wm/notes.txt

Definition of a bend
--------------------

Ends of the line should always be bends, otherwise not
all line vertices are covered by bends (definition elsewhere).

Gentle inflection at the end of the bend
----------------------------------------

The article does not specify how many vertices should be included when
calculating the end-of-bend inflection. We chose the iterative approach -- as
long as the angle is "right" and the distance is (greedily) decreasing, keep
going.

Self-line crossing when cutting a bend
--------------------------------------

The self-line-crossing may happen after a few bends have been skipped. E.g.
ends of A<->B cross the line, but "swallow" a few more in between:

     ,______
    /       \
    |___A   |    \  \
        \   |    B\ |             __
         \  |     | |            /  \
         /  |     | |___,---,___/A  |
        /   |     \_________________|
        \   |
         \  |    \  \
         /  /    B\ |     _       __
    ----/  /      | |    / \     /  \
   / ,____/       | |___/   \___/A  |
  / B|            \_________________|
     |



If a bend with 180+ deg sum of inflection angles is found, its line between
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.

Also, there is another way to remove self-crossing, without removing most of
the bend. E.g. from:

  \  /
  B\ |      __
   | |     /  \
   | |____/A  |
   \__________|

Instead of:

  \  /
   \/ A'
  B

To:

  \  \_
  B\   `-,_.__
   |       A' \
   |          |
   \__________|

But perhaps it doesn't look quite as natural. I will trust the original
article to do the right thing here and remove the bend altogether.

The Context of a Bend
---------------------

Similar bends:

> For example, if bend 1 has four unit areas and bend 2 has six unit areas, the
> average size is five units, and the normalized areas of bends 1 and 2 are
> 4/5=0.8 and 6/5=1.2, respectively.

My comment: everything until this sentence is clear. However, "unit areas" is
misleading: there is little reason to normalize areas, but leave the distances
intact (if we'd like to normalize areas, it would make sense to square-root
them).

Removing that removes changes the meaning of the sentence that **euclidean
distance** is normalized (the composite of the bend properties), rather than
a single component.