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.


Introduction
In 1932 Eugene Wigner introduced the celebrated Wigner function to study quantum corrections to classical statistical mechanics ( [31]).This function relates the wavefunction that appears in Schrödinger's equation to a probability distribution in phase space. The Wigner function of a pure state is defined in the following way where (p, q) ∈ R 2 are momentum and position, and ψ ∈ L 2 C (R) is the wavefunction. In [1] M. Berry studied the semiclassical limit of Wigner's phase-space representation of quantum states. He proved that for 1-dimensional systems, that correspond to smooth (Lagrangian) curves M in the phase space (R 2 , ω = dp ∧ dq), the semiclassical limit of the Wigner function of the classical correspondence M of a pure quantum state takes on high values at points in a neighborhood of M and also in a neighborhood of a singular closed curve, which is called the Wigner caustic of M or the Wigner catastrophe (see [1,23,10,4] for details). Geometrically the Wigner caustic of a planar curve M is the locus of midpoints of chords connecting points on M with parallel tangent lines ( [1,10,11,23]). It is caustic of a Lagrangian map defined in the following way (see [23,11,10] for details).
For the the canonical symplectic form ω = dp ∧ dq on R 2 the map : T R 2 v → ω(v, ·) ∈ T * R 2 is an isomorphism between the bundles T R 2 and T * R 2 . Then ω = * dα = dṗ ∧ dq + dp ∧ dq is a symplectic form on T R 2 , where α be the canonical Liouville 1-form on T * R 2 The linear diffeomorphism Φ 1 2 : R 2 ×R 2 → T R 2 = R 2 ×R 2 , Φ 1 2 (p + , q + , p − , q − ) = (p, q,ṗ,q) = 1 2 p + + p − , q + + q − , p + − p − , q + − q − pullbacks the symplectic formω on T R 2 the canonical symplectic form 1 2 (π * + ω − π * − ω) on the product R 2 × R 2 , where π + , π − : R 2 × R 2 → R 2 are the projections on the first and on the second component, respectively. If M is a smooth regular planar curve then M is an immersed Lagrangian submanifold of (R 2 , ω). Hence Φ 1 2 (M × M ) is an immersed Lagrangian submanifold of (T R 2 ,ω). Let π 1 , π 2 : T R 2 = R 2 × R 2 → R 2 be the projections on the first and on the second component, respectively. Then π 1 and π 2 define Lagrangian fibre bundles with the symplectic structureω. Then the caustic of the Lagrangian map (the set of its critical values) π 1 • Φ 1 2 M ×M is the Wigner caustic [23,11,10,5,6]. On the other hand the Lagrangian map π 2 • Φ 1 2 M ×M is the secant map of M [13]. If M is (locally) described as then the generating family of the Lagrangian submanifold Φ 1 2 (M × M ) has the following form The front of the Legendrian submanifold of the contact manifold (T R 2 ×R, dz+ * α) generated by F is a singular 2-dimensional improper affine sphere, where z is a coordinate on R. The caustic of this front is composed of the curve M and its Wigner caustic. Hence the geometry of the Wigner caustics provides information on singularities of improper affine spheres. In Fig. 1 we present a non-convex planar curve with its Wigner caustic and in Fig. 2 we show the improper affine sphere generated by M in the construction described above (see [5,6] for details). In [8] (see also [3,4,2]) the dynamics of a quantum particle in the optical lattice potential was investigated. The authors analyze evolution of the Wigner function. The function undergoes a number of catastrophic changes. For a semiclassical approximation the Wigner caustic consists of the rainbow diagram (the original curve M ) and a locus of midpoints of chords joining points on rainbow diagram with Figure 2. An improper affine sphere (with different opacities) generated from a curve in Fig. 1 parallel tangent lines. But the catastrophe set of exact Wigner function, in addition, contains a locus of midpoints of chords joining points on neighboring rainbow diagrams with parallel tangents. Hence the Wigner caustic of the curve M should be investigated not only locally but globally too. It turn out that its global geometry is very important for understanding of the quantum-classical correspondence breakdown. It allows to extract important information without using simplifying approximations.
