Beitraege zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 34 (1993), No. 1, 77-85.

Partial Inflation of Closed Polygons in the Plane

Bernd Wegner

Abstract.

Inflation for simply closed regular curves in the plane has been investigated
first by S. A. Robertson [4] and studied in some more detail in [5]. It
consists of an infinite iteration of reflections of parts of the curve at
supporting double tangents, hopefully leading to a convex limit curve which
has the same arc length as the original curve. The same procedure easily can
be defined for simply closed polygons. It provides a special construction
of chord-stretched versions of the given curve.
The aim of this note is to show
that the behaviour of inflation is more comfortable in the piecewise linear
case. It will end after a finite number of steps. This gives a positive
answer to a question posed by T. Kaluza [2].
Furthermore inflation may
lead to some measure for the nonconvexity of a simply closed polygon.