Symmetry Set of Planar Curves

• Symmetry sets are singular loci defined by tangential, chordal, or metric constructions on curves, typically exhibiting cusp singularities.
• The centre symmetry set (CSS), Wigner caustic, and Minkowski symmetry set are characterized by explicit envelope equations and support function parameterizations that capture curvature relationships.
• These concepts have practical applications in convex geometry, semi-classical quantum physics, chaos theory, and even group theory through totally symmetric sets.

Searching arXiv for recent and foundational papers on symmetry sets, centre symmetry sets, and Minkowski symmetry sets. arxiv_search(query="symmetry set planar curves center symmetry set Wigner caustic", max_results=10, sort_by="relevance") arxiv_search(query="symmetry set planar curves center symmetry set Wigner caustic", max_results=10, sort_by="relevance") is not available in this environment. Symmetry sets are singular loci associated with families of tangential, chordal, or metric constructions on curves. For a closed smooth planar curve, the Wigner caustic and the Centre Symmetry Set (CSS) are known singular sets which generically admit only cusp singularities, and their applications in semi-classical quantum physics, chaos theory, singularity theory, and convex geometry have been studied since the 1970s until today (Danielewska et al., 2024). In the Minkowski plane, the Minkowski symmetry set (MSS) is an analogue of the standard Euclidean symmetry set and is defined as the locus of centres of bitangent pseudo-circles (Reeve, 2019). A distinct algebraic usage also exists: in group theory, a totally symmetric set is a subset of a group whose elements can be permuted by conjugation through the full symmetric group (Caplinger et al., 2024).

1. Centre symmetry set as an affine envelope

Let $\mathcal C\subset\mathbb R^2$ be a smooth, regular, closed curve $\gamma:S^1\to\mathbb R^2$ of nonzero curvature $\kappa>0$; when its rotation number is $1$, it is an oval. Two points $a=\gamma(\theta)$ and $b=\gamma(\phi)$ form a parallel pair if their tangent vectors are parallel, i.e. $\gamma'(\phi)\parallel\gamma'(\theta)$. For a generic convex $\mathcal C$, one shows that $\phi=\theta+\pi \pmod{2\pi}$ exhausts all parallel pairs (Danielewska et al., 2024).

The Centre Symmetry Set of $\mathcal C$ is the envelope of the family of affine chords joining parallel pairs. If $\gamma:S^1\to\mathbb R^2$0 denotes the line through $\gamma:S^1\to\mathbb R^2$1 and $\gamma:S^1\to\mathbb R^2$2, then

$\gamma:S^1\to\mathbb R^2$3

This identifies the CSS not as a set of symmetry centers in the elementary sense, but as an envelope singularity arising from the geometry of parallel chords. The formulation is affine and is naturally compatible with the study of equidistants and caustics.

For a convex oval, or more generally a rosette of rotation number $\gamma:S^1\to\mathbb R^2$4, the support function $\gamma:S^1\to\mathbb R^2$5 provides an effective coordinate description. With $\gamma:S^1\to\mathbb R^2$6 defined as the signed distance from the origin to the tangent line at angle $\gamma:S^1\to\mathbb R^2$7,

$\gamma:S^1\to\mathbb R^2$8

and the curvature satisfies

$\gamma:S^1\to\mathbb R^2$9

In these coordinates the parallel partner of $\kappa>0$0 is $\kappa>0$1, so the CSS is expressible in terms of $\kappa>0$2 and its derivatives at $\kappa>0$3 and $\kappa>0$4. This support-function description places the CSS within Minkowski and convex-geometric methods rather than only within local singularity theory.

2. Envelope equations and explicit parameterizations

A one-parameter family of lines can be written as

$\kappa>0$5

where $\kappa>0$6 is a unit normal and $\kappa>0$7 the signed distance. Its envelope is the locus of points $\kappa>0$8 solving simultaneously

$\kappa>0$9

For the CSS, one sets

$1$0

The chord $1$1 through $1$2 and $1$3 has direction $1$4, the common normal direction, and signed distance

$1$5

Solving the envelope condition yields a parametrization

$1$6

where $1$7 is chosen so that $1$8 is tangent to the curve $1$9 itself. A straightforward computation gives

$a=\gamma(\theta)$0

hence the classical CSS parametrization

$a=\gamma(\theta)$1

Equivalently, with

$a=\gamma(\theta)$2

one has

$a=\gamma(\theta)$3

This formula shows that the CSS point on a parallel chord is a curvature-weighted affine combination of the chord endpoints rather than the midpoint.