Singularities of the Wigner caustic for ovals occur exactly from an antipodal pair (the tangent lines at the two points are parallel and the curvatures are equal). The well-known Blaschke-Süss theorem states that there are at least three pairs of antipodal points on an oval ( [22,26]). The absolute value of the oriented area of the Wigner caustic gives the exact relation between the perimeter and the area of the region bounded by closed regular curves of constant width and improves the classical isoperimetric inequality for convex curves ( [34,35,37,38,39]). Furthermore this oriented area improves the isoperimetric defect in the reverse isoperimetric inequality ( [7]). Recently the properties of the middle hedgehog, which is a generalization of the Wigner caustic for non-smooth convex curves, were studied in [29,30]. The Wigner caustic in the literature regarding hedgehogs is known also as a projective hedgehog (see [27,28] and the literature therein). The Wigner caustic could be generalized to obtain an affine λ-equidistant, which is the locus of points of the above chords which divide the chord segments between base points with a fixed ratio λ. The singular points of affine equidistants create the Centre Symmetry Set, the natural generalization of the center of symmetry, which is widely studied in [11,16,18,20,24]. The geometry of an affine extended wave front, i.e. the set is an affine λ-equidistant of a manifold M , was studied in [11,15].
Local properties of singularities of the Wigner caustic and affine equidistants were studied in many papers [5,9,10,11,12,19,25,23]. In this paper we study global properties of the Wigner caustic of a generic planar closed curve. In [1] Berry proved that if M is a convex curve, then generically the Wigner caustic is a parametrized connected curve with an odd number of cusp singularities and this number is not smaller than 3. It is not true in general for any closed planar curve. If M is a parametrized closed curve with self-intersections or inflexion points then the Wigner caustic has at least two branches (smoothly parametrized components). We present a decomposition of a curve into parallel arcs and thanks to this decomposition we are able to describe the geometry of branches of the Wigner caustic. In general the geometry of the Wigner caustic of a regular closed curve is quite complicated (see Fig. 3). In Section 2 we briefly sketch the known results on the Wigner caustic and affine equidistants.
Section 3 contains the algorithm to describe branches of the Wigner caustic and affine equidistants of any generic regular parameterized closed curve. Subsection 3.1 provides an example of an application of this algorithm to a particular curve.
In the beginning of Section 4 we present global propositions on the number of cusps and inflexion points of the Wigner caustic. We show that the procedure based on a decomposition presented in Section 3 can be applied to obtain the number of branches of the Wigner caustic, the number of inflexion points and the parity of the number of cusp singularities of each branch. After that we study global properties of the Wigner caustic on shell, i.e. the branch of the Wigner caustic which connects two inflexion points of a curve. We present the results on the parity of the number of cusp points of the branches of the Wigner caustic on shell. We also prove that each such branch has even number of inflexion points and there are even number of inflexion points on a path of the original curve between the endpoints of this branch.
In Section 5 we use the decomposition introduced in Section 3 to study the geometry of the Wigner caustic of generic regular closed parameterized curves with non-vanishing curvature and of some generic regular closed parameterized curves with two inflexion points.
All the pictures of the Wigner caustic in this manuscript were made in the application created by the second author [36] and in Mathematica [32].

Preliminaries
Let M be a smooth parameterized curve on the affine plane R 2 , i.e. the image of the C ∞ smooth map from an interval to R 2 . A smooth curve is closed if it is the image of a C ∞ smooth map from S 1 to R 2 . A smooth curve is regular if its velocity does not vanish. A regular curve is simple if it has no self-intersection points. A regular simple closed curve is convex if its signed curvature has a constant sign. Let (s 1 , Let A be a subset of R 2 , then clA denotes the closure of A. where λ = 1 2 . Definition 2.4 is different from definitions in papers [5,10,11,12,18,19,25,33,37,38]. The closure in the definition is needed to include inflexion points of M in E λ (M ). For details see Remark 2.18.
