, int(round(max(n))))
         thetas = numpy.linspace(2 * pi, 0, num_vertices, endpoint=False)
 
         sin_th, cos_th = (numpy.sin(thetas), numpy.cos(thetas))
diff --git a/masque/test/test_shape_advanced.py b/masque/test/test_shape_advanced.py
index ab42b4b..fc76fcb 100644
--- a/masque/test/test_shape_advanced.py
+++ b/masque/test/test_shape_advanced.py
@@ -109,6 +109,19 @@ def test_rotated_arc_bounds_match_polygonized_geometry() -> None:
     bounds = arc.get_bounds_single()
     poly_bounds = arc.to_polygons(num_vertices=8192)[0].get_bounds_single()
     assert_allclose(bounds, poly_bounds, atol=1e-3)
+
+
+def test_curve_polygonizers_clamp_large_max_arclen() -> None:
+    for shape in (
+            Circle(radius=10),
+            Ellipse(radii=(10, 20)),
+            Arc(radii=(10, 20), angles=(0, 1), width=2),
+            ):
+        polys = shape.to_polygons(num_vertices=None, max_arclen=1e9)
+        assert len(polys) == 1
+        assert len(polys[0].vertices) >= 3
+
+
 def test_path_edge_cases() -> None:
     # Zero-length segments
     p = MPath(vertices=[[0, 0], [0, 0], [10, 0]], width=2)