Hexagonal Circular 3-Webs with Reducible Polar Curves of Degree Three
The paper reports the progress with the classical problem, posed by Blaschke and Bol in 1938. We present new examples and new classifications of natural classes of hexagonal circular 3-webs. The main results are the classification of hexagonal circular 3-webs with reducible polar curves of degree 3...
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| description | The paper reports the progress with the classical problem, posed by Blaschke and Bol in 1938. We present new examples and new classifications of natural classes of hexagonal circular 3-webs. The main results are the classification of hexagonal circular 3-webs with reducible polar curves of degree 3 and the description of hexagonal circular 3-webs admitting a one-parameter Möbius symmetry.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 21 (2025), 043, 31 pages
Hexagonal Circular 3-Webs
with Reducible Polar Curves of Degree Three
Sergey I. AGAFONOV
Department of Mathematics, São Paulo State University-UNESP, São José do Rio Preto, Brazil
E-mail: sergey.agafonov@gmail.com
Received November 18, 2023, in final form June 04, 2025; Published online June 13, 2025
https://doi.org/10.3842/SIGMA.2025.043
Abstract. The paper reports the progress with the classical problem, posed by Blaschke
and Bol in 1938. We present new examples and new classifications of natural classes of
hexagonal circular 3-webs. The main results is the classification of hexagonal circular 3-
webs with reducible polar curves of degree 3 and description of hexagonal circular 3-webs
admitting a one-parameter Möbius symmetry.
Key words: circular hexagonal 3-webs
2020 Mathematics Subject Classification: 53A60
1 Introduction
The problem to describe hexagonal 3-webs formed by circles in the plane appeared in the first
monograph on the web theory published by Blaschke and Bol in 1938 (see [4, p. 31]). The authors
presented an example with 3 elliptic pencils of circles, each pair of pencils sharing a common
vertex, and observed that one can construct hexagonal circular 3-webs from hexagonal linear
3-webs, completely described by Graf and Sauer [7] as being formed by tangents to a fixed
curves of third class. The construction involves a central projection from a plane to a unit
sphere followed by stereographic projection to a plane. The corresponding circular 3-webs were
described earlier by Volk [14] and Strubecker [13].
Stereographic projection puts the problem into a natural framework of the Möbius geometry:
instead of planar circular webs we study circular webs on the unit sphere, thus treating circles
and straight lines on equal footing. Möbius geometry assigns points outside the unit sphere to
circles on this sphere: the assigned point is the polar point of the plane that cuts the circle on
the sphere. This sphere is called also Darboux quadric. Thus any circular 3-web on the unit
sphere determines locally 3 curve arcs outside the Darboux quadric, one arc per web foliation.
Globally these arcs may belong to one irreducible algebraic curve. In what follows we call this
set of polar points a polar curve of the web. For example, the polar curve of the hexagonal
circular 3-web obtained from a linear 3-web is a planar cubic, possibly reducible. The polar
curve of the cited example from [4] splits into 3 non-coplanar lines.
In the same year as the book [4] appeared, Wunderlich published a new remarkable example
of hexagonal circular 3-web. Its polar curve splits into 3 conics lying in 3 different planes. Since
through a point p on the unit sphere pass the circles whose polar points are intersection of the
web polar curve with the plane tangent to the unit sphere at p, the Wunderlich web is actually
6-web, containing 8 hexagonal 3-subwebs.
Wunderlich gave also a construction of hexagonal 3-webs whose polar curve splits into 3 non-
coplanar lines, two being dual with respect to the Darboux quadric and the third joining them.
These webs were later rediscovered by other authors.
Further, he presented the following way to construct hexagonal 3-webs: for any one-para-
metric group acting in the plane, choose 2 transversally intersecting curve arcs that are also
mailto:sergey.agafonov@gmail.com
https://doi.org/10.3842/SIGMA.2025.043
2 S.I. Agafonov
transversal to the group orbits; acting on the arcs by the group one gets 2 foliations; the third is
composed by the group orbits. These 3 foliations compose a (local) hexagonal 3-web. Choosing
a one-parameter group either of translations, or of dilatations, or of rotations and taking two
intersecting circles (a straight line counts as a circle), we get circular hexagonal 3-webs.