Note that, for any given λ ∈ R, we have E λ (M ) = E 1−λ (M ). Thus, the case λ  [11,16]). If M is a generic convex curve, then CSS(M ), the Wigner caustic and E λ (M ) for a generic λ are smooth closed curves with at most cusp singularities ( [1,16,20,24]), cusp singularities of all E λ (M ) are on regular parts of CSS(M ) ( [20]), the number of cusps of CSS(M ) and E 1 2 (M ) is odd and not smaller than 3 ( [1,16], see also [22]), the number of cusps of CSS(M ) is not smaller than the number of cusps of E 1 2 (M ) ( [11]). Let us denote by κ M (p) the signed curvature of a smooth regular curve M at p. Let a, b be a parallel pair of M . Assume that κ M (b) = 0. Let us fixed local arc length parameterizations of M nearby the points a, b by f : (s 0 , s 1 ) → R 2 and by g : (t 0 , t 1 ) → R 2 , respectively. Let's assume that the parameterizations at a and b are in opposite directions, i.e. the velocities at a, b are opposite. Then there exists a function t : (s 0 , s 1 ) → (t 0 , t 1 ) such that It is easy to see that by the implicit function theorem the function t is smooth and Then by we will denote a local natural parameterization of the Wigner caustic. Whenever we will write about singular points of the Wigner caustic we will denote these points as the singular points of the parameterization given by (2.4).
By direct calculations we get the following lemma. (ii) the curvature of E 1 2 (M ) at p is equal to Lemma 2.7(ii) implies the following propositions.   The right hand side of (2.5) is positive, then κ M (a) and κ M (b) have the same sign, therefore M is curved in the different sides at a and b.
We denote by C ∞ (S 1 , R 2 ) the set of C ∞ mappings from S 1 to R 2 , i.e. the set of smooth closed parameterized planar curves, and by · the dot product in R 2 .
, where κ f denotes the signed curvature of f with respect to the parameterization of f , and κ f denotes the derivative of the curvature with the respect to the arc length parameter ( [10]).
Theorem 2.13. Let G be the subset of C ∞ (S 1 , R 2 ) such that each curve f in G satisfies the following conditions: (i) f is a regular curve with only non-degenerate inflexion points and no undulation points, (ii) f has only transversal self crossings, where κ denote the derivative of the curvature with respect to the arc length parameter.
Then G is a generic subset of C ∞ (S 1 , R 2 ) with Whitney C ∞ topology and the Wigner caustic of f ∈ G is the finite union of smooth curves with at most cusp singularities.
Proof. Since the intersection of two generic subsets is still a generic subset, it is enough to show that properties from each point are generic. The set of smooth regular closed curves is an open and dense subset of C ∞ (S 1 , R 2 ) because the set of 1-jets of smooth non-regular closed curves is a smooth submanifold of J 1 (S 1 , R 2 ) of codimension 2. Let f : S 1 → R 2 be smooth and regular. Having only nondegenerate inflexion points and no undulation points is equivalent to the following property: By the Thom Transversality Theorem (e.g. see Theorem 4.9 in [21]) Property (i) is generic.
To prove generecity of the conditions (ii-iv) we will use the Thom Transversality Theorem for multijets (e.g. see Theorem 4.13 in [21] for details). We denote by J k s (S 1 , R 2 ) the s-fold k-jet bundle and by (S 1 ) (2) the set S 1 × S 1 \{(s, s) | s ∈ S 1 }. Genericity of (ii) follows from transversality of j 1 Generecity of Property (iii) follows from transversality of the second multijet Now we assume that f satisfies (iii). Genericity of Property (iv) follows from the transversality of j 3 By direct calculations one can show that this means that if j 3 2 f (s 1 , s 2 ) ∈ W , then κ f (s 1 ) = κ f (s 2 ), which is equivalent to the condition for cusp singularity in a singular point of the Wigner caustic (see Remark 2.12).
