In general, the problems polygon
models are used to represent are real-world applications, and thus
it is appropriate to consider motions between configurations in 3 dimensions.
To this end, we introduce the notion of motion in terms of * pivots*
, where a polygonal chain of polygon P in R^{d} is reflected
across a Hyperplane H which supports the convex hull of P and contains
at least two vertices of P. Since we are considering the case where d=3
(i.e. R^{3} or 3D), the H is simply a halfplane which contains 2
points on the convex hull of P:

__Theorem 2__: Any planar
convex polygon P can be reconfigured into any other planar convex
polygon P' using pivots.

**Proof****:**
Locate a quadrangle V_{1}V_{2} V_{3}
V_{ 4} where the vertices are labelled -, +, -, +, respectively,
in P. Then, step through the following motions:

**Lemma 4:** Let V_{1}
V_{ 2}V_{3}V_{4} be a planar convex quadrangle.
After two pivots, suppose the quadrangle is again planar, resulting
in a quadrangle, V_{1}V_{2} V_{3}V_{4}
. Then

< V_{2}V_{1} V_{4} will be at least
the original value of the expression |** <**V_{2} V_{
1} V_{3} - *<* V_{4} V_{1}
V_{ 3} |.

**Proof:** If both pivots are on the diagonal
V_{2}V_{ 4}, then the angle at V_{ 1}
has not changed, and so we must only consider the remaining cases,
where the first (or both) pivot is on V_{1}V_{3}, and
where the first pivot is on V_{ 2}V_{4} and the second
is on V_{ 1} V_{ 3} .

In the former case, the pivot must occur on a planar quadrangle, with the vertex of the quadrangle on the moving subpolygon--say it is V_{ 2}-- tracing a circle around V_{1}V_{
3} as it rotates, and thus the angle V_{2}V_{1}
V_{ 3} remains constant throughout the pivot. Since
V_{ 4} remains fixed during the motion, the angle V_{4}
V_{ 1} V_{3} also remains constant, and so the
angle after the pivot must be at least the difference |**<**
V_{ 2} V_{ 1}V_{ 3} - * <* V_{
4} V_{ 1} V_{ 3}|.

In the latter case, the two pivots must bring the quadrangle into a planar configuration, and the distance V_{2}V_{
4} is increasing during the motion, so by the law of sines,
< V_{2}V_{1}V_{4} must also have increased,
and so is greater than |**<**V_{2}V_{1} V_{
3} - *<* V_{4}V_{1}V_{3}
|.

This leads to the main theorem for this part:

**Theorem 3:** There exist
polygons which require arbitrarily many pivots
to achieve a goal configuration.

**Proof:** Consider
the case of a pair of pivots that result in a planar non-intersecting
quadrangle after the motions depicted in the figure below. The parallelogram
has the property that <V_{1} can be made arbitrarily close
to zero, and since it is not a rhombus,

Also, since for small angles <x, sin x__~__
x, and so as <V_{1} decreases, for every two pivots <V_{
1} is only reduced to an angle <V_{1}^{'}
where

- Pivot on V
_{1}V_{3}, rotating the entire subpolygon until V_{2}is over V_{1}V_{3}. - Pivot on V
_{2}V_{4}to bring the quadrangle into a planar configuration. - Pivot on V
_{4}V_{1}to bring the subpolygon defined by those two vertices into the plane of the quadrangle. - Pivot on the remaining three edges to bring each of their subpolygons
into the plane of the quadrangle..

<

In the former case, the pivot must occur on a planar quadrangle, with the vertex of the quadrangle on the moving subpolygon--say it is V

In the latter case, the two pivots must bring the quadrangle into a planar configuration, and the distance V

This leads to the main theorem for this part:

Also, since for small angles <x, sin x

And so since <V_{1}approaches
but does not attain zero (since we disallow the "flat" self-intersecting
configuration that would result), a goal configuration with an arbitrarily
low <V_{ 1}can be chosen, so that an arbitrary number
of pivots are needed to reconfigure the polygon. This is easily extended
for arbitrary pivots by considering any arbitrary pivot as a pair of
pivots on the same diagonal, where the first pivot brings the quadrangle
into a planar non-intersecting position and the second produces the original
pivot desired.

So it is observed that there does indeed exist a motion between any two planar convex polygons using only pivots, but this motion is not bounded by the number of edges of the polygon-- it can be achieved with arbitrarily many pivots if we choose a configuration as in Theorem 3.