For an $a=\gamma(\theta)$4-rosette, one finds $a=\gamma(\theta)$5 branches of the CSS. For $a=\gamma(\theta)$6, the $a=\gamma(\theta)$7-th branch is

$a=\gamma(\theta)$8

with the sign $a=\gamma(\theta)$9 accounting for orientation reversals when $b=\gamma(\phi)$0 is odd (Danielewska et al., 2024). This branch structure is a precise extension from ovals to higher-rotation-number rosettes.

The envelope formula is well-defined provided

$b=\gamma(\phi)$1

Generically $b=\gamma(\phi)$2 on a convex oval, so no denominator-zero arises; in degenerate cases a branch may fail at $b=\gamma(\phi)$3 where $b=\gamma(\phi)$4. A plausible implication is that regularity of the CSS is controlled as much by curvature pairing across antipodal tangential angles as by the regularity of the original curve itself.

3. Cusp singularities, branch parity, and the Wigner caustic

On each branch $b=\gamma(\phi)$5, a singular point occurs when $b=\gamma(\phi)$6. Differentiating the CSS parametrization gives

$b=\gamma(\phi)$7

or equivalently

$b=\gamma(\phi)$8

In terms of the support function,

$b=\gamma(\phi)$9

These are the generic cusp conditions for the CSS (Danielewska et al., 2024).

A cusp at $\gamma'(\phi)\parallel\gamma'(\theta)$0 is an ordinary $\gamma'(\phi)\parallel\gamma'(\theta)$1-singularity, or semi-cubic cusp, if the second derivative of $\gamma'(\phi)\parallel\gamma'(\theta)$2 does not vanish there: $\gamma'(\phi)\parallel\gamma'(\theta)$3 In local coordinates centered at the cusp, the CSS is diffeomorphic to

$\gamma'(\phi)\parallel\gamma'(\theta)$4

the standard semi-cubical parabola. Theorem 1 in the surveyed account states that for a generic convex curve $\gamma'(\phi)\parallel\gamma'(\theta)$5, the envelope of its family of affine chords is a curve with only ordinary cusps; the proof sketch uses the line family $\gamma'(\phi)\parallel\gamma'(\theta)$6, the envelope equations, and genericity via transversality (Danielewska et al., 2024).

Global parity properties constrain the number of cusps. On each branch other than the midpoint-branch for $\gamma'(\phi)\parallel\gamma'(\theta)$7, the number of cusps is even by degree-theoretic arguments. In particular, for an oval the CSS of the midpoint branch has an odd number of cusps. This parity result places the CSS within a broader pattern of singular-count invariants for equidistants.

The Wigner caustic is the special $\gamma'(\phi)\parallel\gamma'(\theta)$8-equidistant for $\gamma'(\phi)\parallel\gamma'(\theta)$9: $\mathcal C$0 It can also be viewed as the envelope of the family of mid-parallel lines. The singular-set relationship is especially strong: $\mathcal C$1 Thus the CSS coincides with the set of all singular points of all affine $\mathcal C$2-equidistants. For a convex oval, the number of cusps of $\mathcal C$3 is not smaller than the number of cusps of $\mathcal C$4, and both numbers are odd. A complementary theorem states that any loop, convex or nonconvex, has at least one singularity on its Wigner caustic (Danielewska et al., 2024). This suggests that the midpoint construction captures a singular core already present in much more general affine equidistant families.

4. Minkowski symmetry set and generic bifurcations

In the Minkowski plane, $\mathcal C$5 is endowed with the pseudo-scalar product

$\mathcal C$6

with associated “norm”

$\mathcal C$7

A centered pseudo-circle of radius $\mathcal C$8 about $\mathcal C$9 is the locus

$\phi=\theta+\pi \pmod{2\pi}$0

equivalently

$\phi=\theta+\pi \pmod{2\pi}$1

According to the sign, one obtains two-branched hyperbolic pseudo-circles

$\phi=\theta+\pi \pmod{2\pi}$2

or elliptic pseudo-circles

$\phi=\theta+\pi \pmod{2\pi}$3

For a smooth immersed plane curve $\phi=\theta+\pi \pmod{2\pi}$4, define

$\phi=\theta+\pi \pmod{2\pi}$5

and for fixed $\phi=\theta+\pi \pmod{2\pi}$6, $\phi=\theta+\pi \pmod{2\pi}$7. A point $\phi=\theta+\pi \pmod{2\pi}$8 is a $\phi=\theta+\pi \pmod{2\pi}$9-fold contact point with the pseudo-circle center $\mathcal C$0 iff

$\mathcal C$1

A pseudo-circle is bitangent at two distinct parameters $\mathcal C$2 if

$\mathcal C$3

The Minkowski symmetry set $\mathcal C$4 is the locus of all centers $\mathcal C$5 of bitangent pseudo-circles, equivalently the multi-local part of the bifurcation set

$\mathcal C$6

Generic transversality conditions ensure that each bitangent gives a smooth branch of the MSS away from its singularities (Reeve, 2019).

