On Polygonal Paths with Bounded Discrete-Curvature: The Inflection-Free Case - Discrete and Computational Geometry and Graphs - page 54

The proof of Theorem 2 proceeds by induction on the number of discrete circular
arcs in the path. Since the basis of the induction is obvious, it suces to prove
the following:
Lemma 4. Any dcc-path, joining two specified endpoint configurations, that
consists of three non-degenerate discrete circular arcs of the same orientation,
joined by (possibly degenerate) bridges, can be replaced by a shorter path, joining
the same endpoint configurations, that consists of at most two non-degenerate
discrete circular arcs.
Proof. Suppose we are given a sub-path consisting of three non-degenerate dis-
crete circular arcs joined by two (possibly degenerate) line segments that we
refer to as the first and second bridge. We will refer to the circles supporting the
three arcs as the first, second and third circles (coloured blue, red and green,
respectively, in all of our figures). We consider several cases depending on the
nature of the two bridges (degenerate or not) and the total turn of the second arc
(essentially whether or not it exceeds ˀ ). With only one explicitly noted excep-
tion, we use one of two continuous shortening transformations both of which
involve moving all (or most) of the vertices on the second arc while keeping the
other arcs fixed: (i) a pivot rotates all (except possibly the opposite endpoint)
of the vertices on the second arc about one of the arc endpoints; and (ii) a
slide translates all (except possibly the opposite endpoint) of the vertices on the
second arc along one of the non-degenerate bridges. Since both transformations
move the second arc in a rigid fashion, we need only consider vertices in the
neighbourhood of the two bridges to confirm that the transformations preserve
the dcc-path property.
To simplify the analysis that follows we will assume throughout that if a
transformation leads to a co-linearity event : one of the bridges becomes co-linear
with one of its adjacent edges (equivalently, the turn at some bridge endpoint
becomes zero), then we will stop the transformation at this point and combine
the two co-linear edges into a new bridge. Obviously, this results in a simpler path
(with one fewer edge) and a possible degeneration of one of the discrete circular
arcs. With this exception, all of our transformations terminate with either (i) the
second arc becoming co-circular with the first or third (a co-circularity event ),
in which case a bridge has been eliminated, (ii) a formerly non-degenerate bridge
becoming degenerate (a bridge degeneration event ), or (iii) a bridge intersecting
one of its associated circles in a chord of length d ʸ (a maximal chord event ). In
the second event, the resulting path is simpler in the sense that a path with a
narrow second arc (total turn less than ˀ ) gets measurably narrower, and one
with a wide second arc (total turn greater than ˀ ) gets measurably wider. The
third event is treated differently, depending on the intersection of the bridge
with it second associated circle. In all cases, the transformations are easily seen
to not only shorten the path, they also arguably leave it in a form that is simpler
than it was to start, from which it immediately follows that the full reduction
consists of only finitely many transformation steps.