Regularizing transformations of polygons

We start with a generic n-gon Q0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_0$$\end{document} with vertices qj,0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{j,0}$$\end{document} (j=0,⋯,n-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j = 0, \dots , n-1$$\end{document}) in the d-dimensional Euclidean space Ed\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {E}}^d$$\end{document}. Additionally, m+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m+1$$\end{document} real numbers u0,…,um∈R(m<n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_0, \ldots , u_m \in {\mathbb {R}} \, (m < n)$$\end{document} with ∑μ=0muμ=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{\mu = 0}^m u_\mu = 1$$\end{document} are given. From these initial data we iteratively define generations of n-gons Qk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_k$$\end{document} in Ed\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {E}}^d$$\end{document} for k∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \in {\mathbb {N}}$$\end{document} with vertices qj,k:=∑μ=0muμqj+μ,k-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{j,k}:= \sum _{\mu = 0}^m u_\mu \, q_{j+\mu , k-1}$$\end{document}. We can show that this affine iteration generally regularizes in an affine sense.


Introduction
, Ziv [7], Nicollier [2] and Donisi et al. [1] studied geometric iteration processes starting with a generic n-gon Q 0 in E 2 . They use homotheties to construct vertices of a next generation polygon Q 1 . Reiterating this process creates a series of generations Q k . This iteration, in general, has a regularizing effect on the polygon. Surprisingly, the result for n-gons in the plane E 2 presented by Roeschel in [5] is also valid for n-gons in higher dimensions. In [5] the proof for E 2 is based on the fact that the space of planar n-gons is spanned by the planar prototype n-gons of E 2 . As this does not hold for higher dimensions the proof for E d with d ≥ 3 demands another approach with different arguments. We prove an affine regularization theorem: these iterations in higher dimensions also deliver generations Q k approaching the affine shape of regular planar polygons.

The spatial affine iteration
We use vectors in R d to describe points of the d-dimensional Euclidean space E d (d > 2) with respect to a Cartesian coordinate frame {O; x 1 , . . . , x d }. We start with some spatial n-gon Q 0 ⊂ E d with vertices {q 0,0 , q 1,0 , . . . , q n−1,0 } On the other hand in an m-dimensional affine space R m (0 < m < n) with a simplex S := {a 0 , . . . , a m } we choose a reference point z * with respect to S: Let z * := m μ=0 u μ a μ be given by its barycentric coordinates (u 0 , . . . , u m ) ∈ R 1×(m+1) with m μ=0 u μ = 1. Let α j,1 be the affine mappings from the ordered reference simplex vertex set S to ordered sets of m consecutive vertices q j,0 , . . . , q j+m,0 of Q 0 (j ∈ J := {0, . . . , n − 1}; first index mod n). Each of these n affine mappings is applied to the reference point z * ; this way we get n image points q j,1 := α j,1 (z * ) = m μ=1 u μ q j+μ−1,0 which form a new n-gon Q 1 called the generation 1 polygon. The same process can now be applied, in turn, to the polygon Q 1 with the same reference simplex S and the same reference point z * , creating a subsequent polygon Q 2 . Iteration yields a series of polygons. Q k := {q 0,k , . . . , q n−1,k } is the k th generation polygon with vertices The procedure is a d-dimensional generalisation of the geometric iteration presented in [5]. Figure 1 shows the first iteration step for an example with n = 8 and m = d = 3.

The iteration process
The nth complex roots of unity ∈ C shall be termed ζ j := exp(i 2jπ n ) = cos 2jπ n + i sin 2jπ n (j ∈ Z). We define the vectors and have P j · M = P j m μ=0 u μ ζ μ j and M · P t n−j = ( m μ=0 u μ ζ μ j ) P t n−j . Thus, the vectors P j and P t n−j (j ∈ J) are left and right eigenvectors of M . The corresponding eigenvalue is As (u 0 , . . . , u m ) ∈ R 1×(m+1) and ζ μ j = ζ μ n−j we have λ j = λ n−j for all j ∈ J\{0}.
We now regard two matrices out of C n×n L and R are symmetric and regular for n > 1 (see [3,5,6] and [7]). We have: L = R and L · R = I n,n with the n × n-unit matrix I n,n ; the matrices L and R are unitary n × n-matrices in C n×n . We have L · M · R = D(λ 0 , . . . , λ n−1 ) with the diagonal matrix D(λ 0 , . . . , λ n−1 ) ∈ C n×n containing the eigenvalues λ j of M as its elements in the main diagonal. This yields M = R·D(λ 0 , . . . , λ n−1 )·L and Then (3.5) yields ν=0 q ν,0 for all k ∈ N\{0}: All polygons Q k have the same center of gravity.
From now on let the initial polygon Q 0 have its center of gravity in the origin O := (0, . . . , 0) t . So we can be sure that for all k ∈ N 1 n As the matrix R is regular the initial polygon Q 0 can explicitly be retrieved from the d × n-matrix Thus, we do not alter the recursion in any way if we replace the diagonal matrix With this in mind, the iteration process can be described by

