Hexagonal Circular 3-Webs

• Hexagonal circular 3-webs are configurations of three families of circular arcs or lines that locally mimic parallel lines, exhibiting vanishing Blaschke curvature.
• The analysis employs Blaschke connections, Möbius transformations, and polar curve techniques to bridge planar web geometry with algebraic and projective geometry.
• Explicit constructions using confocal conics, Apollonian circles, and Darboux cyclides, along with symmetry and local normal form studies, clarify their classification and structural properties.

Searching arXiv for the cited works and closely related papers on hexagonal circular 3-webs. Hexagonal circular 3-webs are planar or spherical $3$-webs whose leaves are arcs of circles or lines and which are hexagonal, or flat, in the sense of web geometry: locally they are equivalent to three families of parallel straight lines. In the classical Blaschke–Bol problem, they are studied through vanishing Blaschke curvature and, in Möbius geometry, through the polar curve formed by the polar points of the web circles. This viewpoint connects planar web geometry, the projective geometry of the unit sphere, and the algebraic geometry of polar curves (Agafonov, 2023, Agafonov, 16 Jul 2025).

1. Definitions and hexagonality

A planar $3$-web is a superposition of three foliations $\mathcal F_1,\mathcal F_2,\mathcal F_3$ by smooth curves. At a regular point the three line fields are pairwise transverse. Locally one may choose $1$-forms $\sigma_i$ defining the foliations and normalize them by

$\sigma_1+\sigma_2+\sigma_3=0.$

The Blaschke connection form $\gamma$ is then defined by

$d\sigma_i+\gamma\wedge \sigma_i=0,\qquad i=1,2,3,$

and the Blaschke curvature is $K_B=d\gamma$. A planar $3$-web is hexagonal if locally it is equivalent to three families of parallel straight lines; equivalently, $3$0 (Agafonov, 2023).

A circular $3$1-web is a $3$2-web whose leaves are arcs of circles and/or straight lines. Under stereographic projection, planar circular webs become circular webs on the unit sphere, and conversely. In the spherical formulation, if $3$3 are local first integrals of the three foliations and $3$4, then hexagonality is equivalent to

$3$5

In the projective-polar description, hexagonality is also characterized by the existence of parameters $3$6 on the three polar components such that

$3$7

for the three circles through a generic point of the sphere (Agafonov, 16 Jul 2025).

2. Möbius geometry and polar curves

The natural global setting is Möbius geometry of the unit sphere $3$8. In homogeneous coordinates $3$9, the sphere is the quadric

$\mathcal F_1,\mathcal F_2,\mathcal F_3$0

A circle on $\mathcal F_1,\mathcal F_2,\mathcal F_3$1 is the intersection with a non-tangent plane

$\mathcal F_1,\mathcal F_2,\mathcal F_3$2

and this plane has polar point $\mathcal F_1,\mathcal F_2,\mathcal F_3$3. Thus a one-parameter family of circles corresponds to a curve of polar points in $\mathcal F_1,\mathcal F_2,\mathcal F_3$4, and the union of the three relevant curves is the polar curve of the web (Agafonov, 16 Jul 2025).

In planar tetracyclic coordinates, a circle

$\mathcal F_1,\mathcal F_2,\mathcal F_3$5

has polar point

$\mathcal F_1,\mathcal F_2,\mathcal F_3$6

This makes it possible to pass explicitly between plane equations of circles and projective equations of polar curves. Möbius transformations preserve circles, so classification is naturally taken up to Möbius equivalence. Line components of the polar curve correspond to pencils of circles, and their position relative to the Darboux quadric distinguishes hyperbolic, elliptic, and parabolic pencils (Agafonov, 2023).

3. Classical constructions and explicit planar families

The classical background begins with the Blaschke–Bol problem: classify all hexagonal $\mathcal F_1,\mathcal F_2,\mathcal F_3$7-webs in the plane formed by circles. Standard sources of examples include the Blaschke web of three elliptic pencils of circles, the Graf–Sauer theorem for linear hexagonal $\mathcal F_1,\mathcal F_2,\mathcal F_3$8-webs, the Volk–Strubecker construction obtained by Darboux transformations of linear webs, Wunderlich’s constructions from a $\mathcal F_1,\mathcal F_2,\mathcal F_3$9-parameter Möbius group, and Shelekhov’s classification of webs formed by three circle pencils (Nilov, 2013).

Several explicit planar families are built from confocal conics and Apollonian circles. In the positive quadrant $1$0, with foci

$1$1

define

$1$2

Here $1$3 gives the hyperbolic Apollonian pencil, $1$4 the elliptic Apollonian pencil, $1$5 the confocal hyperbolas, and $1$6 the confocal ellipses. Four triples were shown to form trivial $1$7-webs, hence hexagonal $1$8-webs: both Apollonian pencils together with confocal hyperbolas; both Apollonian pencils together with confocal ellipses; confocal ellipses and hyperbolas together with the hyperbolic Apollonian pencil; and confocal ellipses and hyperbolas together with the elliptic Apollonian pencil. In each case an explicit diffeomorphism sends the web to three families of parallel lines (Akopyan, 2016).

