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 $\lambda$--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.

arXiv comment15 pages, 13 figures. The main part of this preprint was previously Section 3 and the part of Section 2 in arXiv:1605.05361v3

References in corpus (10)

• The Wigner caustic on shell and singularities of odd functions
• The improved isoperimetric inequality and the Wigner caustic of planar ovals
• Singularities of equidistants and global centre symmetry sets of Lagrangian submanifolds
• Even Dimensional Improper Affine Spheres
• A note on Hurwitz's inequality
• The Gauss-Bonnet Theorem for coherent tangent bundles over surfaces with boundary and its applications
• The Constant Width Measure Set, the Spherical Measure Set and isoperimetric equalities for planar ovals
• Singularities of affine equidistants: extrinsic geometry of surfaces in 4-space
• The Geometry of the Wigner Caustic and a Decomposition of a Curve Into Parallel Arcs
• The Geometry of the Secant Caustic of a Planar Curve

Cited by in corpus (4)

• The Gauss-Bonnet Theorem for coherent tangent bundles over surfaces with boundary and its applications
• Singular Improper Affine Spheres from a given Lagrangian Submanifold
• Isoperimetric equalities for rosettes
• The Geometry of the Secant Caustic of a Planar Curve