The geometry of the Wigner caustic and a decomposition of a curve into parallel arcs

Abstract

In this paper we study global properties of the Wigner caustic of parameterized closed planar curves. We find new results on its geometry and singular points. In particular, we consider the Wigner caustic of rosettes, i.e. regular closed parameterized curves with non-vanishing curvature. We present a decomposition of a curve into parallel arcs to describe smooth branches of the Wigner caustic. By this construction we can find the number of smooth branches, the rotation number, the number of inflexion points and the parity of the number of cusp singularities of each branch. We also study the global properties of the Wigner caustic on shell (the branch of the Wigner caustic connecting two inflexion points of a curve). We apply our results to whorls—the important object to study the dynamics of a quantum particle in the optical lattice potential.

The Geometry of the Centre Symmetry Set of a Planar Curve

“…In this section we study global properties of the Centre Symmetry Set which follow from maximal glueing schemes introduced in Section 3 and methods similar to the ones presented in [10,13,15]. Theorem 3.22 along with Lemma 3.21 yield the following fact.…”

“…This observation leads to the fact that every ordinary inflexion point is the beginning of a component of CSS(M ) ( [16]). This component of CSS(M ) we will call a CSS on shell (similarly to the notion of the Wigner caustic on shell -see [9,15]).…”

“…Let's assume that M is a generic regular closed curve. In this section we will adapt an algorithm of decomposition of M into parallel arcs (for further details see Section 3 in [15]) that will help us to study the geometry of Centre Symmetry Set of M .…”

“…The Centre Symmetry Set of a curve can be viewed also as the set of all singular points of a family of affine λ-equidistants of the curve (see for instance [8,9,12,15,20,28,30] and the references given there. See also [5,7] for the Centre Symmetry Set in a non-smooth, polygonal case).…”

“…In Section 3 we adapt the decomposition of a curve into parallel arcs from [15]. This decomposition is necessary to prove some of the global properties of the Centre Symmetry Set of a planar curve in the last section.…”