Blaschke was well aware of the difficulty of the posed problem and, in his last book on the
web geometry [3], discussed the simpler problem of classifying hexagonal circular 3-webs whose
polar curve splits into 3 non-coplanar lines. Note that to a line corresponds a pencil of circles
that is hyperbolic if the line spears the Darboux quadric, elliptic if the line completely misses
the Darboux quadric, or parabolic if the line touches the Darboux quadric.
By the year 1977, the list of 6 types (one from [4] and five indicated in [15]) of circular
hexagonal 3-webs whose polar curve splits into 3 non-coplanar lines was completed by Erdoǧan [5]
and Lazareva [8]. The first attempt to prove that the list is actually complete was published in
1989 by Erdoǧan [6]. Based on direct computational approach, it did not provide the crucial
computation: in fact, a modern computer systems for symbolic computations shows that there
must be a mistake in the proof presented in [6] (see concluding remarks for further detail).
The Erdoǧan’s claim was proved only in 2005 by Shelekhov [12]. His insight was to look into
the singular set of the webs: defined globally, the webs under study inevitably have singularities.
Shelekhov considered the simplest possible singularities where two of the three circular foliations
are tangent. It turns out that hexagonality imposes a strong restriction: locally, such singular
set is either a circle arc of the 3rd foliation or the common circle arc of the first two. The
restriction was rigid enough to obtain all the types on the list.
Five new types of hexagonal circular 3-webs were presented by Nilov in 2014 [10]. Polar
curves for four of them split into a line and a conic. The fifth example may be viewed as a 5-web
whose polar curve is a union of a line and two conics. Taking the line and two arcs on different
conics as the polar curve, one gets a hexagonal 3-subwebs.
One can not help to observe that the polar curves of all the known examples are algebraic.
Motivated also by the dual reformulation of the Graf and Sauer theorem, we consider the fol-
lowing natural class of 3-webs: hexagonal circular 3-webs with polar curve of degree three. The
main result of the paper is the complete classification of such webs with reducible polar curves.
The case of planar polar curve follows immediately from the Graf and Sauer theorem: 3 points
on the polar curve corresponding to 3 circles through a point p on the sphere are the ones where
the plane, tangent to the sphere at p meets the polar curve. This plane cuts the polar curve
plane along the line. On the polar curve plane we get the configuration dual to the Graf and
Sauer theorem.
The case of non-planar set of 3 lines was finally settled by Shelekhov [12].
We obtain a classification of 3-webs whose planar polar curve splits into a line and a smooth
conic. Up to Möbius transformation, there are 15 types, most of them depending on one pa-
rameter. Four types of five in Nilov’s paper [10] are webs of this list, namely, of the types 6,
10, 11 and 15, presented in Section 5. (In fact, Nilov has found only one Möbius orbit from
one-parametric family of orbits of our type 6.)
Another natural class that we study in this paper is the set of hexagonal circular 3-webs
symmetric by action of one-parameter subgroup of the Möbius group. We also give a complete
classification of such webs.
To select candidates for hexagonal webs we exploit further the above mentioned observation
of Shelekhov on simplest singularities of hexagonal 3-webs. The proof of the observed property
in [12], based on considering the normal form of the web function is not complete: this normal
form often does not exists at singular points (see concluding remarks for more detail). We make
precise the ideas about the type of singularities and then prove the key singularity property.
For completeness, we also present the classification of hexagonal webs with 3 non-coplanar
polar lines. The proof mainly follows the line taken by Shelekhov in [12].
Hexagonal Circular 3-Webs with Reducible Polar Curves of Degree Three 3
2 Hexagonal 3-webs, Blaschke curvature, singularities
A planar 3-web W3 in a planar domain is a superposition of 3 foliations Fi, which may be given
by integral curves of three ODEs σ1 = 0, σ2 = 0, σ3 = 0, where σi are differential one-forms. At
non-singular points, where the kernels of these forms are pairwise transverse, we normalize the
forms so that σ1 + σ2 + σ3 = 0. The connection form of the web W3 is a one-form γ determined
by the conditions dσi+γ∧σi = 0, i = 1, 2, 3. The connection form depends on the normalization
of the forms σi, the Blaschke curvature dγ does not.
Definition 2.1. A 3-web is hexagonal if for any non-singular point there are a neighbourhood
and a local diffeomorphism sending the web leaves of this neighbourhood in 3 families of parallel
line segments.
