1 Introduction

Ziv (2002) and Donisi et al. (2016) studied a geometric iteration process starting with a generic planar polygon \(Q_0\) with \(n > 2\) vertices \(\{ q_{0,0}, \dots , q_{n-1,0}\}\) (\(q_{j,0} \in \mathbb C\)). This process places homothetic copies of a given fixed triangle XYZ on the sides of \(Q_0\). This way the third point Z of XYZ gets them the points of a next generation polygon \(Q_1\). Reapplying this process over and over again with the same triangle XYZ creates a series of generations \(Q_k\). Surprisingly, this iteration process, in general, has a regularizing effect on the polygon: The generation \(Q_k\) approaches the shape of a regular n-gon—see Donisi et al. (2016) and Ziv (2002). A general cyclic iteration process was introduced by Schoenberg (1950). He reports conditions and results on the existence of a limit point of these iterations. We will show that a regularizing effect generally shows up for affine iteration processes of this type. In order to prove this surprising property we set up an appropriate definition of ‘affine regularization’.

2 The affine iteration process

We use complex numbers \(z \in \mathbb C\) with \(z = x + i \, y \, (x,y \in \mathbb R)\) to describe points of the Euclidean plane \(\mathbb R^2\) and start with a generic planar polygon \(Q_0\) with \(n > 2\) vertices \(\{ q_{0,0}, q_{1,0}, \dots , q_{n-1,0}\}\) with \(q_{j,0} \in \mathbb C\). Additionally, we choose a reference triangle abc and a reference point \(z^*:= u \, a + v \, b + w \, c\) with fixed barycentrics \((u,v,w) \in \mathbb R^3\) (\(u+v+w = 1\)) with respect to abc.

We now define the affine mappings \(\alpha _{j,1}\) from the ordered vertex set abc of the reference triangle to ordered triples of consecutive vertices \(q_{j, 0}, \, q_{j+1, 0}, \, q_{j+2, 0}\) of \(Q_0\) (\(j \in \mathbb N; \, j \ mod \ n\)). The point \(z^*\) is mapped into points \(\alpha _{j,1} (z^*) := q_{j,1} = u \, q_{j, 0} + v \, q_{j+1, 0} + w \, q_{j+2, 0}\) (\(j \in \mathbb N, \ j \ mod \ n\)). These points form vertices of the generation 1 polygon \(Q_1\). Iterations of this process yield affine mappings \(\alpha _{j,k}\) and points

$$\begin{aligned} \alpha _{j,k} (z^*): z^*= & {} u \, a + v \, b + w \, c \longrightarrow \alpha _{j,k} (z^*) := q_{j,k} \ \ \mathrm{with} \nonumber \\ q_{j,k}= & {} u \, q_{j, k-1} + v \, q_{j+1, k-1} + w \, q_{j+2, k-1} \ (k \in \mathbb N - \{ 0 \}) \end{aligned}$$
(1)

(\(j \in \mathbb N, \ j \ mod \ n\)) forming a series of polygons \(Q_k := ( q_{0,k}, q_{1,k}, \dots , q_{n-1,k} )\) called the \(k\mathrm{th}\) generation polygons. This procedure generalizes the geometric iteration process presented and discussed in Radcliffe (2016). In the sequel \(Q_k\) also denotes the vectors \(Q_k := (q_{0,k}, \dots , q_{n-1,k})^t \in \mathbb C^n\). Figure 1 shows an example of the first iteration of procedure (1) for \(n = 5\).