Nilov added five further planar examples. They include tangent lines to a circle counted twice together with a parabolic pencil of circles with vertex at the center of the circle; tangent lines to a general conic counted twice together with a hyperbolic pencil with limiting points at the foci of the conic; tangent lines to a general conic, a pencil of lines with vertex at a focus, and circles doubly tangent to the conic whose centers lie on the minor axis; tangent lines to a parabola counted twice together with a hyperbolic pencil whose limiting points are the focus and an arbitrary point on the directrix; and circles doubly tangent to an ellipse of eccentricity $1$9, counted twice, together with the elliptic pencil whose vertices are the foci (Nilov, 2013).

4. Algebraic polar curves of degree three

A major organizing principle is the degree and reducibility of the polar curve. For degree $\sigma_i$0, reducible cases are either three non-coplanar lines or one line plus one smooth conic. The three-line case is the circle-pencil case; the line-plus-conic case produces a much larger list. One classification gives $\sigma_i$1 Möbius types for three non-coplanar lines and $\sigma_i$2 Möbius types for a line plus a smooth conic, and states that if the polar curve of a hexagonal circular $\sigma_i$3-web is algebraic of degree three, then either it is planar or the web is Möbius equivalent to one of those reducible cases (Agafonov, 2023).

A later paper treats the remaining irreducible non-planar cubic case and describes hexagonal circular $\sigma_i$4-webs on the unit sphere such that the polar points lie on a twisted cubic. Up to a Möbius transformation, the polar twisted cubic is

$\sigma_i$5

and the paper states that this gives a two-parameter family of Möbius inequivalent twisted cubics and completes the classification of hexagonal circular $\sigma_i$6-webs with algebraic polar curves of degree three (Agafonov, 16 Jul 2025).

The sources represented here therefore record two different conclusions about the twisted-cubic case. One source states that there is no hexagonal circular $\sigma_i$7-web whose polar curve is a rational normal curve, whereas another classifies a two-parameter family with polar twisted cubic (Agafonov, 2023, Agafonov, 16 Jul 2025).

5. Symmetry, singularities, and local normal forms

A separate but nearby line of work studies symmetries of planar $\sigma_i$8-webs around a point. In that terminology, a simple symmetry preserves each foliation, a mirror symmetry preserves one foliation and swaps the other two, and a circular symmetry cyclically permutes the three foliations. Hexagonal planar $\sigma_i$9-webs always have simple, mirror, and circular symmetries. In suitable local coordinates,

$\sigma_1+\sigma_2+\sigma_3=0.$0

and the web is flat, or hexagonal, if and only if $\sigma_1+\sigma_2+\sigma_3=0.$1. The same source also shows that circular symmetry alone does not force flatness: there exist non-flat $\sigma_1+\sigma_2+\sigma_3=0.$2-webs with a circular symmetry around a point (Dufour, 28 Jan 2025).

For singular web geometry, there is a complete classification of hexagonal singular $\sigma_1+\sigma_2+\sigma_3=0.$3-web germs in the complex plane satisfying two conditions: first, the Chern connection remains holomorphic at the singular point; second, the web admits at least one infinitesimal symmetry at this point. As a by-product, a classification of hexagonal weighted homogeneous $\sigma_1+\sigma_2+\sigma_3=0.$4-webs is obtained. That work does not explicitly use the phrase “circular $\sigma_1+\sigma_2+\sigma_3=0.$5-web,” but it gives strong structural information that can be applied to circular webs or to webs locally equivalent to circular ones (Agafonov, 2011).

Flat local models also arise from Frobenius geometry. Webs constructed from semi-simple geometric Frobenius $\sigma_1+\sigma_2+\sigma_3=0.$6-folds are hexagonal and biholomorphic to characteristic webs on the solutions of the corresponding associativity equation, and they admit at least one infinitesimal symmetry at each singular point. This places hexagonal circular $\sigma_1+\sigma_2+\sigma_3=0.$7-webs within a broader class of flat $\sigma_1+\sigma_2+\sigma_3=0.$8-webs sharing the same local Chern-connection formalism (Agafonov, 2011).

6. Global settings, extensions, and terminological boundaries

Hexagonal circular $\sigma_1+\sigma_2+\sigma_3=0.$9-webs also appear on surfaces in $\gamma$0. Darboux cyclides carry up to six real families of circles, and the possible hexagonal circular $\gamma$1-webs on a nontrivial irreducible Darboux cyclide were completely classified. Three circle families form a $\gamma$2-web except in one forbidden configuration, and exactly five types occur: three non-single families with no two paired; two special paired families together with one additional family that has a paired partner; one single family with two paired families; one single family with two non-single families that are not paired together; and two single families together with one additional family that has a paired partner (Pottmann et al., 2011).

A geometric generalization replaces circles by geodesics. A surface carries a hexagonal $\gamma$3-web of geodesics if and only if the geodesic flow on the surface admits a cubic first integral. This suggests a useful bridge: circular webs can often be studied through metrics or projective models in which the relevant circles become geodesics, while the hexagonality condition remains the vanishing-curvature condition of web geometry (Agafonov, 2018).

The term “$\gamma$4-web” is also used in representation theory and skein theory for planar trivalent graphs. In uniformly random reduced $\gamma$5 webs, almost all interior faces far from the boundary are hexagons, while in the reduced $\gamma$6-skein algebra every reduced $\gamma$7-web can be canonically decomposed into unions of pyramid formations of hexagons and disjoint arcs with possible additional crossbars. This is a different notion from planar foliation $\gamma$8-webs, but it explains why “hexagonal webs” appears in two mathematically distinct literatures (Kogan, 1 Oct 2025, Frohman et al., 2020).