From now one, when we will talk about generic curves, we will mean a curve from the set G. Furthermore, generecity of f implies the following geometric properties of f . Proposition 2.14. [13] If f ∈ C ∞ (S 1 , R 2 ) has only non-degenerate inflexion points and has no undulation points, then the number of inflexion points of f and the rotation number of f are finite.  2 has an inflexion point at p (as the limit point) and is tangent to M at p. The Wigner caustic is tangent to M at p too and it has an endpoint there. The Wigner caustic and the Centre Symmetry Set approach p from opposite sides ( [1,10,16,19]). This branch of the Wigner caustic is studied in Section 4.
If M is a generic regular closed curve then E 1 2 (M ) is a union of smooth parametrized curves. Each of these curves we will call a smooth branch of the Wigner caustic of M . In Fig. 5 we illustrate a non-convex curve M , E 1 2 (M ), and different smooth branches of E 1 2 (M ).

A Decomposition of a Curve into Parallel Arcs
In this section we assume that M is a generic regular closed curve. We will present a decomposition of M infto parallel arcs which will help us to study the geometry of the smooth branches of the Wigner caustic of M .
The angle function has the following properties.   Proof. It is a consequence of the fact that the number of local extrema of a generic smooth function from S 1 to S 1 is even.
Let ϕ M be the angle function of M . In the sequence of division points there are points s ∈ S 1 such that f (s), p is a parallel pair, where p is an inflexion point of M (if M has not inflexion point, then p = f (0)). In Fig. 6 we illustrate an example of a closed regular curve M , the angle function ϕ M and the sequence of division points. Let us notice that the images of points in the sequence of division points divide the curve M in arcs. Some of these arcs (say A 1 and A 2 ) have the property that for any point a i ∈ A i there exists a point a j ∈ A j such that a i , a j is a parallel pair for i = j ∈ {1, 2}. Such arcs we will call parallel arcs. The set of arcs splits into subsets such that any two arcs in the same subset are parallel (see Definition 3.9).
Let α belong to (0, π). First we assume that α is not equal to a local extremum of a path P. Then a line ϕ = α intersects the path P an even number of times if P is a path from 0 to 0 or from π to π, since both the beginning and the end of P are on the same side of the line (see Fig. 7(i)). This line intersects P an odd number of times if P is a path from 0 to π or from π to 0, since the beginning and the end of P are on different sides of the line (see Fig. 7(ii)).

Figure 7. Continuous paths
Now we assume that α is equal to a local extremum of P. In this case the line ϕ = α intersects a path P an odd number of times if P is a path from 0 to 0 or from π to π (see Fig. 7(iii)) and this line intersects P an even number of times if P is a path from 0 to π or from π to 0 (see Fig. 7(iv)), since by a small local vertical perturbation around the extremum point we obtain the previous cases and the numbers of intersection points have a difference ±1 (see Fig. 8). Let us notice that a path from 0 to 0 or from π to π corresponds to an arc of a curve with the rotation number equals 0 and a path from 0 to π or from π to 0 corresponds to an arc of a curve with t he rotation number equals ± 1 2 . Since the rotation number of M is an integer, the number of paths from 0 to π or from π to 0 in the graph of ϕ M is even. Each path of this type intersects every horizontal line ϕ = α at least once. Thus the number of intersections of ϕ M and the line ϕ = ϕ M (f −1 (a)) is odd. But the number of points b = a such that a, b is a parallel pair is one less than the number of intersection points of the graph of ϕ M and the line ϕ = ϕ M (f −1 (a)).
The number of inflexion points of a generic regular closed curve is even. Thus by Proposition 3.7 we have the following corollary. In the following definition indexes i in ϕ i are computed modulo 2n, indexes j, j + 1 in p j p j+1 and p j+1 p j are computed modulo 2m.
where p k = f (s k ) and p k p l = f s m2m(k,l) , s M2m(k,l) .
