TL;DR
This paper explores isoperimetric equalities for rosettes, revealing precise relationships between their length and area through advanced geometric constructs, and investigates affine equidistants and unions of rosettes.
Contribution
It introduces new isoperimetric equalities for rosettes and analyzes the geometry of affine equidistants and unions of these curves.
Findings
Exact relations between length and oriented area of rosettes.
New results on affine equidistants of rosettes.
Insights into the geometry of unions of rosettes.
Abstract
In this paper we study the isoperimetric-type equalities for rosettes, i.e. regular closed planar curves with non-vanishing curvature. We find the exact relations between the length and the oriented area of rosettes based on the oriented areas of the Wigner caustic, the Constant Width Measure Set and the Spherical Measure Set. We also study and find new results about the geometry of affine equidistants of rosettes and of the union of rosettes.
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