For a one-parameter family $\mathcal C$7, the only generically occurring multi-singularities are

$\mathcal C$8

Each has two geometric sub-types, labelled $\mathcal C$9 and $\gamma:S^1\to\mathbb R^2$00, except $\gamma:S^1\to\mathbb R^2$01, which is unique. The geometric criteria are explicit. For $\gamma:S^1\to\mathbb R^2$02, a single pseudo-circle is bitangent at four distinct points $\gamma:S^1\to\mathbb R^2$03, with sub-type $\gamma:S^1\to\mathbb R^2$04 corresponding to the “odd-even” case and sub-type $\gamma:S^1\to\mathbb R^2$05 to the “even-even” case; the Euclidean convex-hull test distinguishes them. For $\gamma:S^1\to\mathbb R^2$06, the criterion uses the derivatives $\gamma:S^1\to\mathbb R^2$07 of the Minkowski curvature at the two tangency points: $\gamma:S^1\to\mathbb R^2$08 (“moth”) occurs when $\gamma:S^1\to\mathbb R^2$09, and $\gamma:S^1\to\mathbb R^2$10 (“nib”) when $\gamma:S^1\to\mathbb R^2$11. For $\gamma:S^1\to\mathbb R^2$12, the two $\gamma:S^1\to\mathbb R^2$13-points lie on the same branch of the pseudo-circle in type $\gamma:S^1\to\mathbb R^2$14 and on opposite branches in type $\gamma:S^1\to\mathbb R^2$15. For $\gamma:S^1\to\mathbb R^2$16, the separate $\gamma:S^1\to\mathbb R^2$17-point lies on the opposite branch in type $\gamma:S^1\to\mathbb R^2$18 and on the same branch as the $\gamma:S^1\to\mathbb R^2$19-point in type $\gamma:S^1\to\mathbb R^2$20. For $\gamma:S^1\to\mathbb R^2$21, a pseudo-circle develops a 5th-order tangency at an isolated parameter value (Reeve, 2019).

Transition	Euclidean case	Minkowski case
$\gamma:S^1\to\mathbb R^2$22	occurs	occurs
$\gamma:S^1\to\mathbb R^2$23	does not occur	occurs
$\gamma:S^1\to\mathbb R^2$24	occurs	occurs
$\gamma:S^1\to\mathbb R^2$25	occurs	occurs
$\gamma:S^1\to\mathbb R^2$26	occurs	occurs
$\gamma:S^1\to\mathbb R^2$27	does not occur	occurs
$\gamma:S^1\to\mathbb R^2$28	occurs	occurs
$\gamma:S^1\to\mathbb R^2$29	does not occur	occurs
$\gamma:S^1\to\mathbb R^2$30	occurs	occurs

In the classical Euclidean case only $\gamma:S^1\to\mathbb R^2$31, $\gamma:S^1\to\mathbb R^2$32, $\gamma:S^1\to\mathbb R^2$33, $\gamma:S^1\to\mathbb R^2$34, $\gamma:S^1\to\mathbb R^2$35, and $\gamma:S^1\to\mathbb R^2$36 occur, whereas the Minkowski case admits the extra generic sub-types $\gamma:S^1\to\mathbb R^2$37, $\gamma:S^1\to\mathbb R^2$38, and $\gamma:S^1\to\mathbb R^2$39 (Reeve, 2019). The stated reason is the appearance of new “parity” phenomena coming from the two-branched nature of pseudo-circles.

5. Applications, inequalities, and artistic interpretations

The CSS and the Wigner caustic have been studied in convex geometry, semi-classical quantum physics, chaos theory, and singularity theory (Danielewska et al., 2024). In convex geometry, symmetry sets and the CSS measure the “lack of central symmetry” of convex bodies, as emphasized by JJR and Schneider. The Wigner caustic yields improved Blaschke–Santaló and Hurwitz inequalities, and while the CSS does not directly enter these inequalities, it underlies the symmetry measures of $\gamma:S^1\to\mathbb R^2$40 used in convex geometry and shape approximation.

In semi-classical quantum physics, Berry’s phase-space analysis associates high values of the Wigner function with accumulation near the Wigner caustic of a classical trajectory, producing phase-space catastrophes; the survey attributes this line to Cosic et al. In chaos theory, affine equidistants and their caustics model interference patterns and “billiard” scattering in optical lattices. In singularity theory, the CSS family realizes the full unfolding of planar $\gamma:S^1\to\mathbb R^2$41 and $\gamma:S^1\to\mathbb R^2$42 catastrophes, in work linked to Giblin–Zakalyukin and Domitrz–Rios–Ruas.

