diff --git a/masque/shapes/path.py b/masque/shapes/path.py
index feb1700..43cff34 100644
--- a/masque/shapes/path.py
+++ b/masque/shapes/path.py
@@ -449,11 +449,15 @@ class Path(Shape):
                                         for v in normed_vertices])
 
         # Reorder the vertices so that the one with lowest x, then y, comes first.
-        x_min = rotated_vertices[:, 0].argmin()
-        if not is_scalar(x_min):
-            y_min = rotated_vertices[x_min, 1].argmin()
-            x_min = cast('Sequence', x_min)[y_min]
-        reordered_vertices = numpy.roll(rotated_vertices, -x_min, axis=0)
+        x_min_val = rotated_vertices[:, 0].min()
+        x_min_inds = numpy.where(rotated_vertices[:, 0] == x_min_val)[0]
+        if x_min_inds.size > 1:
+            y_min_val = rotated_vertices[x_min_inds, 1].min()
+            tie_breaker = numpy.where(rotated_vertices[x_min_inds, 1] == y_min_val)[0][0]
+            start_ind = x_min_inds[tie_breaker]
+        else:
+            start_ind = x_min_inds[0]
+        reordered_vertices = numpy.roll(rotated_vertices, -start_ind, axis=0)
 
         width0 = self.width / norm_value
 
diff --git a/masque/shapes/polygon.py b/masque/shapes/polygon.py
index c9617a9..34a784b 100644
--- a/masque/shapes/polygon.py
+++ b/masque/shapes/polygon.py
@@ -417,11 +417,15 @@ class Polygon(Shape):
                                         for v in normed_vertices])
 
         # Reorder the vertices so that the one with lowest x, then y, comes first.
-        x_min = rotated_vertices[:, 0].argmin()
-        if not is_scalar(x_min):
-            y_min = rotated_vertices[x_min, 1].argmin()
-            x_min = cast('Sequence', x_min)[y_min]
-        reordered_vertices = numpy.roll(rotated_vertices, -x_min, axis=0)
+        x_min_val = rotated_vertices[:, 0].min()
+        x_min_inds = numpy.where(rotated_vertices[:, 0] == x_min_val)[0]
+        if x_min_inds.size > 1:
+            y_min_val = rotated_vertices[x_min_inds, 1].min()
+            tie_breaker = numpy.where(rotated_vertices[x_min_inds, 1] == y_min_val)[0][0]
+            start_ind = x_min_inds[tie_breaker]
+        else:
+            start_ind = x_min_inds[0]
+        reordered_vertices = numpy.roll(rotated_vertices, -start_ind, axis=0)
 
         # TODO: normalize mirroring?