Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 34 (1993), No. 1, 77-85.

Partial Inflation of Closed Polygons in the Plane

Bernd Wegner

Fachbereich Mathematik, TU Berlin, Straße des 17. Juni 135, W-1000 Berlin 12

Abstract. Inflation for simply closed regular curves in the plane has been investigated first by S. A. Robertson [4] and studied in some more detail in [5]. It consists of an infinite iteration of reflections of parts of the curve at supporting double tangents, hopefully leading to a convex limit curve which has the same arc length as the original curve. The same procedure easily can be defined for simply closed polygons. It provides a special construction of chord-stretched versions of the given curve. The aim of this note is to show that the behaviour of inflation is more comfortable in the piecewise linear case. It will end after a finite number of steps. This gives a positive answer to a question posed by T. Kaluza [2]. Furthermore inflation may lead to some measure for the nonconvexity of a simply closed polygon.

1. Basic definitions and observations

Let {p_{i}| i = 1, …, n} be the vertex set of a simply closed polygon P in the Euclidean plane. Choose the labelling in such a way that the straight line segment \overline{p_{i}p_{i+1}} is a side of P for all i = 1, …, n. We shall consider all roman subscripts mod n in the future. Let α_{i} ∈ (0, π] be the angle enclosed by the directed line segments \overrightarrow{p_{i}p_{i−1}} and \overrightarrow{p_{i}p_{i+1}}. Clearly we may assume α_{i} ≠ π for all i = 1, …, n. Furthermore let d_{i} be the length of \overline{p_{i}p_{i+1}}.

The simply connected closed domain D(P) having the polygon P as its boundary has a compact hull C(P) which is bounded by a closed convex polygon C(P). The vertices of C(P) coincide with vertices of the original polygon P. A side of C(P) has two choices: it coincides with a side of P or it connects two vertices p_{i} and p_{i+k} of P such that the straight line p_{i} ∨ p_{i+k} is a supporting line of D(P) and the open polygonal subarc P(p_{i}, p_{i+k}) of P from p_{i} to p_{i+k} lies entirely in the interior of C(P). Let us call this latter kind of side of C(P) a bridge of P.

If we reflect P(p_{i}, p_{i+k}) at the line p_{i} ∨ p_{i+k} generated by this bridge, then the image of P(p_{i}, p_{i+k}) will fit together with the remaining closed polygonal subarc of P forming a new simply closed polygon again which we call a partial inflation I(P) of P. If this is

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1993 Heldermann Verlag, Berlin

done for all bridges of P simultaneously and can be repeated infinitely many times (see Figure 1), then we get a full inflation of P (see [4]). But generally the first or some later step of this procedure may produce self-intersections for the image as is shown by Figure 2. Therefore without making any further restrictions it only will make sense to consider partial inflations (see also [5]). In contrast to the differentiable case we shall see that there is no need to choose the bridges for the iterated partial inflations in a special way in order to obtain a convex polygon finally.

Figure 1: Fully inflatable hexagon  Figure 2: Overlapping partial inflations

If we define I(P) = P for a convex polygon P, then partial inflation may be repeated infinitely many times. For a sequence of partial inflations we denote by p_{i ν+1} the image of the vertex p_{iν} under the partial inflation I of I^{ν}(P) (where all partial inflations are denoted by the same symbol I), ν ∈ N, starting with p_{i0} := p_{i}, i = 1, …, n. The sides of I^{ν}(P) are given by s_{iν} = \overline{p_{iν}p_{i+1 ν}} and have length d_{i} again, i = 1, …, n. In particular we have

Observation 1.1 The side lengths of a simply closed polygon are invariant under partial inflation.

Furthermore the definition of a partial inflation implies immediately

Observation 1.2 The sequences D(I^{ν}(P)), C(I^{ν}(P)), area(D(I^{ν}(P))), area(C(I^{ν}(P))), ν ∈ N, are strictly monotonically increasing until ν possibly has reached some value ν_{0} ∈ N such that I^{ν_{0}}(P) is convex. If such a ν_{0} exists, then all these sequences stabilize at constant values.

