Full and partial inflation of plane curves.

*(English)*Zbl 0829.53003
Böröczky, K. (ed.) et al., Intuitive geometry. Proceedings of the 3rd international conference held in Szeged, Hungary, from 2 to 7 September, 1991. Amsterdam: North-Holland. Colloq. Math. Soc. János Bolyai. 63, 389-401 (1994).

The first author [Inflation of plane curves, Geometry and topology III, Proc. Workshop, Leeds/UK 1990, 264-275 (1991; Zbl 0727.53004)] introduced the notion of inflation as the simultaneous reflection of the interior parts of a simply closed curve at its bisupporting straight lines. Unfortunately, this procedure may destroy the simple closedness of the curve. Hence, arriving at a convex curve after an infinite number of iterations will only be possible under special assumptions.

This drawback has been overcome in the paper under review by considering only one bisupporting line for every iteration. Particular strategies are described in the smooth case for which infinite iterations lead to a convex limit curve. Since every iteration represents a chord stretching, this limit is a convex chord stretched version of the original curve. Furthermore an example of a nonconvex curve is given for which a circle can be obtained as its limit. The corresponding procedure for piecewise linear curves arrives at a convex polygon after finitely many iterations, as has been shown by the second author in Beitr. Algebra Geom. 34, No. 1, 77-85 (1993; Zbl 0772.52004).

For the entire collection see [Zbl 0809.00022].

This drawback has been overcome in the paper under review by considering only one bisupporting line for every iteration. Particular strategies are described in the smooth case for which infinite iterations lead to a convex limit curve. Since every iteration represents a chord stretching, this limit is a convex chord stretched version of the original curve. Furthermore an example of a nonconvex curve is given for which a circle can be obtained as its limit. The corresponding procedure for piecewise linear curves arrives at a convex polygon after finitely many iterations, as has been shown by the second author in Beitr. Algebra Geom. 34, No. 1, 77-85 (1993; Zbl 0772.52004).

For the entire collection see [Zbl 0809.00022].

Reviewer: Bernd Wegner (Berlin)

##### MSC:

53A04 | Curves in Euclidean and related spaces |