A cyclographic approach to the vertices of plane curves. (English) Zbl 0807.53003
The centers of circles inscribed to a smooth Jordan curve \(C\) and touching \(C\) at least twice form a tree whose end points correspond to vertices of \(C\) with osculating circles enclosed by \(C\). Using this idea a theorem is proved which gives a lower bound for the number of such vertices. The four-vertex theorem is a special case. A similar result for ovals was obtained by R. C. Bose [On the number of circles of curvature perfectly enclosing or perfectly enclosed by a closed convex oval, Math. Z. 35, 16–24 (1932; Zbl 0003.40903)]and generalized for smooth Jordan curves by O. Haupt [Verallgemeinerung eines Satzes von R. C. Bose über die Anzahl der Schmiegekreise eines Ovals, die vom Oval umschlossen werden oder das Oval umschließen, J. Reine Angew. Math. 239/240, 339–352 (1969; Zbl 0183.511)]. Haupt works in the frame of order geometry so that the circles may be replaced by more general curves. But the author’s proof is considerably simpler than Haupt’s.
Reviewer: Erhard Heil (Darmstadt)
MSC:
| 53A04 | Curves in Euclidean and related spaces |
| 51L15 | \(n\)-vertex theorems via direct methods |
| 53C75 | Geometric orders, order geometry |
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