Prototype polygons
The Gaussian plane of complex numbers C can be interpreted as a Euclidean plane E 2 with a Cartesian coordinate frame {O; 1, i}. We embed E 2 into E d by identifying 1 and i with the d-dimensional unit vectors e 1 := (1, 0, 0, . . . , 0) t and e 2 := (0, 1, 0, . . . , 0) t , respectively. The elements of P j (3.2) can be viewed as a collection of n points ζ ν j (ν ∈ J) equally distributed on the unit circle of Its points can be written as T j forms the so-called 'regular prototype n-gon of jth kind'. The regular n-gon T n−j is symmetric to T j w.r.t. the axis e 1 and thus affinely equivalent to T j . If j and n are relatively prime the polygon T j is either a regular n-gon or an Vol. 108 (2017) Regularizing transformations of polygons 795 n-sided regular star. If j is a divisor of n with n = j p the polygon T j is either a regular p-gon or an ordinary regular star with p vertices, each of the vertices being multiply counted (j times).

The concept of affine regularization
An affine mapping of E d keeping the origin O in its place is described by The affine image of the polygon Our iteration (2.1) seems to regularize for certain (u 0 , . . . , u m ) ∈ R 1×(m+1) irrespective of the choice of the starting polygon Q 0 . In order to examine this interesting peculiarity we compare the n-gons Q k with a regular prototype n-gon T j (4.1) of j th kind 1 : Definition 5.1. We call the iteration (2.1) affinely regularizing of kind j with 1 ≤ j ≤ n/2 if, for any generic initial polygon Q 0 , there exist affine mappings with the property that the series Δ k of sums of the squared distances of respective vertices of β(Q k ) and of the regular prototype polygon T j of jth kind is a null series: lim k→∞ Δ k = 0.

The affine regularization theorem
The shape of the polygons Q k depends on the input data set Q 0 and on the barycentric coordinates (u 0 , . . . , u m ) of the reference point z * with m μ=0 u μ = 1. The latter determine the matrix M (3.1), the eigenvalues λ j and the diagonal matrix D * = D(0, λ 1 , . . . , λ n−1 ). The norms n j := |λ j | of λ j for j ∈ J * are given by but in general still may change its shape and its position from generation to generation. For N > 1 the series Q k gradually expands for increasing k.
We will prove that for any N > 0, the algorithm is-in general-affinely regularizing. We divide the set of indices into two distinct subsets: According to (3.3), for any j * ∈ J 1 the index n − j * is also contained in J 1 ; for even n and j * = n/2 these two indices coincide. We have Equations (3.10) yield Regardless of the input data b j (3.8) the coefficients ( λν N ) k form null series for all ν ∈ J 2 and k → ∞; the coefficients ( λν N ) k for all ν ∈ J 1 are complex numbers of norm 1 for all k ∈ N.
Q k and any homothetic image ρ k (Q k ) have the same affine shape. Following Definition 5.1 we can apply homotheties ρ k : With reference to the cardinal number of the index set J 1 we have three cases: Case A: The index set J 1 contains just one element. This can only happen if n is an even integer and the barycentrics (u 0 , . . . , u m ) lead to J 1 = {n/2}. We have ζ n/2 = −1, and λ n/2 = m μ=0 u μ (−1) μ ∈ R. As N = λ n/2 > 0 and therefore λ n/2 = ±N = 0 formula (6.5) reads as For every k we apply a further homothety σ k : We have b n/2 = n−1 ν=0 q ν,0 ζ ν n/2 = n−1 ν=0 (−1) ν q ν,0 ∈ R d . For a generic input polygon Q 0 we can assume b n/2 = o d . In this case we choose an affine mapping τ with fixed point O and b n/2 → e 1 ∈ R d . The mapping τ induces an affine Vol. 108 (2017) Regularizing transformations of polygons 797 vectors b * ν do not depend on k. The affine mapping β k := τ • σ k • ρ k places the k th generation polygon Q k into The distance vectors d j,k of the vertices of β k (Q k ) to the respective vertices of the prototype polygon T n/2 = e 1 P n/2 (4.1) are the columns of . (6.9) The vectors b * ν ζ j ν are independent from k. As the norms of ( λν N ) k form null series for all ν ∈ J 2 we can be sure that lim k→∞ d j,k = o d for all j ∈ J. The sum of the squared distances Δ k := n−1 j=0 d j,k 2 is a null series: lim k→∞ Δ k = 0.
Thus, according to our Definition 5.1 the iteration process in case A is affinely regularizing of kind n/2. For generic input Q 0 the polygons Q k approach the shape of the n-gon T n/2 . The straight lines approximating the polygons Q k tend towards the straight line through O with direction vector b n/2 .

