Limiting Forms of Iterated Circular Convolutions of Planar Polygons

Abstract

We consider a complex representation of an arbitrary planar polygon $$\mathcal {P}$$ centered at the origin. Let $$\mathcal {P}^{(1)}$$ be the normalized polygon obtained from $$\mathcal {P}$$ by connecting the midpoints of its sides and normalizing the complex vector of vertex coordinates. We say that $$\mathcal {P}^{(1)}$$ is a normalized average of $$\mathcal {P}$$ . We identify this averaging process with a special case of a circular convolution. We show that if the convolution is repeated many times, then for a large class of polygons the vertices of the limiting polygon lie either on an ellipse or on a star-shaped polygon. We derive a complete and compact analytical description of the limiting elliptical envelope using discrete Fourier transforms and circular convolutions. One of the key insights of this approach is the realization that the repeated circular convolution removes all higher harmonic pairs leaving only the principal harmonic pair from the discrete Fourier transform of the original polygon to dominate the Fourier transform of the repeatedly convolved polygon thereby controlling the limiting behavior.