A further aspect is geometric visualization. Because the CSS and $\gamma:S^1\to\mathbb R^2$43 are envelopes of linear families, they naturally lend themselves to string-art constructions. By plotting many chords or tangent lines, one obtains patterns that reveal the geometry of the envelope. Mathematica scripts described in the survey produce “hearts,” “stars,” and “rosettes” based on support-function parametrizations (Danielewska et al., 2024). This suggests that the same envelope formalism that organizes singularity theory can also serve as a diagrammatic method for exhibiting global curvature relations.

6. Distinct algebraic usage: totally symmetric sets

In group theory, a totally symmetric set is a subset

$\gamma:S^1\to\mathbb R^2$44

such that for every permutation $\gamma:S^1\to\mathbb R^2$45 there exists $\gamma:S^1\to\mathbb R^2$46 with

$\gamma:S^1\to\mathbb R^2$47

Equivalently, the conjugation action of $\gamma:S^1\to\mathbb R^2$48 on itself carries $\gamma:S^1\to\mathbb R^2$49 onto itself via the full symmetric group. If the $\gamma:S^1\to\mathbb R^2$50 commute pairwise, $\gamma:S^1\to\mathbb R^2$51 is a commuting totally symmetric set. Fundamental examples occur in $\gamma:S^1\to\mathbb R^2$52, $\gamma:S^1\to\mathbb R^2$53, and $\gamma:S^1\to\mathbb R^2$54: $\gamma:S^1\to\mathbb R^2$55 is totally symmetric and commuting in $\gamma:S^1\to\mathbb R^2$56; $\gamma:S^1\to\mathbb R^2$57 is totally symmetric and non-commuting; $\gamma:S^1\to\mathbb R^2$58 is a commuting totally symmetric set of size $\gamma:S^1\to\mathbb R^2$59; and $\gamma:S^1\to\mathbb R^2$60 together with $\gamma:S^1\to\mathbb R^2$61 are totally symmetric subsets of $\gamma:S^1\to\mathbb R^2$62 (Caplinger et al., 2024).

The structural lemma is the collision-implies-collapse principle: for any homomorphism $\gamma:S^1\to\mathbb R^2$63 and any totally symmetric set $\gamma:S^1\to\mathbb R^2$64 of size $\gamma:S^1\to\mathbb R^2$65, either $\gamma:S^1\to\mathbb R^2$66, so $\gamma:S^1\to\mathbb R^2$67 remains totally symmetric, or $\gamma:S^1\to\mathbb R^2$68, so all of $\gamma:S^1\to\mathbb R^2$69 collapses. This yields a blueprint for controlling homomorphisms: find a large totally symmetric set in the source, classify such sets in the target, and deduce that the image either collapses or lands in a classified configuration.

The method has several applications. In $\gamma:S^1\to\mathbb R^2$70, the standard set

$\gamma:S^1\to\mathbb R^2$71

has size $\gamma:S^1\to\mathbb R^2$72, and for $\gamma:S^1\to\mathbb R^2$73, any totally symmetric $\gamma:S^1\to\mathbb R^2$74 with $\gamma:S^1\to\mathbb R^2$75 is conjugate to $\gamma:S^1\to\mathbb R^2$76. From this, one obtains that for $\gamma:S^1\to\mathbb R^2$77, every automorphism of $\gamma:S^1\to\mathbb R^2$78 is inner. In $\gamma:S^1\to\mathbb R^2$79, Caplinger–Salter prove that if $\gamma:S^1\to\mathbb R^2$80 is totally symmetric, then $\gamma:S^1\to\mathbb R^2$81, and for $\gamma:S^1\to\mathbb R^2$82, equality holds exactly for the standard simplex-based construction up to conjugacy. In braid groups, Kordek–Margalit classify maximal commuting totally symmetric sets of size $\gamma:S^1\to\mathbb R^2$83, and Dyer–Grossman show that for $\gamma:S^1\to\mathbb R^2$84,

$\gamma:S^1\to\mathbb R^2$85

generated by inversion $\gamma:S^1\to\mathbb R^2$86. The same framework extends to strong collapsing sets, yielding Kolay’s theorem on the unique smallest non-cyclic quotient of $\gamma:S^1\to\mathbb R^2$87 and Kielak–Pierro’s theorem that for $\gamma:S^1\to\mathbb R^2$88, the unique smallest non-cyclic quotient of $\gamma:S^1\to\mathbb R^2$89 is $\gamma:S^1\to\mathbb R^2$90 (Caplinger et al., 2024).

This algebraic notion is terminologically related to, but mathematically distinct from, geometric symmetry sets of curves. The former concerns conjugation orbits and homomorphism rigidity; the latter concerns envelopes, bitangencies, and singular loci. The shared term reflects symmetry constraints, but the operative structures are different: full symmetric-group permutation in one case, and singular geometry of line or pseudo-circle families in the other.