Let α_{iν} denote the angles of I^{ν}(P) defined in the same way as the α_{i} introduced above. If a partial inflation is carried out with respect to a bridge with end points p_{iν} and p_{i+k ν}, then all angles of I^{ν}(P) will be preserved with exception of those at p_{iν} and p_{i+k ν}. An elementary geometric consideration shows that just these two angles increase, though they are forced to remain in the range (0, π]. Hence we have Observation 1.3 If I^{ν}(P) is inflated at the bridge with end points p_{iν} and p_{i+k ν}, then α_{i ν+1} > α_{iν}, α_{i+k ν+1} > α_{i+k ν} and α_{j ν+1} = α_{jν} for all j = 1, …, n, j ≠ i, j ≠ i + k.

2. The finiteness theorem

The aim of this section is to show that for every simply closed polygon P and every choice of iterated partial inflations we shall get a convex polygon after a suitable finite number of steps. This is in contrast to the differentiable case, where for every nonconvex simply closed C^{1}-curve an infinite number of carefully chosen partial inflations is needed to obtain a convex limit curve (see [5]).

Proposition 2.1 Let I^{ν}(P), ν ∈ N, be a sequence of partial inflations of the simply closed polygon P. Then there is some subsequence ν_{ρ}, ρ ∈ N, such that the sequence I^{ν_{ρ}}(P), ρ ∈ N, of polygons converges towards a limit polygon L(P). Furthermore we have

C(L(P)) = \overline{⋃_{ν∈N} C(I^{ν}(P))}.

Proof. Since the length of C(I^{ν}(P)) does not exceed that of I^{ν}(P) we get from the Observations 1 and 2 that C(I^{ν}(P)), ν ∈ N, is a monotonically increasing sequence of compact convex domains for which the lengths of the boundaries are bounded by that of P. Hence we get from the theory of convex bodies (see [3]) that

C := \overline{⋃_{ν∈N} C(I^{ν}(P))}

is a compact convex domain for which the length of the boundary is bounded by that of P again. This implies for every i = 1, …, n that the sequences {p_{iν}}_{ν∈N} remain in the compact set C. Hence there exist converging subsequences {p_{iν_{ρ}}}_{ρ∈N} having limits

q_{i} := lim_{ρ→∞} p_{iν_{ρ}} ∈ C, i = 1, …, n.

Let L(P) be the polygon generated by the vertices {q_{i}| i = 1, …, n}. Clearly it is the limit of the sequence I^{ν_{ρ}}(P), ρ ∈ N. From Observation 2 we conclude D(I^{ν}(P)) ⊂ D(L(P)) for all ν ∈ N which implies C(I^{ν}(P)) ⊂ C(L(P)) for all ν ∈ N, i.e.,

⋃_{ν∈N} C(I^{ν}(P)) ⊂ C(L(P)).

On the contrary it is easy to see that

D(L(P)) ⊂ \overline{⋃_{ν∈N} D(I^{ν}(P))} ⊂ \overline{⋃_{ν∈N} C(I^{ν}(P))}

which implies

C(L(P)) ⊂ \overline{⋃_{ν∈N} C(I^{ν}(P))}.

From both inclusions we get the proposed equality because C(L(P)) is closed.

Theorem 2.2 Let I^{ν}(P), ν ∈ N, be a sequence of partial inflations of the simply closed polygon P. Then there is some ν_{0} ∈ N such that I^{ν_{0}}(P) is convex.

Proof: Let β_{i} be the angle between \overrightarrow{q_{i}q_{i−1}} and \overrightarrow{q_{i}q_{i+1}} of the polygon L(P) established in Proposition 1. Then we get from the monotonicity condition in Observation 3 that

β_{i} = lim_{ρ→∞} α_{iν_{ρ}} = lim_{ν→∞} α_{iν} for all i = 1, …, n,

where β_{i} is approached from below. Hence for a given ε > 0 there is some m_{ε} ∈ N such that

α_{iν} ≤ β_{i} ≤ α_{iν} + ε for all ν ≥ m_{ε},

i = 1, …, n. Similarly Observation 2 implies that for a given η > 0 there is some l_{η} ∈ N such that

area(D(I^{ν}(P))) ≤ area(D(L(P))) ≤ area(D(I^{ν}(P))) + η for all ν ≥ l_{η}.

