Affine Properties of Convex EqualArea Polygons
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Abstract
In this paper we discuss some affine properties of convex equalarea polygons, which are convex polygons such that all triangles formed by three consecutive vertices have the same area. Besides being able to approximate closed convex smooth curves almost uniformly with respect to affine length, convex equalarea polygons admit natural definitions of the usual affine differential geometry concepts, like affine normal and affine curvature. These definitions lead to discrete analogous to the sixvertex theorem and an affine isoperimetric inequality. One can also define discrete counterparts of the affine evolute, parallels and the affine distance symmetry set preserving many of the properties valid for smooth curves.
Keywords
Equalarea polygons Discrete sixvertex theorem Discrete isoperimetric inequality Discrete affine evolute Discrete affine distance symmetry set1 Introduction
A convex equalarea polygon is a simple polygon bounding a convex planar domain such that all triangles formed by three consecutive vertices have the same area [10]. As a consequence, each four consecutive vertices form a trapezoid. It is worthwhile to mention that the class of equalarea polygons includes also nonconvex polygons and appears as a set of maximizers of some area functionals (see [7]). Affinely regular polygons are affine transforms of convex regular polygons and thus are convex equalarea polygons. The affinely regular polygons appear as maximizers of area functionals and in many other different contexts (see [2]).
We begin with the study of the space \({\mathcal{T}}_{n}\) of convex equal area polygons with n vertices, which can be thought of as a subset of the (2n)dimensional affine space. The space \({\mathcal{T}}_{n}\) has dimension (n−5) and in Sect. 3 we describe in detail its structure. We also consider the problem of approximating a closed convex curve by polygon in \({\mathcal{T}}_{n}\). We show how to construct a convex equalarea ngon whose vertices, for n large, are close to a uniform distribution of points on the curve with respect to affine arclength.
For convex equalarea polygons, we define in Sect. 4 the affine tangent vector, affine normal vector at each vertex and the affine curvature μ at each edge in a straightforward way. In this context, we define a sextactic edge as an edge where Δμ is changing sign. Then we show that there are at least six sextactic edges in a convex equalarea polygon, which is a discrete analog of the wellknown sixvertex theorem.
In Sect. 5 we define the affine evolute and the λparallel curves for convex equalarea polygons. As in the smooth case, the sixvertex theorem may be rephrased as the existence of at least six cusps on the affine evolute [13, 16]. Besides, we show that cusps of the parallels belong to the evolute, as in the smooth case. The set of selfintersections of the parallels is called the affine distance symmetry set (ADSS) [5]. Again, we prove here two properties analogous to the smooth case: cusps of the ADSS occur at points of the affine evolute and endpoints of the ADSS are exactly the cusps of the evolute [6].
Finally in Sect. 6 we prove an isoperimetric inequality by using a Minkowski mixed area inequality for parallel polygons. This inequality has a wellknown smooth counterpart saying that the integral of the affine curvature with respect to affine arclength is at most \(\frac{L^{2}}{2A}\), where L is the affine perimeter of the convex curve and A is the area that it bounds, with equality holding only for ellipses (see [14]). In the convex equalarea polygon case, the equality holds only for affinely regular polygons.
We have used the free software GeoGebra [4] for all figures and many experiments during the preparation of the paper. Applets of some of these experiments are available at [17]. We would like to thank the GeoGebra team for this excellent mathematical tool.
2 Review of Some Concepts of Affine Geometry of Convex Curves
In this section we review the basic affine differential concepts associated with convex curves. Since in this paper we deal with a discrete model, these concepts are not strictly necessary for understanding the paper. Nevertheless, they are the inspiration for most definitions and results.
Affine ArcLength, Tangent, Normal and Curvature
Affine Evolute, Parallels and Affine Distance Symmetry Set
The envelope of the family of affine normal lines is called the affine evolute of γ. For a fixed λ∈ℝ, the set of points of the form γ(s)+λγ″(s), s∈[a,b], is called the λparallel. The ADSS is the locus of selfintersection of parallels [5, 6].
SixVertex Theorem
A wellknown theorem of global affine planar geometry says that any convex closed planar curve must have at least six vertices [1]. Equivalently, the affine evolute must have at least six cusps.
