diff --git a/IV/notes.txt b/IV/notes.txt
new file mode 100644
index 0000000..fc3a506
--- /dev/null
+++ b/IV/notes.txt
@@ -0,0 +1 @@
+Definition of a bend: ends of the line should always be bends, otherwise not all line vertices are covered by bends (definition elsewhere).
diff --git a/IV/tests.sql b/IV/tests.sql
index 2ccac64..7c66c87 100644
--- a/IV/tests.sql
+++ b/IV/tests.sql
@@ -17,20 +17,25 @@ insert into figures (name, way) values ('fig3',ST_GeomFromText('LINESTRING(0 0,1
 insert into figures (name, way) values ('fig3-1',ST_GeomFromText('LINESTRING(0 0,12 0,13 4,20 2,20 0,32 0,33 10,38 16,43 15,44 10,44 0)'));
 insert into figures (name, way) values ('fig5',ST_GeomFromText('LINESTRING(0 39,19 52,27 77,26 104,41 115,49 115,65 103,65 75,53 45,63 15,91 0,91 0)'));
 
+drop table if exists bends;
+create table bends (way geometry);
+insert into bends select unnest(detect_bends((select way from figures where name='fig3')));
+insert into bends select unnest(detect_bends((select way from figures where name='fig5')));
+
 do $$
 declare
   bends geometry[];
 begin
   select detect_bends((select way from figures where name='fig3')) into bends;
-  perform assert_equals(3, array_length(bends, 1));
-  perform assert_equals(4::bigint, (select count(1) from (select (st_dumppoints(detect_bends[1])).path from bends) a));
-  perform assert_equals('LINESTRING(12 0,13 4,20 2,20 0)', st_astext(bends[1]));
-  perform assert_equals('LINESTRING(20 2,20 0,32 0,33 10)', st_astext(bends[2]));
-  perform assert_equals('LINESTRING(32 0,33 10,38 16,43 15,44 10,44 0)',st_astext(bends[3]));
+  perform assert_equals(5, array_length(bends, 1));
+  perform assert_equals('LINESTRING(0 0,12 0,13 4)', st_astext(bends[1]));
+  perform assert_equals('LINESTRING(12 0,13 4,20 2,20 0)', st_astext(bends[2]));
+  perform assert_equals('LINESTRING(20 2,20 0,32 0,33 10)', st_astext(bends[3]));
+  perform assert_equals('LINESTRING(32 0,33 10,38 16,43 15,44 10,44 0)', st_astext(bends[4]));
 
 
-  perform assert_equals(3, array_length(detect_bends((select way from figures where name='fig3-1')), 1));
+  perform assert_equals(4, array_length(detect_bends((select way from figures where name='fig3-1')), 1));
 
   select detect_bends((select way from figures where name='fig5')) into bends;
-  perform assert_equals(2, array_length(bends, 1));
+  perform assert_equals(3, array_length(bends, 1));
 end $$ language plpgsql;
diff --git a/IV/wm.sql b/IV/wm.sql
index 374fe97..c652db8 100644
--- a/IV/wm.sql
+++ b/IV/wm.sql
@@ -16,7 +16,11 @@ declare
 begin
   pi = radians(180);
 
-  for p in (select (dp).geom from st_dumppoints(line) as dp) loop
+  for p in (
+    (select (dp).geom from st_dumppoints(line) as dp order by (dp).path[1] asc)
+    union all
+    (select (dp).geom from st_dumppoints(line) as dp order by (dp).path[1] desc limit 1)
+  ) loop
     p3 = p2;
     p2 = p1;
     p1 = p;
@@ -25,7 +29,11 @@ begin
     end if;
     cur_sign = sign(pi - st_angle(p1, p2, p2, p3));
 
-    bend = st_linemerge(st_union(bend, st_makeline(p3, p2)));
+    if bend is null then
+      bend = st_makeline(p3, p2);
+    else
+      bend = st_linemerge(st_union(bend, st_makeline(p3, p2)));
+    end if;
 
     if prev_sign + cur_sign = 0 then
       if bend is not null then
@@ -36,10 +44,8 @@ begin
     prev_sign = cur_sign;
   end loop;
 
-  -- the last bend may be lost if there is no "final" inflection angle.
-  -- to avoid that, return the last bend if the last accumulation has >3
-  -- vertices.
-  if (select count(1) from ((select st_dumppoints(bend) as a)) b) >= 3 then
+  -- the last bend may be lost if there is no "final" inflection angle. Add it.
+  if (select count(1) from ((select st_dumppoints(bend) as a)) b) >= 2 then
     bends = bends || bend;
   end if;
 end