remove incorrect direction approximation rules

IV/notes.txt

@@ -40,11 +40,6 @@ inflection angles (AB in our examples) must be crossed with all the other bends
 to detect a possible line-crossing. This is O(N*M), where N is the total number
 of line segments, and M is the number of qualifying bends. It is expensive.
 
-This may be simplified: if other bend's endpoints (A' and B') are in the same
-sub-plane as divided by AB, then the bend can be skipped from checking if it
-intersects with AB. Some intersections may be missed (see the last example),
-but they will be eliminated by joining A and B anyway.
-
 Also, there is another way to remove self-crossing, without removing most of
 the bend. E.g. from:
 
@@ -69,4 +64,4 @@ To:
    \__________|
 
 But perhaps it doesn't look quite as natural. I will trust the original
-article to do the right thing here.
+article to do the right thing here and remove the bend altogether.

IV/tests.sql

@@ -18,7 +18,7 @@ create table figures (name text, way geometry);
 --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)'));
 insert into figures (name, way) values ('fig6',ST_GeomFromText('LINESTRING(84 47,91 59,114 64,122 80,116 92,110 93,106 106,117 118,136 107,135 76,120 45,125 39,141 39,147 32)'));
---insert into figures (name, way) values ('fig6-rev',ST_Reverse(ST_Translate((select way from figures where name='fig6'), 80, 0)));
+insert into figures (name, way) values ('fig6-rev',ST_Reverse(ST_Translate((select way from figures where name='fig6'), 80, 0)));
 --insert into figures (name, way) values ('inflection-1',ST_GeomFromText('LINESTRING(110 24,114 20,133 20,145 15,145 0,136 5,123 7,114 7,111 2)'));
 
 drop table if exists debug;

IV/wm.sql

@@ -178,8 +178,6 @@ declare
   p3 geometry;
   a geometry;
   b geometry;
-  s2 real;
-  s3 real;
   bend geometry;
   this geometry;
   multi geometry;
@@ -224,20 +222,10 @@ begin
 
       raise notice 'j: %, i: %', j, i;
 
-      -- are p2 and p3 on the different sides of line(p0,p1)? vector
-      -- multiplication; https://stackoverflow.com/questions/1560492/
-      s2 = (st_x(p0)-st_x(p1)*(st_y(p2)-st_y(p1))-
-            (st_y(p0)-st_y(p1))*(st_x(p2)-st_x(p1)));
-      s3 = (st_x(p0)-st_x(p1)*(st_y(p3)-st_y(p1))-
-            (st_y(p0)-st_y(p1))*(st_x(p3)-st_x(p1)));
-      raise notice 'sign(s2): %, sign(s3): %', sign(s2), sign(s3);
-      continue when sign(s2) = sign(s3);
-
       -- do end vertices of bend[i] cross bend[j]?
       a = st_pointn(bends[i], 1);
       b = st_pointn(bends[i], -1);
       multi = st_split(bends[j], st_makeline(a, b));
-      insert into debug (i, way) values (0, st_geometryn(multi, 1));
       raise notice 'bends[i]: %', st_astext(bends[i]);
       raise notice 'bends[j]: %', st_astext(bends[j]);
       raise notice 'a: %, b: %', st_astext(a), st_astext(b);