An Affine Isoperimetric Inequality
3 Convex EqualArea Polygons
3.1 Basic Definitions
The following lemma will be useful:
Lemma 3.1
Consider a trapezoid ABCD with AD∥BC. Then A and D belong to a hyperbola whose asymptotes are the lines parallel to AB and CD passing through C and B, respectively. As a consequence, in a convex equalarea polygon, P _{ i } and P _{ i+3} belong to a hyperbola whose asymptotes are P _{ i−1} P _{ i+2} and P _{ i+1} P _{ i+4}, for any 1≤i≤n.
Proof
3.2 The Space of Convex EqualArea Polygons
Denote by \({\mathcal{T}}_{n}\) the space of convex equalarea ngons modulo affine equivalence. By taking the coordinates of the vertices of the polygon, we can consider \({\mathcal{T}}_{n}\subset\mathbb{R}^{2n}\). For a polygon to be equalarea, conditions (3.1) must be satisfied. Since these conditions are cyclic, one of them is redundant, and thus they amount to (n−1) equations. Also, the affine group is 6dimensional and thus we expect \({\mathcal{T}}_{n}\) to be a (n−5)dimensional space. In fact, this is proved in [10] for the space of (not necessarily convex) equalarea polygons.
In this section we study the space \({\mathcal{T}}_{n}\) in some detail. We prove here the following proposition:
Proposition 3.2
There exists an open set U⊂ℝ^{ n−5} and W⊂∂U with injective smooth maps \(\phi_{1}:U\cup W\to{\mathcal{T}}_{n}\), \(\phi_{2}:U\cup W\to{\mathcal{T}}_{n}\) such that \(\mathrm{Im}(\phi_{1})\cup \mathrm{Im}(\phi_{2})={\mathcal{T}}_{n}\) and ϕ _{1}_{ W }=ϕ _{2}_{ W }. Moreover, there exists an open set U _{0}⊂U, \(\overline{U_{0}}\cap W=\emptyset\), such that \(\phi_{1}_{U_{0}}=\phi_{2}_{U_{0}}\).
Proof
It is now clear how to define the smooth maps ϕ _{1} and ϕ _{2}. For μ∈U _{0}∪U _{1}∪W, ϕ _{1}=ϕ _{2} corresponds to the unique convex equalarea polygon associated with μ. For μ∈U _{2}, let ϕ _{1} and ϕ _{2} correspond to the two convex equalarea polygons associated with μ. To conclude the proof of the proposition, take U=U _{0}∪U _{1}∪U _{2}. □
Next lemma says that the affinely regular ngon corresponds to μ∈U:
Lemma 3.3

For n=5,6,7,8, \(\mu\in\overline{U_{0}}\).

For n≥9, the ratio R(μ) defined by Eq. (3.3) is strictly greater than 4 and thus \(\mu\in U\overline{U_{0}}\).
Proof
We conjecture that the set \({\mathcal{T}}_{n}\) is connected, as suggested by experiments done with GeoGebra.
3.3 Approximating a Convex Curve by a Convex EqualArea Polygon
In this section we propose an algorithm for approximating a closed convex curve by convex equalarea polygons. Although the resulting polygons may be neither inscribed nor circumscribed, they are asymptotically close to the inscribed polygon whose vertices are equally spaced with respect to the affine arclength of the curve. For asymptotic results concerning optimal approximations of convex curves by inscribed or circumscribed polygons, we refer to [11] and [12]. For surveys on approximation of convex curves by polygons, we refer to [8] and [9].
Given a closed convex planar curve γ, we consider the following algorithm: Fix any three points P _{1},P _{2},P _{3} in a positive orientation at γ such that \(d(P_{1},P_{2})=d(P_{2},P_{3})=\frac{L}{n}\), where d(⋅,⋅) denotes affine distance along γ and L is the affine perimeter of γ. Then P _{4} is obtained as the intersection of a parallel to P _{2} P _{3} at P _{1} with γ. Proceeding in this way we obtain P _{ k }, k=1,…,m, at the curve, and we continue until condition (2) in the proof of Proposition 3.2 holds with R(μ)<4.
Denote by o(s ^{ k }) any quantity satisfying \(\lim_{s\to0}\frac{o(s^{k})}{s^{k}}=0\).
Lemma 3.4
Assume that d(P _{ i−1},P _{ i })=s and d(P _{ i },P _{ i+1})=s+o(s ^{2}). Then d(P _{ i+1},P _{ i+2})=s+o(s ^{2}).