Fig. 1
figure 1

The case \(n=5\): the polygon \(Q_0\) and the first generation polygon \(Q_1\) for \(z^*= u \, a + v \, b + w \, c\)

Full size image

As we are interested in affine properties of the iteration process we describe affine mappings of the plane \(\mathbb R^2\) by complex numbers: With \(z = x + i \, y\) (\(x, y \in \mathbb R\), let the conjugate number to z be named \(\bar{z} := x- i \, y\)) an affine mapping \(\beta : \mathbb R^2 \longrightarrow \mathbb R^2\) is given by \((x,y) \longrightarrow \beta (x,y) = (x^*, y^*) := (d_1 + a_1 x - b_2 y, d_2 + a_2 x + b_1 y)\) with \(d_1, d_2, a_1, a_2, b_1, b_2 \in \mathbb R\). With the complex numbers \(d := d_1 + i \, d_2\), \(e := (a_1 + i \, a_2)/2\) and \(f := (b_1 + i \, b_2)/2\) in \(\mathbb C\) this yields

$$\begin{aligned} \beta : z \longrightarrow \beta (z) = \, d \, + \, z \, (e \, + \, f) \, + \, \bar{z} \, (e \, - \, f). \end{aligned}$$
(2)

We write \(\beta (Q_k) := ( \beta (q_{0,k}), \dots , \beta (q_{n-1,k}) )^t\). The affine mapping \(\beta \) is regular for \(a_1 b_1 + a_2 b_2 \ne 0\) or, equivalently, \(e \, \bar{f} + \bar{e}\, f \ne 0\).

3 The iteration process and regular n-gons

Formula (1) can formally be written as

$$\begin{aligned} \begin{array}{ccc} \left( \begin{array}{c} q_{0,k} \\ \vdots \\ q_{n-1,k} \end{array} \right) = &{} \left( \begin{array}{cccccc} u &{} v &{} w &{} 0 &{} \dots &{} 0 \\ 0 &{} u &{} v &{} w &{} \dots &{} 0 \\ \vdots &{} \vdots &{}\ddots &{} \ddots &{}\ddots &{} \vdots \\ 0 &{} &{} \dots &{} u &{} v &{} w \\ w &{} 0 &{} \dots &{} \dots &{} u &{} v \\ v &{} w &{} 0 &{} \cdots &{} 0 &{} u \end{array} \right) &{} \left( \begin{array}{c} q_{0,k-1} \\ \vdots \\ q_{n-1,k-1} \end{array} \right) \ \Leftrightarrow \\ &{} &{} \\ &{} Q_k \, = \, M \, . \ Q_{k-1} &{} \\ \end{array} \end{aligned}$$
(3)

with the \(n \times n\)- matrix \(M \in \mathbb R^{n,n}\). As M is a so-called circulant matrix, discrete Fourier transformations and the \(n\mathrm{th}\) roots of unity \(\zeta _j := exp(i \frac{2 j \pi }{n}) = \cos \frac{2 j \pi }{n} + i \sin \frac{2 j \pi }{n}\) (\(j \in \mathbb Z\)) play a pivotal role for us (see Pech 2002, Schoenberg 1950 and Ziv 2002). For them we have \(\zeta _j = \zeta _{j \, mod \, n}, \ \bar{\zeta }_j = \zeta _{n-j}\) (\(j \in \mathbb Z\)). We define the vectors

$$\begin{aligned} P_j := (\zeta _j^0, \dots , \zeta _j^{n-1})^t \ \ j \in \mathbb Z. \end{aligned}$$
(4)

We have \(P_{n-j}^t \, M = (u \, \zeta _j^0 + v \, \zeta _j^1 + w \, \zeta _j^2 ) P_{n-j}^t\) and \(M \, P_{j} = (u \, \zeta _j^0 + v \, \zeta _j^1 + w \, \zeta _j^2 ) P_{j}\) for all \(j \in \mathbb J:= \{ 0, \ldots , n-1 \} \). Therefore the vectors \(P_{n-j}^t\) and \(P_j\) are left and right eigenvectors of the matrix M to the same eigenvalue

$$\begin{aligned} \lambda _j := u \, \zeta _j^0 + v \, \zeta _j^1 + w \, \zeta _j^2 \ (j \in \mathbb J). \end{aligned}$$
(5)

Owing to \((u,v,w) \in \mathbb R^3\) we have \(\bar{\lambda }_j = \lambda _{n-j}\) for all \(j \in \mathbb J\).