Applying Observation 2 again we conclude

0 ≤ area(D(I^{ν+1}(P))) − area(D(I^{ν}(P))) ≤ area(D(L(P))) − area(D(I^{ν}(P))) ≤ η

for all ν ≥ l_{η}. Let q_{i} be a vertex of L(P) which is located on the boundary of C(L(P)) such that the boundary polygon C(L(P)) has an angle γ_{i} < π at that vertex. Then we have 0 < β_{i} ≤ γ_{i} < π. Setting Δ := min{d_{i}| i = 1, …, n} > 0 we conclude with Observation 1 that the area A_{iν} of the triangle with the vertices p_{i−1 ν}, p_{iν} and p_{i+1 ν} can be estimated from below by the value Δ^{2} sin α_{iν} > 0. Furthermore 0 < β_{i} < π implies Δ^{2} sin β_{i} > 0. Then the relations between the α_{iν} and the β_{i} established above imply that there is some k_{i} ∈ N such that for all ν ≥ k_{i}

A_{iν} ≥ Δ^{2} sin α_{iν} > \frac{1}{2}Δ^{2} sin β_{i} > 0

Hence if p_{iν} were moved by the corresponding partial inflation of the given sequence for some ν ≥ k_{i}, then the area of D(I^{ν}(P)) will increase at least by 2A_{iν} > Δ^{2} sin β_{i} > 0. Setting 0 < η < Δ^{2} sin β_{i} in the area estimates given above, this shows that p_{iν} will be stable under all partial inflations of the given sequence which have a number higher than ν_{i} := max{k_{i}, l_{η}}. But then q_{i} = lim_{ρ→∞} p_{iν_{ρ}} implies q_{i} = p_{iν} for all ν ≥ ν_{i}.

This argument applies to all of the (finitely many) extremal points of C(L(P)). Hence there is some τ ∈ N such that these points are vertices of I^{τ} (P). This implies

C(L(P)) ⊂ C(I^{τ} (P)).

Applying Proposition 1 we conclude the equality of these sets. Now, if we assume that I^{τ} (P) is not located on the boundary of C(L(P)), then I^{τ} (P) must have at least one bridge. All these bridges of I^{τ} (P) must be contained in the boundary of C(L(P)). Hence a partial inflation with respect to one of these bridges will lead to a part of I^{τ+1}(P) which is not contained in C(L(P)). This is a contradiction to Proposition 1 showing that our assumption is wrong. Thus we have proved that I^{τ} (P) is convex.

During the preparation of this paper P. Braß brought to our attention that Theorem 2.2 gives an affirmative answer to a problem posed by T. Kaluza in [2]. In addition to this Kaluza asked, if there can be found an upper bound for the number of inflations needed to convexify a polygon depending on the number of the vertices only. This question will be answered negatively in the next section.

Remark 2.3 Let c_{1}, c_{2} : S^{1} ⟶ E^{2} be two rectifiable simply closed oriented plane curves of the same length l. An orientation preserving map σ : S^{1} ⟶ S^{1} is called a chordstretching map from c_{1} to c_{2} if for all pairs s, t ∈ S^{1} the positively oriented arc of c_{1} from c_{1}(s) to c_{1}(t) and that of c_{2} from c_{2}(σ(s)) to c_{2}(σ(t)) have the same arc lengths, but the lengths of the corresponding chords are increasing, i.e.

∥c_{1}(s) − c_{1}(t)∥ ≤ ∥c_{2}(σ(s)) − c_{2}(σ(t))∥.

In this case c_{2} is called a chord-stretched version of c_{1}. According to G. Choquet [1] and G. T. Sallee [6] there exists a convex chord-stretched version for every rectifiable simply closed plane curve.

Obviously every partial inflation of a simply closed smooth curve ([5]) or a closed polygon in the plane transforms the given curve into a chord-stretched version. Hence iterated partial inflation gives a simple procedure to obtain a convex chord-stretched version of the given curve finally. This also shows that generally there may be several convex chord-stretched versions of a given curve. Having reached such a convex chord-stretched version no further chord-stretching will be possible (up to congruence), as has been shown recently by J. Strantzen and J. Brooks in [7]. In the case of convex polygons this can be seen very easily.