Proof
Corollary 3.5
Proof
Corollary 3.5 says that, for n large, the convex equalarea polygon constructed above gives a sampling of the curve γ that is approximately uniform with respect to affine arclength.
It is natural to ask if, given a closed convex curve γ, a point P _{1} on it and n, there exists a convex equalarea ngon inscribed in γ with P _{1} as a vertex. We believe that this is true for odd n, but not for even n. We plan to consider this question in a future work.
4 Discrete Planar Affine Geometry and the Six Sextactic Edges Theorem
4.1 Affine Curvature and Sextactic Edges
4.2 The Six Sextactic Edges Theorem
In this section we prove a discrete analog of the sixvertex theorem [1, 13]. We begin with the following lemma:
Lemma 4.1
Proof
Since a rotation of the plane preserves the convex equalarea property we do not need to consider the term in xy, and so the lemma is proved. □
Theorem 4.2
Any convex equalarea ngon, n≥6, admits at least six sextactic edges.
Proof
Suppose by contradiction that Δμ(i) changes sign four times or less. Then there exists a quadratic function q(x,y) that is positive in a region that contains the vertices where Δμ(i) is positive and negative in a region that contains the vertices where Δμ(i) is negative. In fact, if there are no changes of sign, just take q=constant. If there are just two changes of sign, take q to be a linear function whose zero line divides the vertices with positive Δμ from the vertices with negative Δμ. In the case of four changes of sign, consider lines l _{1} and l _{2} passing through edges where Δμ changes sign and whose intersection occurs inside the polygon. Then take q as the product of linear functions whose zero lines are l _{1} and l _{2}. The existence of such a quadratic function contradicts Lemma 4.1 and so the theorem is proved. □
5 Affine Evolutes, Parallel Polygons and the Affine Distance Symmetry Set
5.1 Parallel Polygons and the Affine Evolute
Proposition 5.1
Every cusp of a parallel belongs to the affine evolute.
The following proposition suggests that affinely regular polygons are the discrete counterparts of ellipses:
Proposition 5.2
The affine evolute of a convex equalarea polygon reduces to a point if and only if the polygon is affinely regular.
Proof
5.2 Cusps of the Affine Evolute
Proposition 5.3
\(Q_{i+\frac{1}{2}}\) is a cusp of the affine evolute if and only if \(\mathbf{v}_{i+\frac{1}{2}}\) is a sextactic edge of the polygon.
Proof
Following [16], the normal lines P _{ i }(λ), λ∈ℝ, form an exact system, which means that the parallel polygons are closed. Then the discrete fourvertex theorem of [16] says that the affine evolute admits at least four cusps. The following corollary, which is a direct consequence of Proposition 5.3 and Theorem 4.2, says that, in the context of convex equalarea polygons, the affine evolute has at least six cusps.
Corollary 5.4
The affine evolute of a convex equalarea polygon has at least six cusps.
5.3 Affine Distance Symmetry Set
Proposition 5.5
\(M_{i+\frac{1}{2}}\) is an endpoint of the ADSS if and only if \(M_{i+\frac{1}{2}}\) is a cusp of the affine evolute.
Proof
\(M_{i+\frac{1}{2}}\) is an endpoint of the ADSS if it is the limit of cusps of the parallels P(λ) at the normal lines P _{ i }(λ), λ∈ℝ, and P _{ i+1}(λ), λ∈ℝ, with λ converging to \((\mu_{i+\frac{1}{2}})^{1}\). It is now easy to see that this occurs if and only if \(M_{i+\frac {1}{2}}\) is a cusp of the affine evolute. □
As a consequence of the above proposition and Corollary 5.4, we conclude that any ADSS has at least three branches.
6 An Affine Isoperimetric Inequality
Denote by A=A _{0} the area bounded by the convex equalarea ngon P. Assuming that \([\mathbf{v}_{i\frac{1}{2}},\mathbf {n}_{i}]=[\mathbf{v}_{i+\frac{1}{2}},\mathbf{n}_{i}]=1\), the affine perimeter is defined as L=n. The following inequality is a discrete counterpart of inequality (2.2).
Theorem 6.1
Notes
Acknowledgements
The first and second authors want to thank CNPq for financial support during the preparation of this manuscript.
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