diff --git a/qsomap.py b/qsomap.py
index 7cc4104..927bd83 100755
--- a/qsomap.py
+++ b/qsomap.py
@@ -65,6 +65,78 @@ def random_country_color():
     return f"#{r:02x}{g:02x}{b:02x}"
 
 
+def is_point_in_polygon(point, polygon):
+    # Idea: draw an infinite line from the test point along the x axis to the
+    # right. Then check how many polygon edges this line intersects. If the
+    # number is even, the point is outside the polygon.
+
+    edges = []  # list of lists, containing two points each
+
+    for i in range(len(polygon)-1):
+        edges.append([polygon[i], polygon[i+1]])
+
+    # the closing edge
+    edges.append([polygon[-1], polygon[0]])
+
+    num_intersects = 0
+
+    test_x, test_y = point
+
+    for edge in edges:
+        start_x = edge[0][0]
+        start_y = edge[0][1]
+        end_x = edge[1][0]
+        end_y = edge[1][1]
+
+        # quick exclusion tests
+        if start_x < test_x and end_x < test_x:
+            continue  # edge is completely left of the test point
+
+        if start_y < test_y and end_y < test_y:
+            continue  # edge is completely below the test point
+
+        if start_y > test_y and end_y > test_y:
+            continue  # edge is completely above the test point
+
+        # calculate the x coordinate where the edge intersects the whole
+        # horizontal line
+        intersect_x = start_x + (end_x - start_x) \
+            * (test_y - start_y) \
+            / (end_y - start_y)
+
+        if intersect_x > test_x:
+            # we found an intersection!
+            num_intersects += 1
+
+    if num_intersects % 2 == 0:
+        return False  # even number of intersects -> outside polygon
+    else:
+        return True  # odd number of intersects -> inside polygon
+
+
+def svg_make_inverse_country_path(doc, map_radius, polygon, **kwargs):
+    # build a closed circle path covering the whole map
+    commands = [f"M 0, {map_radius}",
+                f"a {map_radius},{map_radius} 0 1,0 {map_radius*2},0",
+                f"a {map_radius},{map_radius} 0 1,0 {-map_radius*2},0",
+                "z"]
+
+    # "subtract" the country polygon
+    commands.append(f"M {polygon[0][0]} {polygon[0][1]}")
+
+    # add lines for each polygon point
+    for point in polygon[1:]:
+        commands.append(f"L {point[0]} {point[1]}")
+
+    # ensure straight closing line
+    commands.append(f"L {polygon[0][0]} {polygon[0][1]}")
+
+    # close the inner path
+    commands.append("z")
+
+    return doc.path(commands, **kwargs)
+
+
 def render(ref_lat, ref_lon, output_stream):
     random.seed(0)
 
@@ -127,6 +199,12 @@ def render(ref_lat, ref_lon, output_stream):
     ref_lat = ref_lat * np.pi / 180
     ref_lon = ref_lon * np.pi / 180
 
+    antipodal_lat = -ref_lat
+    antipodal_lon = ref_lon + np.pi
+
+    if antipodal_lon > np.pi:
+        antipodal_lon -= 2*np.pi
+
     R = 500
 
     """
@@ -183,7 +261,6 @@ def render(ref_lat, ref_lon, output_stream):
 
         v['proj_coordinates'] = proj_polys
 
-
     # generate the SVG
 
     doc = svgwrite.Drawing("/tmp/test.svg", size=(2*R, 2*R))
@@ -224,14 +301,26 @@ def render(ref_lat, ref_lon, output_stream):
 
         group = doc.g()
 
-        for poly in v['proj_coordinates']:
+        for i in range(len(v['proj_coordinates'])):
+            poly = v['proj_coordinates'][i]
             points = poly.T + R  # shift to the center of the drawing
 
-            pgon = doc.polygon(points, **{
-                               'class': 'country',
-                               'fill': color})
+            # check if the antipodal point is inside this polygon. If so, it
+            # needs to be "inverted", i.e. subtracted from the surrounding map
+            # circle.
 
-            group.add(pgon)
+            if is_point_in_polygon((antipodal_lon, antipodal_lat),
+                                   v['coordinates'][i].T):
+                print("!!! Found polygon containing the antipodal point!")
+                obj = svg_make_inverse_country_path(doc, R, np.flipud(points), **{
+                                  'class': 'country',
+                                  'fill': color})
+            else:
+                obj = doc.polygon(points, **{
+                                  'class': 'country',
+                                  'fill': color})
+
+            group.add(obj)
 
         group.set_desc(title=v['name'])
         doc.add(group)