CHANGES.txt

MAJOR CHANGES introduced in the version by Günter, 24 Sep 2019
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embedding -> drawing (It is the simpler word, and embedding has the
connotation of being injective.)

IS THERE A REFERENCE for the "1998 Watanabe" question about untangling?

I removed prec_Y (in "proof outline") and replaced it by something
simpler. (Or can we just say that (0,a) is *below* (0,b) without
defining anything?)  I am also not sure if the so-called "partial
order" prec_C in "definitions" is ever used.

I commented out the second paragraph of "Proof Outline", about the
easy case where no edge crosses Y, for two reasons.
 a) A whole paragraph of arguments for dismissing an easy case is too
 much in an *outline*.
 b) Tutte's [23] works only for strictly convex. We would have to
 say something more, somewhere in the paper. This is not actually
 the outline of our actual proof.

I have reformulated the definition of drawing (previously
embedding) in Sec.2. Gamma has been used only in a few places and it was easy
to eliminate those.
Please tell me if you see some problems where this might cause
confusion in some OTHER places.
In particular, I have changed the frightening sentence 
"For a given F\'ary embedding $G$, we often address the problem of
..." to something simpler.

admissible -> proper good. (in accordance with the original definition).

Definition moved together with Thm.2. (first appearance of this
concept).

Thm 5 (restatement of Thm.2) moved to a separate subsection.
PROPOSAL: Move it to Thm.2, or move it closer to the place where it is used?
(Apparently it is used 2x in the whole paper).

Reformulation of Theorem 6.
The outer face is now an afterthought.
Similarly Theorem 7.

Removed the fully specified ordering around a vertex (previous Fig.3).
Moved the figure backward, now Fig.6
(that e1 and e_m are boundary edges is part of the theorem statement.)

The boundary equations are shown as (7), so that we can refer to them.

Changed >epsilon into >=epsilon in the eps-strong constraints.

Lemma 2. Proof rewritten.

Section 3.5 "A Parametric Family of Linear Systems"
Rewritten, including proof of Lemma 5. It is now shorter.
I hope I did not delete anything essential.

changed notation from A_t to A^t, y^t b^t etc. but kept
s(t) to emphasize the functional dependency.

I added a picture for Claim 1 in the "Base Case". p.25.