Remark 3.1

For \(j \in \mathbb J^*:= \{ 1, \dots , n-1 \}\) the elements of \(P_j\) can be interpreted as a collection of n points \(\zeta _j^k\) (\(k \in \mathbb J\)) equally distributed on the unit circle. They determine vertices of a so-called ’regular n-gon of \(j\mathrm{th}\) kind’. The regular n-gon of \((n-j)\mathrm{th}\) kind \(P_{n-j}\) is the reflection of \(P_j\) in the real axis \(y = 0\) and is affinely equivalent to \(P_j\). If j and n are relatively prime the polygon \(P_j\) is either a regular n-gon or an n-sided regular star. If j is a divisor of n with \(n = j \, m\) the polygon \(P_j\) either is a regular m-gon or a regular star with m vertices on the unit circle which has to be counted j times. For \(j \in \mathbb J^*\) these regular n-gons \(P_j\) of \(j\mathrm{th}\) kind share the origin 0 as their center of gravity. Figure 2 displays the situation for \(n = 8\): The regular octogons \(P_7, \, P_6, \, P_5\) coincide with \(P_1, \, P_2, \, P_3\), but have opposite orientation.

Fig. 2
figure 2

The case \(n=8\): the regular octogons \(P_1, \dots P_4\) of first, second, third and forth kind

Full size image

4 The normal form of the matrix representation

We define two matrices

$$\begin{aligned} L := \frac{1}{\sqrt{n}} \, \left( P_{n}, \, P_{n-1}, \dots , P_1 \right) \quad \mathrm{and} \quad R := \frac{1}{\sqrt{n}} \, \left( P_0, \, P_1, \dots , P_{n-1} \right) . \end{aligned}$$
(6)

Both matrices, L and R, are symmetric and regular for \(n > 1\) (see Pech 2002, Schoenberg 1950 and Ziv 2002). Moreover, we have \(L = \bar{R}\) and \(L \, \cdot \, R = I_{n,n}\) with the \(n \times n\)- unit matrix \(I_{n,n}\); the matrices L and R are unitary \(n \times n\)-matrices in \(\mathbb C^{n\times n}\). This way we have

$$\begin{aligned} L \, \cdot \, M \, \cdot \, R = D(\lambda _0, \dots , \lambda _{n-1}) \end{aligned}$$
(7)

with the diagonal matrix \(D(\lambda _0, \dots , \lambda _{n-1})\) containing the eigenvalues \(\lambda _j\) of M as elements of the main diagonal. We now have

$$\begin{aligned} \begin{array}{l} L \, \cdot \, Q_k \, = \, D(\lambda _0, \dots , \lambda _{n-1}) \, \cdot \, L \, \cdot \, Q_{k-1} \ \mathrm{and} \\ L \, \cdot \, Q_k \, = \, D(\lambda _0, \dots , \lambda _{n-1})^k \, \cdot \, L \, \cdot \, Q_{0} \ \ (k \in \mathbb N - \{ 0 \} ). \\ \end{array} \end{aligned}$$
(8)

For the \(j\mathrm{th}\) component of the vectors in (8) this can be written as

$$\begin{aligned} \sum _{m=0}^{n-1} \zeta _{n-j}^{n-m} q_{m,k} \, = \, \lambda _j^k \, \sum _{m=0}^{n-1} \zeta _{n-j}^{n-m} q_{m,0} \, = \, (u \zeta _j^0 + v \zeta _j^1 + w \zeta _j^2)^k \, \sum _{m=0}^{n-1} \zeta _{n-j}^{n-m} q_{m,0} \end{aligned}$$
(9)