3. Inflational complexities

The purpose of this section is to associate certain natural numbers to a simply closed polygon P giving a measure for its complexity with respect to partial inflation, and to study some elementary properties of these numbers. In view of Theorem 1 we can define the maximal inflational complexity c_{max}(P) and the minimal inflational complexity c_{min}(P) of P as follows:

c_{max}(P) := max{ν ∈ N | I^{ν}(P) is convex for some sequence of partial inflations,
but not I^{ν−1}(P)},
c_{min}(P) := min{ν ∈ N | I^{ν}(P) is convex for some sequence of partial inflations,
but not I^{ν−1}(P)}.

If c_{min}(P) = c_{max}(P), then we just speak of the inflational complexity of P. Clearly the inflational complexity of a convex polygon vanishes.

Example 3.1 The examples of heptagons exhibited in Figures 3a-d show that the maximal inflational complexity and the minimal inflational complexity may differ from each other for a given polygon. They also demonstrate that the strategies developed in [5] for choosing suitable bridges generally do not lead to the minimal (or maximal) inflational complexity automatically. For example, choosing bridges enclosing maximal areas gives a sequence of inflations realizing c_{min} for one heptagon, while the same stategy leads to a sequence realizing c_{max} for the other one.

Figure 3a: _{c_{max}(P)=4}  Figure 3b: _{c_{min}(P)=2}

Figure 3c: _{c_{max}(Q)=3}  Figure 3d: _{c_{min}(Q)=2}

Proposition 3.2 For every k ∈ N there exists a simply closed quadrangle P_{k} such that

c_{min}(P_{k}) = c_{max}(P_{k}) = k.

Figure 4

Proof. Since a simply closed quadrangle P can have at most one bridge we immediately get c_{min}(P_{k}) = c_{max}(P_{k}). Assume that P has a bridge between p_{4} and p_{2}. Let η_{0} denote the angle between \overrightarrow{p_{4}p_{2}} and \overrightarrow{p_{4}p_{1}} (see Figure 4). Then, if we assume additionally 2d_{1} ≤ Δ where Δ = min{d_{2}, d_{4}}, we get the estimate

0 ≤ tan η_{0} ≤ \frac{d_{1}}{Δ − d_{1}}

using the notations introduced in Section 1. After the first partial inflation we have p_{i1} = p_{i} for i = 1, 2, 3, α_{11} = α_{1} and

α_{21} = \{ \begin{matrix}α̃_{21} & α̃_{21} ≤ π \\ π − α̃_{21} & α̃_{21} > π \end{matrix}

where α̃_{21} := 2α_{1} + 2η_{0} − α_{2}. The obvious estimate

α̃_{21} < 2α_{1} + 2η_{0}

shows, that the condition 2α_{1} + 2η_{0} < π is sufficient for I(P) to possess a bridge again. This bridge will be between p_{11} and p_{31}. Assuming that this is the case we can consider I^{2}(P). Now p_{21} will play the role of p_{1} in our initial investigations and p_{11} will play the role of p_{2}. Denoting by η_{1} the angle between \overrightarrow{p_{31}p_{21}} and \overrightarrow{p_{31}p_{11}} we get with Observation 1 the estimate

0 ≤ tan η_{1} ≤ \frac{d_{1}}{Δ − d_{1}}

and using the same notation as above

α̃_{12} := 2α_{21} + 2η_{1} − α_{11} < 2α_{21} + 2η_{1} < 4α_{1} + 4η_{0} + 2η_{1}.

This shows, that the condition 4α_{1} + 4η_{0} + 2η_{1} < π is sufficient for I^{2}(P) to possess a bridge. This bridge will be between p_{42} and p_{22}. Continuing with the same argument we easily see that a sufficient condition for P to survive k partial inflations without becoming convex will be

2^{k}α_{1} + ∑_{i=0}^{k−1} 2^{k−i}η_{i} < π,

where the η_{i} are defined in the obvious manner satisfying the uniform estimate

0 ≤ tan η_{i} ≤ \frac{d_{1}}{Δ − d_{1}} , i = 1, …, k − 1.

For a given ε > 0 we can find a nonconvex simply closed quadrangle P_{ε} satifying α_{1} < ε and d_{1} ≤ (Δ − d_{1}) tan ε. The latter condition implies 0 ≤ η_{i} ≤ ε as far as the η_{i} are defined. But then choosing

0 < ε_{k} < \frac{π}{2^{k}(k + 1)}

the sufficient condition established above shows that I^{k}(P_{ε_{k}}) will not be convex. Hence the inflational complexity c of P_{ε_{k}} is greater than k. This shows that I^{c−k}(P_{ε_{k}}) satisfies the requirements of our proposition.