Case B:
The index set J 1 contains exactly two different elements: J 1 = {j * , n− j * } with 1 ≤ j * < n/2. In a way, this could be considered the general case. We put λ j * = Ne i φ and λ n−j * = Ne −i φ with some real angle φ ∈ [0, 2π) and For a generic input n-gon Q 0 the two vectors x, y ∈ R d are linearly independent. Let σ : E d −→ E d be any affine mapping that maps the two vectors x, y into σ(x) := e 1 /2 and σ(y) := −e 2 /2. σ induces an affine mapping C d −→ C d transforming b ν (ν ∈ J 2 ) into σ(b ν ). We have σ(ρ k (Q k )) = (e 1 cos kφ + e 2 sin kφ) P j * + P j * 2 + (−e 1 sin kφ + e 2 cos kφ) We define the complex numbers θ μ, ν := σ(b μ ) t σ(b ν ) for μ, ν ∈ J 2 . The matrices As every τ k preserves scalar products we have τ k (σ(b μ )) t τ k (σ(b ν )) = θ μ, ν for all μ, ν ∈ J 2 . According to (5.2) we compute the sum of squared distances of the vertices of β k (Q k ) to the respective vertices of the prototype polygon T j * and arrive at (6.15) As the values θ μ, n−μ are independent from k and 0 ≤ λμλμ N 2 < 1 for all μ ∈ J 2 the values Δ k (k ∈ N) form a null series. Accordingly, the corresponding iteration process in case B is regularizing of kind j * with 1 ≤ j * < n/2. For generic input n-gons Q 0 the two vectors x and y determine a plane ε * through O. The planes ε k approximating the polygon Q k tend towards ε * .

Case C:
The index set J 1 contains more than two different elements. We have j * , j * * , n − j * ∈ J 1 with 1 ≤ j * < j * * ≤ n/2. According to (6.1) this is characterized by m μ,ν=0 The coefficients of u μ u ν in (6.16) are ζ μ−ν The corresponding barycentrics (u 0 , . . . , u m ) denote points z * ∈ R m which, in general, are positioned on an (m − 1)-dimensional quadric of R m containing the vertices of the simplex S. In this case we cannot prove any regularizing effect of the affine iteration.
If the iteration is affinely regularizing of kind j * then, for a generic input n-gon Q 0 , the shape of Q k gradually approaches the shape of an affinely transformed prototype n-gon T j * .

Remarkable exceptions
For specific initial polygons Q 0 the algorithm may deliver unexpected results. If the coefficient vectors b ν of the regarded eigenvalues λ ν for ν ∈ J 1 in (6.4) vanish the respective eigenvalues have no influence on the regularizing process. So, for such a specific n-gon Q 0 , the algorithm works in the same way as if in (3.10) these eigenvalues λ ν had been replaced by λ ν = 0. The remaining eigenvalues deliver another maximum norm N * < N and a different set J 1 . Now our classification (Sect. 6) reveals the affine shape of the series Q k . Figure 3 shows such an example for n = 8, m = d = 3 where we have (u 0 , u 1 , u 2 , u 3 ) = (0.5, −0.25, 0.5, 0.25). We get n 1 ≈ 0.52, n 2 = 0.5, n 3 ≈ 0.99 < 1, n 4 = 1. Hence N = 1 and we conclude that the algorithm is affinely regularizing of kind 4; Q k is expected to approach the shape of the prototype is T 4 which is a line segment. The special initial octogon Q 0 , however, yields b 4 = o d ; we put λ 4 := 0 and perform a new case study. The affine shape of Q k tends towards the prototype T 3 . Figure 3 displays Q 0 and the following generations up to Q 6 .

Conclusion
We studied affine iterations transforming an initial n-gon Q 0 in E d (d > 1) into successive generations of n-gons Q k . The Affine Regularization Theorem in this paper does not only extend the results in [5] to dimensions d > 2; surprisingly, even for dimensions d > 2 the regularization leads to planar, regular prototypes no matter which generic input n-gon Q 0 we start with. For very specific input n-gons Q 0 , though, the same algorithm seems to regularize in a different way. The understanding of this phenomenon completes the results.