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Case I: both bridges are non-degenerate.
We begin by considering the case where
both bridges are non-degenerate. As we shall see, if one or more of the bridges
is degenerate, a shortening transformation exists that will bring us back to this
case.
There are two sub-cases to consider. In the first sub-case the turn from the
first bridge edge to the second is less than or equal to
ˀ
(see Fig.
11
). First note
that if we slide the middle discrete arc (vertices
c
through
x
) along the first bridge
edge (taking
c
towards
b
) we maintain the discrete bounded curvature property
at
b
and
c
as long as
b
(respectively,
c
) lies outside the second circle (respectively,
first) circle (i.e. until the bridge
bc
becomes degenerate). Meanwhile, the discrete
bounded curvature property is maintained at the endpoints of the second bridge
edge (
xy
) as long as the predecessor (
r
) of the outer point (
y
) lies outside of
the third circle (because of the direction of the translation the successor of the
other bridge endpoint point (
x
) cannot enter the second circle). If this point
r
meets the third circle (a maximal chord event) while outside of the second circle,
then point
r
can replace
y
as the outer point of the second bridge (leading to a
shortening of the second discrete arc), and we can continue in Case I.
By symmetry the analogous properties hold if we slide the middle discrete
arc along the second bridge edge (taking
x
to
y
). Since both of these translations
serve to shorten the curve, we can assume that they have been done until either
or both of the bridges have degenerated (taking us to Case II or Case III below)
or we are left with unresolved maximal chord events on both bridges. In the
latter case, the successor point of
b
(illustrated by
p
in Fig.
12
) must lie on the
first circle, the predecessor point of
y
(illustrated as
r
) must lie on the third
circle, and both
p
and
r
must lie inside the second circle.
c
c
q
a
b
q
a
b
a
b
d
d
c
p
p
d
w
w
w
x
x
x
r
r
s
s
y
y
y
z
z
z
Fig. 11.
Fig. 12.
Fig. 13.
|
must be at least the distance from
p
to the point
q
on the second circle intersected
by the line through
p
with the slope of the second bridge edge, or (ii) the distance
|
To deal with this last situation, we observe that either (i) the distance
|
rx
must be at least the distance from
r
to the point
s
on the second circle
intersected by the line through
r
with the slope of the first bridge edge. (It
is easily confirmed that if neither of these hold, we get a contradiction of the
pc
|
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