If M(ϕ M ) is empty then we define only one set of parallel arcs as follows: The set of parallel arcs has the following property.
Proposition 3.10. Let f : S 1 → R 2 be the arc length parameterization of M . For every two arcs p k p l , p k p l in Φ i the well defined map where the pair p, P (p) is a parallel pair of M , is a diffeomorphism.
Definition 3.11. Let p k1 p k2 , p k2 p l2 belong to the same set of parallel arcs, then p k1 p k2 p l1 p l2 denotes the following set (the arc) In addition p k1 . . . p kn p l1 . . . p ln denotes . We will call this set a glueing scheme.
Remark 3.12. If p k p l belongs to a set of parallel arcs, then there are neither inflexion points nor points with parallel tangent lines to tangent lines at inflexion points of M in p k p l \ {p k , p l }.
Definition 3.13. The 1 2 -point map ( [11]) is the map Let A 1 = p k1 p k2 and A 2 = p l1 p l2 be two arcs of M which belong to the same set of parallel arcs. It is easy to see that E 1 2 A 1 ∪ A 2 consists of one arc p k1 p k2 p l1 p l2 under π 1 2 (see Fig. 9). From this observation we get the following proposition. Proposition 3.15. Let M be a generic regular closed curve which is not convex. If a glueing scheme is of the form p k1 p k2 p l1 p l2 , then this scheme can be prolonged in a unique way to p k1 p k2 p k3 p l1 p l2 p l3 such that (k 1 , l 1 ) = (k 3 , l 3 ).
Proof. Let us consider , p l2 }. By Remark 3.12 A 1 and A 2 must be curved in the same side or in the opposite sides at any parallel pair in A 1 ∪ A 2 (see Fig. 10(i-ii)). Let us consider the case in Fig. 10(i), the other case is similar. Then (3.1) can be prolonged in the following two ways.
(2) One of points p k2 , p l2 is an inflexion point of M . Let us assume that this is p k2 . Then (3.1) can be prolonged to p k1 p k2 p k3 p l1 p l2 p l1 , where k 1 = k 3 (see Fig. 10(iv)).
Since M is generic, at least one of the points p k2 , p l2 is not an inflexion point of M . There is only finite number of arcs from which we can construct branches of E 1 2 (M ). Therefore we can define a maximal glueing scheme. Definition 3.17. A maximal glueing scheme is a glueing scheme which is a maximal element of the set of all glueing schemes equipped with the inclusion relation. Proof. It follows from uniqueness of the prolongation of the glueing scheme (see Proposition 3.15).
Lemma 3.20. Let f : S 1 → R 2 be the arc length parameterization of M . Then (i) for every two different arcs p k1 p k2 , p l1 p l2 in Φ i there exists exactly one maximal glueing scheme containing (ii) every maximal glueing scheme is in the following form where {p k , p l } = {p k , p l } whenever p k = p l and p k = p l . (iii) if p k is an inflexion point of M , then there exists a maximal glueing scheme which is in the form where p l is a different inflexion point of M and p ki = p li for i = 1, 2, . . . , n.
Proof. (i) is a consequence of the uniqueness of the prolongation of a glueing scheme (see Proposition 3.15). The proof of (ii) follows from (i) and the fact that the following equalities hold: To prove (iii) let us prolong p k p k1 p k p l1 to the maximal glueing scheme G.
Any point p l in the sequence of division points S M belongs to exactly two arcs in all sets of parallel arcs. Then by (ii) this maximal glueing scheme is in the following form If (3.2) would contain some other inflexion point p r in the middle, then (3.2) would contain the following part: Proof. Let f : S 1 → R 2 be the arc length parameterization of M .
It is easy to see that (s m2m(k,l) , s M2m(k,l) ) and then where denotes the disjoint union. Then by Proposition 3.10 we obtain that and every arc p k p l p k p l is in exactly one maximal glueing scheme, then every branch of the Wigner caustic is the image of a maximal glueing scheme under the As a summary of this section we present an algorithm to find all maximal glueing schemes.
