explain self-crossing:

mj-msc.tex

@@ -20,6 +20,7 @@
 \usepackage{float}
 \usepackage{tikz}
 \usepackage{minted}
+\usepackage{fancyvrb}
 \input{version.inc}
 \input{vars.inc}
 
@@ -229,6 +230,8 @@ This section defines vocabulary and terms as defined in the rest of the paper.
 
     \item[Baseline] is a line between bend's first and last vertex.
 
+    \item[Sum of inner angles] TBD.
+
 \end{description}
 
 \subsection{Radians and Degrees}
@@ -421,12 +424,36 @@ following the rules of the article.
     \hfill
     \begin{subfigure}[b]{.4\textwidth}
         \includegraphics[width=\textwidth]{fig6-self-crossing-after}
-        \caption{Bend's baseline is crossing another bend}
+        \caption{Self-crossing removed}
     \end{subfigure}
     \caption{Originally Figure 6: self-line crossing}
     \label{fig:fig6-self-crossing}
 \end{figure}
 
+The self-line-crossing may happen not by the neighboring bend, but by any other
+bend in the line. For example, the baseline of bend A<->B may cross different
+bends in between, as depicted in figure~\onpage{fig:ascii-selfcross}.
+
+\begin{figure}[h]
+    \centering
+    \begin{BVerbatim}
+    \  \
+    B\ |     _       __
+     | |    / \     /  \
+     | |___/   \___/A  |
+     \_________________|
+    \end{BVerbatim}
+    \caption{A baseline crossing non-neighboring in-between bends}
+    \label{fig:ascii-selfcross}
+\end{figure}
+
+Naively implemented, checking every bend with every bend is costs $O(n^2)$.
+
+It is possible to optimize this step and skip checking some of the bends. Only
+bends whose sum of inner angles is $\pi$ can ever self-cross. If the value is
+less than $\pi$, it cannot cross other bends. That way, only a fraction of
+bends need to be checked.
+
 \subsection{Attributes of a Single Bend}
 
 \subsection{Shape of a Bend}

wm.sql

@@ -99,7 +99,7 @@ $$ language plpgsql;
 --
 -- The text does not specify how many vertices can be "adjusted"; it can
 -- equally be one or many. This function is adjusting many, as long as the
--- commulative inflection angle small (see variable below).
+-- cumulative inflection angle small (see variable below).
 --
 -- The implementation could be significantly optimized to avoid `st_reverse`
 -- and array reversals, trading for complexity in fix_gentle_inflections1.
@@ -235,9 +235,7 @@ begin
     -- crosses an imaginary line of end-vertices
 
     -- go through each bend in the given line, and see if has a potential to
-    -- cross bends[i]. optimization: we care only about bends which beginning
-    -- and end start at different sides of the plane, separated by endpoints
-    -- of the vertex.
+    -- cross bends[i].
     j = 0;
     while j < array_length(bends, 1) loop
       j = j + 1;
@@ -251,17 +249,16 @@ begin
       continue when st_numgeometries(multi) = 2 and
         (st_contains(bends[j], a) or st_contains(bends[j], b));
 
-      -- stars are aligned, we are changing the bend
+      -- vertices, segments and stars are aligned, we are changing the bend
       mutated = true;
 
-      -- Sincere apologies to someone who will need to debug the block below.
-      -- To understand it, I suggest you take a pencil and paper, draw a
-      -- self-crossing bend (fig6 from the article works well), and figure out
-      -- what happens here, by hand.
+      -- To understand the block below, I suggest you take a pencil and paper,
+      -- draw a self-crossing bend (fig6 from the article works well), and
+      -- figure out what happens here, by hand.
       prev_length = array_length(bends, 1);
       if j < i then
         -- remove first vertex of the following bend, because the last
-        -- segment is always duplicated with the i-th bend.
+        -- segment is always duplicated with the i'th bend.
         bends[i+1] = st_removepoint(bends[i+1], 0);
         bends[j] = st_geometryn(multi, 1);
         bends[j] = st_setpoint(