for all \(k \in \mathbb N - \{ 0 \}\) and \(j \in \mathbb J\). The element \(j = 0\) yields \(\lambda _0^k = (u + v + w)^k = 1 \ \forall k \in \mathbb N\) and therefore \(\sum _{m=0}^{n-1}{q_{m,k}} = \sum _{m=0}^{n-1}{q_{m,0}}\) for all \(k \in \mathbb N - \{ 0 \}\). Thus, all polygons \(Q_k\) have the same center of gravity (\(k \in \mathbb N\)). Without loss of generality, we put this center of gravity into the origin 0 of our coordinate system. Thus, we assume that the initial polygon \(Q_0\) has its center of gravity in 0:

$$\begin{aligned} \frac{1}{n} \, \sum _{m=0}^{n-1}{q_{m,0}} = 0. \end{aligned}$$
(10)

As the matrix L is regular the initial polygon \(Q_0\) can be explicitly recovered from the vector

$$\begin{aligned} B := L \, Q_0 \ \ \mathrm{with} \ B = (0, b_1, \dots , b_{n-1})^t \in \mathbb C^{n-1}. \end{aligned}$$
(11)

Under our assumption (10) the first coordinate of the vectors in (8) is 0. Thus, we do not alter the recursion if we replace the matrix \(D(\lambda _0, \dots , \lambda _{n-1})\) in (8) by the diagonal matrix \(D^*:= D(0, \lambda _1, \dots , \lambda _{n-1})\). With that and (11) in mind, the iteration process is described by

$$\begin{aligned} L \, \cdot \, Q_k \, = \, D^{*k} \cdot \, B \ \Leftrightarrow \ Q_k \, = \, \frac{1}{\sqrt{n}} \, \sum _{j \in \mathbb J^*} \lambda _j^k \, b_j \, P_j \ \ (k \in \mathbb N). \end{aligned}$$
(12)

5 Definition of affine regularization

Just like the iterations defined in Donisi et al. (2016) and Ziv (2002), iteration (1) also seems to regularize ‘in a way’ for certain \((u,v,w) \in \mathbb R^3\) independent of the choice of the initial polgon \(Q_0\). In order to discuss this interesting property we compare the n-gons \(Q_k\) with a regular prototype n-gon \(P_j\) of kind j.

Definition 5.1

We call the procedure (1) affinely regularizing of kind \(j \in \mathbb J^*\) if, for general input data (initial polygon \(Q_0\)), there exist affine mappings \(\beta _k\) transforming \(Q_k = (q_{0,k}, \dots , q_ {n-1,k})^t\) into polygons \(\beta _k (Q_k)\) with the property that for all \(k \in \mathbb N\) the series \(\Delta _{k}\) of sums of the squared distances of the points of \(\beta _k (Q_k)\) to the regular prototype polygon \(P_j\) (4) of kind j is a null series.

Thus, the iterative processes (1) is affinely regularizing of kind j if there exist affine mappings \(\beta _k\) with:

$$\begin{aligned} \Delta _{k} := \sum _{m=0}^{n-1} dist^2(\beta _k(q_{m,k}), \, \zeta _j^m) = (P_j-\beta _k (Q_k))^t \, \cdot \, (\overline{P_j -\beta _k (Q_k)}) \end{aligned}$$
(13)

has the property \(\displaystyle {\lim _{k \longrightarrow \infty } \Delta _k = 0}\).

Remark 5.2

  • It is important to note that definition 5.1 does not require the existence of a limit polygon \(Q_k\) for \(k \longrightarrow \infty \). In general for the iterative processes (1) such a limit polygon will not exist. ‘Affinely regularizing’ in our sense is a property of the affine shape of the series \(Q_k\) alone.

  • In order to be able to measure the sum of the squared distances in Definiton 5.1 we need the Euclidean structure of the plane.

