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Isoptics of a closed strictly convex curve. II. (English) Zbl 0881.53003

[Part I was reviewed in Zbl 0739.53001.]
This second part proves several interesting properties of isoptics of plane closed convex curves \(C\). Let \(C_\alpha\) denote the \(\alpha\)-isotopic of \(C\), i.e., the locus of points from which all of \(C\) is seen under the constant angle \(\pi-\alpha\). A Crofton-type integral formula is proved for the annulus enclosed by \(C\) and \(C_\alpha\), and it is used to derive a differential equation for the dependence of the area of this annulus from \(\alpha\). Relations between tangent directions to \(C_\alpha\) and directions of special chords of \(C\) are contained. The preservation of constant width is investigated. Some technical differential equations are derived at the end of the paper. The methods of proof are elementary.

MSC:

53A04 Curves in Euclidean and related spaces
52A10 Convex sets in \(2\) dimensions (including convex curves)
53C65 Integral geometry

Citations:

Zbl 0739.53001
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References:

[1] K. Benko - W. Cie - S. Góźdź - W. Mozgawa , On isoptic curves, An. St. Univ. Al. I. Cuza, Iaşi , 36 ( 1990 ), pp. 47 - 54 . MR 1109793 | Zbl 0725.52002 · Zbl 0725.52002
[2] T. Bonnesen - W. Fenchel , Theorie der konvexen Körper , Chelsea Publi. Comp ., New York ( 1948 ). MR 372748 | Zbl 0906.52001 · Zbl 0906.52001
[3] W Cieślak - A. Miernowski - W. Mozgawa , Isoptics of a Closed Strictly Convex Curve , Lect. Notes in Math. , 1481 ( 1991 ), pp. 28 - 35 . MR 1178515 | Zbl 0739.53001 · Zbl 0739.53001
[4] L. Santalo , Integral geometry and geometric probability , Encyclopedia of Mathematics and its Applications , Reading, Mass. ( 1976 ). MR 433364 | Zbl 0342.53049 · Zbl 0342.53049
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