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Limiting Forms Of Iterated Circular Convolutions Of Planar Polygons, Boyan Kostadinov
Limiting Forms Of Iterated Circular Convolutions Of Planar Polygons, Boyan Kostadinov
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We consider a complex representation of an arbitrary planar polygon P centered at the origin. Let P(1) be the normalized polygon obtained from P by connecting the midpoints of its sides and normalizing the complex vector of vertex coordinates. We say that P(1) is a normalized average of 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 …