  • Our definiton makes use of affine mappings \(\beta _k\) transforming \(Q_k\) ‘close to some regular prototype polygon’ \(P_j\) of fixed size. In order to define the regulization of the iteration process it would not be enough to do it the other way round: the existence of affine mappings \(\gamma _k\) transforming the prototype polygon \(P_j\) close to the polygons \(Q_k\) (again with sums of squared distances forming a null series) falls short of our requirements. The following counterexample demonstrates that fact: Given a series of polygons \(Q_k := (1/2)^k Q_0\) that are homothetic copies of an initial polygon \(Q_0\). For any \(j \in \mathbb J^*\) the affine mappings \(\gamma _k (P_j) := (1/2)^k P_j\) yield sums of squared distances of corresponding vertices of \(Q_k\) and \(\gamma _k (P_j)\) forming a null series independent from the input data \(Q_0\). As \(Q_k\) is homothetic to a polygon \(Q_0\) ‘of general shape’ this process, in general, is far from being affinely regularizing.

6 The affine shape of \(Q_k\)

The shape of the polygons \(Q_k\) depends on the input data set \(Q_0\) and the barycentric corordinates (u, v, w) of the point z with \(u+v+w = 1\). (u, v, w) uniquely determine the diagonal matrix \(D^*= D(0, \lambda _1, \dots , \lambda _{n-1})\). We compute the norms \(n_j := \sqrt{\lambda _j \bar{\lambda _j}}\) of the eigenvalues \(\lambda _j\) for \(j \in \mathbb J^*\) and have

$$\begin{aligned} n_j^2= & {} \lambda _j \bar{\lambda _j} = u^2 + v^2 + w^2 + v(u + w)(\zeta _j + \zeta _{-j}) + uw (\zeta _j^2 + \zeta _{-j}^2) \nonumber \\= & {} u^2 + v^2 + w^2 + 2 \, v(u + w) \cos (2 \pi j / n) + 2 \, u w \cos (4 \pi j / n) \end{aligned}$$
(14)

(\(j \in \mathbb J^*\)). The spectral radius of the matrix \(D^*\) is the maximal value \(N := \max \, \{ n_1, \dots , n_{n-1} \}\) of the norms of the eigenvalues \(\lambda _j\) for \(j \in \mathbb J^*\). We assume that the barycentrics (u, v, w) with \(u+v+w=1\) are chosen generally in the sense that not all \(\lambda _1, \dots , \, \lambda _{n-1}\) vanish.

It is easy to see that the exceptional case \(\lambda _1 = \dots = \lambda _{n-1} = 0\) can only occour in the case where \(n=3\) and, additionally, \((u,v,w) = (1/3,1/3,1/3)\). Therefore, this case in general yields iterated series of ‘degenerate triangles’ \(Q_k\) that are formed by the center of gravity 0. This trivial case shall be excluded from now on by putting \(n>3\).

In all other cases we have that \(N > 0\). So we can define the two sets of indices

$$\begin{aligned} \mathbb J_1 := \{ j \in \mathbb J^*/ \left| \lambda _j \right| = N \} \ne \emptyset \quad \mathrm{and} \quad \mathbb J_2 := \mathbb J^*- \mathbb J_1. \end{aligned}$$
(15)

Note that, according to (5), for any \(j^*\in \mathbb J_1\) the index \(n-j^*\) is also contained in \(\mathbb J_1\) (only for even n and \(j^*= n/2\) these two indices are coincident) and we have

$$\begin{aligned} \frac{\left| \lambda _j \right| }{N} = 1 \ \forall j \in \mathbb J_1, \quad \frac{\left| \lambda _j \right| }{N} < 1 \ \forall j \in \mathbb J_2. \end{aligned}$$
(16)

Figure 3 shows the positions of \(\lambda _j\) for \(n=8\) and the barycentrics (u, v, w) of \(z^*\) with respect to the triangle abc. For this special case we have \(\mathbb J_1 = \{ 1, 7 \}\).