If there exists a number k such that p k is an inflexion point of M and there exists the set of arcs p k p l1 , p k p l2 or p l1 p k , p l2 p k in Λ, create a glueing scheme p k p l1 p k p l2 , remove the used set of arcs from Λ and go to step (7). Otherwise go to step (8).
(7) If the created glueing scheme is of the form . . . p k1 . . . p l1 and there exists the set of arcs p k1 p k2 , p l1 p l2 or p k2 p k1 , p l2 p l1 in Λ, then prolong the scheme to the following scheme . . . p k1 p k2 . . . p l1 p l2 , remove the used set of arcs from Λ and go to step (7), otherwise the considered glueing scheme is a maximal glueing scheme and then go to step (6).
(8) If Λ is empty, then all maximal glueing schemes for E 1 2 (M ) were created, otherwise find any set of arcs p k1 p l1 , p k2 p l2 in Λ, create a glueing scheme p k1 p l1 p k2 p l2 , remove the used set of arcs from Λ and go to step (7). 3.
1. An example of construction of branches of the Wigner caustic.
Let M be a curve illustrated in Fig. 6. Then the sets of parallel arcs are as follows Then there exist two maximal glueing schemes of M : Therefore this branch of the Wigner caustic has exactly two inflexion points -see Fig. 11(ii). The same conclusion holds for the glueing scheme (3.5) and the branch in Fig. 11(i). In this case we exclude the first and the last parallel pair. Figure 11. A curve M as in Fig. 6 and different branches of E 1 2 (M )

The geometry of the Wigner caustic of regular curves
In this section we start with propositions on numbers of inflexion points and cusp singularities of the Wigner caustic which follows from properties of maximal glueing schemes introduced in Section 3. Lemma 4.2. Let C be a closed smooth curve with at most cusp singularities. If the rotation number of C is an integer, then the number of cusps of C is even and if the rotation number of C is a half-integer, then the number of C is odd.
Proof. A continuous normal vector field to the germ of a curve with the cusp singularity is directed outside the cusp on the one of two connected regular components and is directed inside the cusp on the other component as it is illustrated in Fig.  12. That observation end the proof.  Proof. If the normal vectors to M at p k and p l are opposite, then the rotation number of C is equal to r 2 , where r is an odd integer. By Lemma 4.2 the number of cusps in C is odd. Otherwise the rotation number of C is an integer, therefore the number of cusps of C is even.
By Proposition 2.9, Corollary 3.8 and Proposition 4.1 we get the following corollaries on inflexion points of branches of the Wigner caustic of M .
The vector field n N is a continuous unit normal field to N . The normal vector field around the points of type (4.1) and (4.2) is described in Fig. 13. Thus by the same argument as in the proof of Lemma 4.2 we can get that the total number of cusps and singularities of type (4.1) in N is even, so the number of cusps of C is odd if and only if exactly one of the inflexion points f (t 1 ), f (t 2 ) is of type (4.1).
> 0, (4.15) where κ M (s) denotes the curvature of M at f (s), the pairs f (s), f (t 1 (s)) and f (s), f (t 2 (s)) are parallel pairs such that t i (s) → s i whenever s → s i and s < t i (s) for the left-hand side neighborhood of s i for i = 1, 2.
Proof. By genericity of M we get that f (s 1 ) and f (s 2 ) are ordinary inflexion points. Then the theorem is a consequence of Theorem 4.6 and Proposition 4.7.
Now we study inflexion points on the Wigner caustic on shell.
Since f (t 1 ) is an inflexion point then ϕ M (t 1 ) is a local extremum. Without loss of generality we assume that ϕ M (t 1 ) is a local minimum. To prove that the number of inflexion points in f (t 1 , t 2 ) is even it is enough to show that ϕ M (t 2 ) is a local maximum.