Topologically, hexagonality means the following incidence property that has given its name
to the notion: for any point m, each sufficiently small curvilinear triangle with the vertex m
and sides formed by the web leaves, may be completed to the curvilinear hexagon, whose sides
are web leaves and whose “large” diagonals are the web leaves meeting at m (see the gallery
of pictures illustrating hexagonal webs in the next section). Computationally, hexagonality
amounts to vanishing of the Blaschke curvature [4].
Up to a suitable affine transformation, the forms σi may be normalized as follows:
σ1 = (Q−R)(dy − Pdx), σ2 = (R− P )(dy −Qdx), σ3 = (P −Q)(dy −Rdx),
where P (x, y), Q(x, y), R(x, y) are the slopes of the tangent lines to the web leaves at (x, y).
Vanishing of the curvature writes as
(R−Q)[Pxx + (Q+R)Pxy +QRPyy] + (P −R)[Qxx + (P +R)Qxy + PRQyy]
+ (Q− P )[Rxx + (P +Q)Rxy + PQRyy]
+
(Q−R)(2P −Q−R)
[
P 2
x + (Q+R)PxPy +QRP 2
y
]
(P −Q)(P −R)
+
(R− P )(2Q− P −R)
[
Q2
x + (P +R)QxQy + PRQ2
y
]
(Q−R)(Q− P )
+
(P −Q)(2R− P −Q)
[
R2
x + (P +Q)RxRy + PQR2
y
]
(R−Q)(R− P )
+
(2R− P −Q)PxQx
P −Q
+
(2P −Q−R)QxRx
Q−R
+
(2Q−R− P )RxPx
R− P
+
(
R2 − PQ
)
[PxQy + PyQx]
(P −Q)
+
(
P 2 −QR
)
[QxRy +QyRx]
Q−R
+
(
Q2 − PR
)
[RxPy +RyPx]
R− P
+
(
2PQR− (P +Q)R2
)
PyQy
Q− P
+
(
2PQR− (Q+R)P 2
)
QyRy
R−Q
+
(
2PQR− (P +R)Q2
)
RyPy
P −R
= 0. (2.1)
If only one slope, say R, is given as an explicit functions of x, y and P , Q are roots of
a quadratic equation P 2 +AP +B = 0, then one finds the first derivatives of P , Q by differen-
tiating the Vieta relations
P +Q = −A, PQ = B, (2.2)
as a functions of P , Q and the first derivatives of A, B. Differentiating these expressions, one
gets also the second derivatives. Finally, excluding P and Q with the help of (2.2), one can
rewrite (2.1) in terms of A, B, R and their derivatives. The result is presented in Appendix A.
4 S.I. Agafonov
The webs considered later will inevitably have singularities: some kernels of the forms σi can
be not transverse or the forms can vanish at some points. We call a singular 3-web hexagonal if
its Blaschke curvature vanishes identically at regular points. The simplest type of singularities
of hexagonal 3-webs have the following remarkable property, first observed by Shelekhov [12].
Lemma 2.2. Suppose that a hexagonal 3-web, defined by three analytic direction fields ξ1, ξ2, ξ3,
has a singular point p0 such that
(1) all ξi are well defined at p0,
(2) ξ1 and ξ2 are transverse at p0,
(3) ξ1 = ξ3 at p0,
then either the leaves of ξ1 and ξ3 through p0 coincide or ξ1 = ξ3 along the leaf of ξ2 through p0.
Proof. The property is a consequence of separation of variables for hexagonal webs. The second
condition implies that we can rectify ξ1 and ξ2, i.e., choose some local coordinates u, v so that
ξ1 = ∂v and ξ2 = ∂u. Then ξ3 = −f(u, v)∂u + ∂v with f(u0, v0) = 0, where p0 = (u0, v0).
Consider the analytic functions φ(u) := f(u, v0).
If φ(u) ≡ 0, then f(u, v0) = 0 and the leaves of ξ1 and ξ3 are tangent along the line v = v0,
which is the leaf of ξ2.