Fig. 3
figure 3

The case \(n=8\): the eigenvalues \(\lambda _j\) for \(n=8\) and \(z^*= u \, a + v \, b + w \, c\)

Full size image

Equation (12) yields

$$\begin{aligned} Q_k \, = \, \frac{N^k}{\sqrt{n}} \, \left( \sum _{j \in \mathbb J_1} (\frac{\lambda _j}{N})^k \, b_j \, P_j + \sum _{j \in \mathbb J_2} (\frac{\lambda _j}{N})^k \, b_j \, P_j \right) . \end{aligned}$$
(17)

Independent of the input data \(b_j\) the coefficients \((\frac{\lambda _j}{N})^k \, b_j\) of \(P_j\) in the second sum in (17) form null series for all \(j \in \mathbb J_2\) and \(k \longrightarrow \infty \). The coefficients in the first sum are finite complex numbers for all \(k \in \mathbb N\). Depending on the spectral radius N formula (17) delivers three possibilities:

  • For \(N < 1\) the series \(Q_k\) decreases for increasing k and tends towards the center of gravity 0.

  • For \(N = 1\) the series \(Q_k\) remains finite, but in general changes its position for different k.

  • For \(N > 1\) the series \(Q_k\) increases for increasing k and seems to explode.

For all these possibilities, though, the series \(Q_k\) seems to be affinely regularizing. This is to be discussed in three cases:

Case A The index set \(\mathbb J_1\) contains exactly one element.

As stated before, this is only possible for even n and special barycentrics (u, v, w) such that \(\mathbb J_1 = \{ n/2 \}\), \(\mathbb J_2 = \mathbb J^*- \{ n/2 \}\). We have \(\zeta _{n/2}= -1\), and therefore \(\lambda _{n/2} = u - v + w \in \mathbb R\). This yields \(\left| \lambda _{n/2} \right| / N = (-1)^s\) with \(s \in \{ 0,1\}\) depending on the sign of \(u - v + w\). The only element in the first sum of (17) has the coefficient \(c_k^*:= (-1)^{s \, k}\, b_{n/2}\). For a general input polygon \(Q_0\) we have \(b_{n/2} \ne 0\) and therefore \(c_k^*\ne 0\). We can define the following affine image of the \(k\mathrm{th}\) generation \(Q_k\):

$$\begin{aligned} \beta _k (Q_k) := \frac{\sqrt{n}}{c_k^*\, N^k} \, Q_k = P_{n/2} + \frac{1}{c_k^*} \sum _{j \in \mathbb J_2} \left( \frac{\lambda _j}{N}\right) ^k \, b_j \, P_j. \end{aligned}$$
(18)

With (17) we compute the sum of squared distances of \(\beta _k (Q_k)\) from \(P_{n/2}\) via

$$\begin{aligned} \begin{array}{l} \Delta _k = n \, \left( \sum _{j \in \mathbb J_2} (\frac{\bar{\lambda }_j \, \lambda _{j}}{N^2})^k \, \bar{b}_{j} \ b_j \right) \, / ({\bar{b}_{n/2} \, b_{n/2}}). \end{array} \end{aligned}$$
(19)

As stated before, for all \(j \in \mathbb J_2\) the value \((\frac{\bar{\lambda }_j \lambda _j}{N^2})^k\) decreases for increasing k and \(\displaystyle {\lim \nolimits _{k\longrightarrow \infty } (\frac{\bar{\lambda }_{j} \lambda _j}{N^2})^k = 0}\). Thus, according to our Definition 5.1 the iteration process in Case A for general input data is affinely regularizing of kind n / 2. For large k generation \(Q_k\) will approach the shape of the degenerate regular n-gon \(P_{n/2}\): It consists of two points, each of them counted n / 2 times.