Theorem 3.3 For every n ∈ N, n > 3, and every k ∈ N there exists a simply closed n-gon P in the plane such that

c_{min}(P) = c_{max}(P) = k.

Proof. We take the quadrangle P_{k} which has been constructed in the proof of Proposition 2. It has one side which remains unchanged under all partial inflations of P_{k}. Then we replace this side by a suitable convex polygonal arc which is so flat that it also remains unchanged under these partial inflations. Clearly we may assume that this arc has n − 2 vertices. This will give the example we need for our theorem.

Remark 3.4 It is easy to show that for every nonconvex quadrangle Q there is an affine transformation A such that the inflational complexity of A(Q) equals to 1. Hence we may consider the following affine invariants for a simply closed polygon P:

a_{min} := min{c_{min}(A(P)) | A is a nonsingular affine transformation},
a_{max} := min{c_{max}(A(P)) | A is a nonsingular affine transformation}.

For quadrangles these numbers will be equal and attain the values 0 or 1 only, depending on the convexity of the quadrangle. For a pentagon these numbers also can be shown to coincide. Their range is the set {0, 1, 2} in this case. There are the following open problems: Do a_{max} and a_{min} coincide for every n-gon? Is there always a bound for a_{max} depending on the number of the vertices only? Determine the shape of the n-gons realizing a_{min} resp. a_{max} as their inflational complexity.

Remark 3.5 In contrast to the differentiable case there is a construction for polygons which may be considered as an inverse to partial inflation. We may call it deflation. The procedure goes as follows: Assume that we have two nonconsecutive vertices p_{i} and p_{i+k} of the polygon P such that p_{i} ∨ p_{i+k} meets P in these vertices only without being a line of support, and that the polygonal subarc P(p_{i}, p_{i+k}) of P between p_{i} and p_{i+k} meets the reflection of the polygonal subarc P(p_{i+k}, p_{i}) at the line p_{i} ∨ p_{i+k} at its end points only. Then this reflection can be joined with P(p_{i}, p_{i+k}) to a simply closed polygon I^{−1}(P). We may call p_{i} ∨ p_{i+k} a cut of P and the polygon I^{−1}(P) a deflation of P. Obviously P can be reconstructed from I^{−1}(P) by a suitable partial inflation.

There are polygons which do not admit a cut. They have reached their final state of iterated deflation. For other polygons it will be interesting how such a final state looks like and after how many steps it may be reached. Generally this final state is not uniquely determined. Again we have two deflational complexities. It is conjectured that they are finite.

Remark 3.6 In view of Remark 2.3 the deflation of a closed polygon (as considered in the preceding remark) may be interpreted as a transition to a chord-shrinking version of the given polygon. To remain within the set of polygons under consideration, we have to require additionally that the chord-shrinking map preserves the line segments of the polygon and their lengths. Then the question may be studied if the final states of iterated deflations of a polygon are also final states under the procedure of (restricted) chord-shrinking.

References

[1] Choquet, G.: Variétés et corps convexes. Ass. Française pour l’Avancement des Sciences, 64ème Session 1945, Mathématiques, 91 - 93.

[2] Kaluza, T.: Problem 2: Konvexieren von Polygonen. Math. Semesterber., Neue Folge 28 (1981), 153 - 154.

[3] Leichtweiß, K.: Konvexe Mengen. (Berlin et al.: Springer-Verlag, 1980).

[4] Robertson, S. A.: Inflation of plane curves. Geometry and Topology of Submanifolds, III, 264 - 275 (Singapore et al.: World Scientific, 1991).

[5] Robertson, S. A.; Wegner, B.: Partial inflation of plane curves. (to appear in Proc. Conf. Intuitive Geometry, Coll. of the Janos Bolyai Soc.).

[6] Sallee, G. T.: Stretching chords of space curves. Geom. Dedicata 2 (1973), 311 - 315.

[7] Strantzen, J.; Brooks, J.: A chord stretching map of a convex loop is an isometry. Geom. Dedicata 41 (1992), 51 - 62.