Singular Points of the Wigner Caustic and Affine Equidistants of Planar Curves

Abstract

In this paper we study singular points of the Wigner caustic and affine λ-equidistants of planar curves based on shapes of these curves. We generalize the Blaschke-Süss theorem on the existence of antipodal pairs of a convex curve.2010 Mathematics Subject Classification. 53A04, 53A15, 58K05, 81Q20.

The Gauss–Bonnet Theorem for Coherent Tangent Bundles over Surfaces with Boundary and Its Applications

“…Proof. It is a direct consequence of the formula of the singular curvature and the formula of the curvature of the Centre Symmetry Set (see Lemma 2.6 in [9]). By Theorem 1.6 in [35] we know that the singular curvature does not depend on the orientation of the parameter θ, the orientation of M , the choice of ν, nor the orientation of the singular curve.…”

“…Remark 5.6. In [9,43] we study in details the geometry of affine λ-equidistants of rosettes. We show among other things that there exist m branches of E 0.5 (R m ) and 2m − 1 branches of E λ (R m ) for λ = 0, 0.5, 1.…”

Isoperimetric equalities for rosettes

“…, m be different branches of the Wigner caustic of R m . Then by [9] the parameterization of E 0.5,k (R m ) is as follows.…”

“…The affine λ-equidistant is the set of points divided chords connecting points on M where tangent lines to M are parallel in the ratio λ. There are many papers considering affine equidistants, see [4,6,7,8,9,14,16,24,32,36]. In [4,14] the Wigner caustic is known as the area evolute and in [24] is known as the symmetry defect.…”

“…The Wigner caustic also leads to one of two constructions of improper affine spheres ( [5]). In [9] we study the global geometry of the Wigner caustic and equidistants of plane curves, we find many global results, for instance we presented an algorithm to describe smooth branches of equidistants of parametrized closed planar curves. By this algorithm we can find among other things the exact number of smooth branches, the number of inflexion points, the parity of the number of cusps of each branch.…”