Case B The index set \(\mathbb J_1\) contains exactly two different elements: \(\mathbb J_1 = \{ j^*, n-j^*\}\) with \(j^*\ne n/2\) (in Fig. 3 we have an example for \(n=8\) with \(j^*=1\)). We put \(\lambda _{j^*} = N e^{i \, \phi }\) and \(\lambda _{n-j^*} = N e^{- i \, \phi }\) with some real angle \(\phi \in [0, 2 \pi )\). Then (17) yields

$$\begin{aligned} Q_k \, = \, \frac{N^k}{\sqrt{n}} \, \left( e^{i \, k \phi } \, b_{j^*} \, P_{j^*} + e^{-i \, k \phi } \, b_{n-j^*} \, \bar{P}_{j^*} + \sum _{j \in \mathbb J_2} \left( \frac{\lambda _j}{N}\right) ^k \, b_j \, P_j \right) . \end{aligned}$$
(20)

We define \(\sigma := b_{j^*} \bar{b}_{j^*} - b_{n - j^*} \bar{b}_{n-j^*} \ (\sigma \in \mathbb R)\). For general input data we have \(\sigma \ne 0\) and we can define affine images of the \(k\mathrm{th}\) generation \(Q_k\):

$$\begin{aligned} \beta _k (Q_k):= & {} \frac{\sqrt{n}}{\sigma \, N^k} e^{-i k \phi } \left( \bar{b}_{j^*} \, Q_k - b_{n-j^*} \, \bar{Q}_k \right) \nonumber \\= & {} P_{j^*} + \frac{1}{\sigma } e^{-i k \phi } \sum _{j \in \mathbb J_2} (\frac{\lambda _j}{N})^k \, (b_j \bar{b}_{j^*} - \bar{b}_{n-j} b_{n-j^*}) \, P_j \end{aligned}$$
(21)

Now we compute the sum \(\Delta _k := (P_{j^*} - \beta _k(Q_k))^t \, \cdot \, \overline{(P_{j^*} - \beta _k(Q_k))}\) of squared distances of respective points of \(\beta _k(Q_k)\) and \(P_{j^*}\). Short computation yields

$$\begin{aligned} \Delta _k = \frac{n}{\sigma ^2} \sum _{j \in \mathbb J_2} \left( \frac{\bar{\lambda }_j \, \lambda _{j}}{N^2}\right) ^k \, (b_j \bar{b}_{j^*} - \bar{b}_{n-j} b_{n - j^*}) \, (\bar{b}_j b_{j^*} - b_{n-j} \bar{b}_{n - j^*}). \end{aligned}$$
(22)

The only factors in (22) depending on k are \((\frac{\bar{\lambda }_j \, \lambda _{j}}{N^2})^k\). For each \(j \in \mathbb J_2\) these terms form null series for \(k \in \mathbb N\). The finite sum \(\Delta _k\) of null series delivers a null series again and we have \(\displaystyle {\lim _{k \longrightarrow \infty } \Delta _k = 0}\). Thus, the corresponding iteration process is affinely regularizing of kind \(j^*\).

Case C The index set \(\mathbb J_1\) contains more than two different elements \(j^{*}, j^{**}, n- j^*\) with \(j^{**} \ne j^{*}\) and \(j^{**} \ne {n-j^{*}}\). According to (14) we have

$$\begin{aligned}&v (u + w) (\zeta _{j^*} + \zeta _{-j^*} - \zeta _{j^{**}} - \zeta _{-j^{**}}) \nonumber \\&\quad +\, uw (\zeta _{j^*}^2 + \zeta _{-j^*}^2 - \zeta _{j^{**}}^2 - \zeta _{-j^{**}}^2) = 0. \end{aligned}$$
(23)

In general this is the equation of a conic section in the plane of the triangle abc. For these special choices of the barycentrics (u, v, w) we cannnot give any affine mappings like in Cases A and B for general input polygon \(Q_0\) which deliver a null series for the corresponding distance functions.

We have:

Theorem 6.1

For a general input polygon \(Q_0\) and general barycentrics (u, v, w) the iteration process (1) is affinely regularizing.

