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

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.

Generalized evolutes of planar curves

The evolutes of regular curves in the Euclidean plane are given by the caustics of regular curves. In this paper, we define the generalized evolutes of planar curves which are spatial curves, and the projection of generalized evolutes along a fixed direction are the evolutes. We also prove that the generalized evolutes are the locus of centers of slant circles of the curvature of planar curves. Moreover, we define the generalized parallels of planar curves and show that the singular points of generalized parallels sweep out the generalized evolute. In general, we cannot define the generalized evolutes at the singular points of planar curves, but we can define the generalized evolutes of fronts by using moving frames along fronts and curvatures of the Legendre immersion. Then we study the behaviors of generalized evolutes at the singular points of fronts. Finally, we give some examples to show the generalized evolutes.

Intersection points of planar curves can be computed

Consider two paths ϕ , ψ : [ 0 ; 1 ] → [ 0 ; 1 ] 2 in the unit square such that ϕ ( 0 ) = ( 0 , 0 ), ϕ ( 1 ) = ( 1 , 1 ), ψ ( 0 ) = ( 0 , 1 ) and ψ ( 1 ) = ( 1 , 0 ). By continuity of ϕ and ψ there is a point of intersection. We prove that from ϕ and ψ we can compute closed intervals S ϕ , S ψ ⊆ [ 0 ; 1 ] such that ϕ ( S ϕ ) = ψ ( S ψ ).