The numbers of local maxima and local minima of ϕ M are equal. Thus the difference between the number of local maxima and local minima of ϕ M be a local extremum of ϕ M such that there are no extremum on ϕ M s 1 (0,t) and ϕ M s 2 (0,t) . Since ϕ M (t 1 ) is a local minimum then ϕ M (s i (t)) is a local maximum and ψ j (t − ε,t + ε) for j = i changes the orientation int (see Fig. 15).    Fig. 16 we illustrate a closed curve M and branches of the Wigner caustic between inflexion points of M . In Fig. 17 we illustrate a closed curve M such that the branch of the Wigner caustic which connects two inflexion points of M has no inflexion points.  Proof. If C is regular, i.e. has no cusp singularities, then by Lemma 3.4 we get that C has an even number of inflexion points. If C has cusp singularities, then we change C nearby each cusp in the way illustrated in Fig. 18 creating two more inflexion points. After this transformation of C we obtain a regular closed curve C such that the parity of the numbers of inflexion points of C and C are equal. Therefore the number of inflexion points of C is even. Figure 18. A curve C with the cusp singularity at x and a curve C with inflexion points at p and q Proof. Let us notice that all branches of E 1 2 (M ) except the branches of the Wigner caustic which connect two inflexion points of M are closed curves. So the result for these branches follows from Lemma 4.11. Otherwise it follows from Theorem 4.10.

The Wigner caustic of closed curves with at most 2 inflexion points
In this section we study the geometry of the Wigner caustic of closed regular curves with non-vanishing curvature (rosettes) and of closed regular curves with exactly two inflexion points.

The Wigner cuastic of whorls
In [3] waves with vacuum wavenumber k, travelling in the ξ direction, incident normally on a medium that varies periodically and weakly in the η direction were studied. This problem describes the diffraction of light by ultrasound and diffraction of beams of atoms by beams of light and dynamics of quantum particle in optical lattice potential ( [8]).
Catastrophic manifolds of the semiclassical Wigner catastrophes are formed by the Wigner caustic of a fixed whorl and by the whorl by itself ( [8]). It is worth mentioning that by its construction ( [3]), whorls are π-periodic in the y-value (see Figure 24).
We illustrate the Wigner caustic of the periodic whorl from Figure 24 in Figure  25. Notice that the centers of symmetry of the π-whorl belong to the Wigner caustic. Now, we explain why the Wigner caustic of the whorl for x = π has singular points. We apply a result on existence of singular points of the Wigner caustic ( [14]).     [14]). Let F 0 and F 1 be embedded regular curves with endpoints p, q 0 and p, q 1 , respectively. Let 0 be the line through q 1 parallel to T p F 0 and let 1 be the line through q 0 parallel to T p F 1 . Let c = 0 ∩ 1 , b 0 = 0 ∩ T p F 1 , b 1 = 1 ∩ T p F 0 . Let us assume that (i) T p F 0 T q1 F 1 and T q0 F 0 T p F 1 , (ii) the curvature of F i for i = 0, 1 does not vanish at any point, (iii) absolute values of rotation numbers of F 0 and F 1 are the same and smaller than 1 2 , (iv) for every point a i in F i there is exactly one point a j in F j such that a i , a j is a parallel pair for i = j, (v) F 0 , F 1 are curved in the different sides at every parallel pair a 0 , a 1 such that a i ∈ F i for i = 0, 1. Let ρ max (respectively ρ min ) be the maximum (respectively the minimum) of the set . If ρ max < 1 or ρ min > 1, then the Wigner caustic of F 0 ∪ F 1 has a singular point.  In Figure 26 we present a π-whorl with tangent lines for parameters: t = −0.125, t ≈ −1.40562, t = −0.4, t ≈ −1.4511, together with parallel arcs with endpoints at these points. In Figure 27 we illustrate translated parallel arcs from Figure 26, which fulfil assumptions of Proposition 6.1. Therefore, the Wigner caustic created from parallel arcs in Figure 26 has a singular point. This method can be applied for other whorls, too.
For more figures of the whorls and its Wigner caustics see [8].