Remark 6.2

In general, the shape of \(Q_k\) gradually approaches that of an affinely transformed regular n-gon \(P_{j^*}\) of kind \(j^*\). The kind \(j^*\) of this affine regularization belongs to those indices \(j^*\) which determine the spectral radius \(N = \left| \lambda _{j^*} \right| \) of the transformation matrix \(D^*= D(0, \lambda _1, \dots , \lambda _{n-1})\).

The exceptional cases are addressed in

Theorem 6.3

In the following exceptional cases the iteration process (1), in general, fails to regularize:

  • Case A We have even n, special barycentrics (u, v, w) and get case A with \(c_k^*= 0\). The initial polygon \(Q_0\) has a special shape with \(b_{n/2}=0\).

  • Case B The input polygon \(Q_0\) has a special shape with \(\sigma := b_{j^*} \bar{b}_{j^*} - b_{n - j^*} \bar{b}_{n-j^*} = 0\).

  • Case C The input barycentrics \((u,v,w)\in \mathbb R^3\) with \(u+v+w=1\) comply with (23) for suitable triples \((j^{*}, j^{**}, n- j^*)\) consisting of three different indices from the index set \(\mathbb J_1\).

Remark 6.4

As long as we are not in the realm of the exceptional cases listed in Theorem 6.3 the regularizing property of Theorem 6.1 remains valid, even if some of the data points of the initial polygon \(Q_0\) or of some further generation are coincident.

Figure 4 shows an example for \(n=7\) with the barycentrics \((u,v,w) = (0.5, -0.2, 0.7)\) of \(z^*\). We get \((n_1^2, \, n_2^2, \, n_3^2) \approx (0.57, \, 0.51, \, 1.28)\) and therefore we have \(\mathbb J_1 = \{ 3, 4\}\). This way we have an example for case B. The figure shows \(Q_0\) and the \(10\mathrm{th}\) generation \(Q_{10}\) which approaches the affine image of the regular star 7-gon \(P_3\).

Fig. 4
figure 4

The case \(n=7\) with \(\mathbb J_1 = \{ 3, 4 \}\)

Full size image

7 Generalisations

The considerations from above can be modified to obtain further affinely regularizing iteration processes. Instead of the planar affine mappings used in (1) we can also use affine mappings from an n-dimensional affine space with basic points in the vertices of a simplex \(a_0, \dots , a_{n-1}\). A point \(z^*:= \sum _{\mu =0}^{n-1} u_\mu a_\mu \) with barycentrics (\(u_0, \dots , u_{n-1}\)) and \(\sum _{\mu =0}^{n-1} u_\mu = 1\) with respect to that simplex then is mapped to the plane of the initial polygon \(Q_0\) via affine mappings

$$\begin{aligned} \alpha _{j,k} (z^*): z^*= & {} \sum _{\mu =0}^{n-1} u_\mu a_\mu \longrightarrow q_{j,k} \ \ \mathrm{with} \nonumber \\ q_{j,k}:= & {} \sum _{\mu =0}^{n-1} u_\mu q_{j + \mu (mod \, n), k-1 } \ (j \in \mathbb J; \, k \in \mathbb N - \{ 0 \}). \end{aligned}$$
(24)

Such an iteration process has been studied by Schoenberg (1950). The author did not state results on the affine shape of \(Q_k\). In this case the eigenvalues \(\lambda _j\) are given by

$$\begin{aligned} \lambda _j := \sum _{\mu = 0}^{n-1} u_\mu \, \zeta _j^\mu \ (j \in \mathbb J). \end{aligned}$$
(25)

These generalized iteration processes show the same behavior as that described in Theorems 6.1 and 6.3. Again, for general input data \(Q_0\) and general \(u_\mu \) (\(\mu = 0, \dots , n-1\)) with \(\sum _{\mu =0}^{n-1} u_\mu = 1\) the corresponding iteration processes are still affinely regularizing. Our results could be viewed as a nice add-on to I. J. Schoenberg’s paper (Schoenberg 1950).