From Random Polygon to Ellipse: An Eigenanalysis

Adam N. Elmachtoub

doi:10.1137/090746707

From Random Polygon to Ellipse: An Eigenanalysis

2010-01-01

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DOI：10.1137/090746707

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Suppose x and y are unit 2-norm n-vectors whose components sum to zero. Let P(x, y) be the polygon obtained by connecting (x(1), y(1)), ..., (x(1), y(1)), (x(1), y(1)) in order. We say that P(x, y) is the normalized average of (P) over cap((x) over cap, (y) over cap) if it is obtained by connecting the midpoints of its edges and then normalizing the resulting vertex vectors (x) over cap and (y) over cap so that they have unit 2-norm. If this process is repeated starting with P(0) =P(x((0)),y((0))), then in the limit the vertices of the polygon iterates P(x((k)),y((k))) converge to an ellipse epsilon that is centered at the origin and whose serniaxes are tilted forty-five degrees from the coordinate axes. An eigenanalysis together with the singular value decomposition is used to explain this phenomenon. The problem and its solution is a metaphor for matrix-based research in computational science and engineering.

Keyword:

power method

eigenvalue analysis

ellipse

polygon

From Random Polygon to Ellipse: An Eigenanalysis

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