If φ(u) ̸≡ 0, then φ(u) = (u − u0)
nψ(u) with natural n and ψ(u0) ̸= 0. Thus for any u1
close to u0 with u1 ̸= u0 holds f(u1, v0) = (u1 − u0)
nψ(u1) ̸= 0. Now the hexagonality amounts
to ∂v
(∂uf
f
)
= 0 hence the germ f̃ of f at (u1, v0) factors f̃(u, v) = a(u)b(v) with analytic germs
a, b at u1 and v0 respectively. One can choose b(u) so that b(v0) = 1. Then a(u) = ψ(u) =
(u−u0)
nψ(u) is analytic also at u0, the function of two variables a(u)b(v) is analytic at (u0, v0)
and coincides with f(u, v) at some neighborhood of (u1, v0) included in the domain of f . Then
by uniqueness f(u, v) = a(u)b(v). Now observe that a(u0) = 0 which implies f(u0, v) ≡ 0 and
the integral curves of ξ1 and ξ3 passing through p0 coincide. ■
3 Projective model of Möbius geometry
Following Blaschke [2], we call the subgroup PSO(3, 1) of projective transformations of RP3,
leaving invariant the quadric X2 + Y 2 + Z2 − U2 = 0, the Möbius group. For the reference, we
present here infinitesimal generators of Möbius group in homogeneous coordinates [X : Y : Z : U ]
in P3, affine coordinates x = X
U , y = Y
U , z = Z
U in R3 and cartesian coordinates (x̄, ȳ) in R2 related
to points (x, y, z) on the unit sphere via stereographic projection x = 2x̄
1+x̄2+ȳ2
, y = 2ȳ
1+x̄2+ȳ2
,
z = 1−x̄2−ȳ2
1+x̄2+ȳ2
. There are 3 rotations around the affine axes:
Rz = Y ∂X −X∂Y = y∂x − x∂y = ȳ∂x̄ − x̄∂ȳ,
Rx = Z∂Y − Y ∂Z = z∂y − y∂z = x̄ȳ∂x̄ +
1
2
(
1− x̄2 + ȳ2
)
∂ȳ,
Ry = X∂Z − Z∂X = x∂z − z∂x = −1
2
(
1 + x̄2 − ȳ2
)
∂x̄ − x̄ȳ∂ȳ,
and 3 boosts (or “hyperbolic rotations”)
Bx = U∂X +X∂U = ∂x − x(x∂x + y∂y + z∂z) =
1
2
(
1− x̄2 + ȳ2
)
∂x̄ − x̄ȳ∂ȳ,
By = U∂Y + Y ∂U = ∂y − y(x∂x + y∂y + z∂z) = −x̄ȳ∂x̄ +
1
2
(
1 + x̄2 − ȳ2
)
∂ȳ,
Bz = U∂Z + Z∂U = ∂z − z(x∂x + y∂y + z∂z) = −x̄∂x̄ − ȳ∂ȳ.
Hexagonal Circular 3-Webs with Reducible Polar Curves of Degree Three 5
The identity component of PSO(3, 1) is well known to be isomorphic to the group PSL2(C),
the isomorphism being given by the action A(V ) = AV A∗ of A ∈ SL2(C) on the vector space
of matrices
V =
(
X + U Y + iZ
Y − iZ U −X
)
with real X, Y , Z, U . This action preserves determinant of V , which is U2−X2−Y 2−Z2. By
this isomorphism, the generators are represented by the following matrices:
Rx =
1
2
(
−i 0
0 i
)
, Ry =
1
2
(
0 −i
−i 0
)
, Rz =
1
2
(
0 1
−1 0
)
,
Bx = iRx, By = iRy, Bz = iRz.
Two points p1 = [X1 : Y1 : Z1 : U1] and p2 = [X2 : Y2 : Z2 : U2] in RP3 determine a line with
the Plücker coordinates
a := X1U2 −X2U1, b := Y1U2 − Y2U1, c := Z1U2 − Z2U1,
f := Y1Z2 − Y2Z1, g := Z1X2 − Z2X1, h := X1Y2 −X2Y1.
By direct computation, one proves the following fact.
Lemma 3.1. All points of a line with Plücker coordinates [a : b : c : f : g : h] are stable with
respect to subgroup with the infinitesimal generator aRx + bRy + cRz + fBx + gBy + hBz.
Observe that the line dual to [a : b : c : f : g : h] is the one with coordinates [−f : −g : −h :
a : b : c], which corresponds to multiplication by i of the corresponding matrix representation of
the generator.
A line in RP3 can be hyperbolic, elliptic or parabolic with the respect to the Darboux quadric.
Considering simple representatives of these classes one sees that
(1) For hyperbolic line, the action of the corresponding operator on the dual line is Möbius
conjugate to the action of Rz on the line Z = U = 0 at infinity thus not having extra
stable points in RP3.
(2) For elliptic line, the action of the corresponding operator on the dual line is Möbius
conjugate to the action of Bz on the line X = Y = 0 (which action leaves invariant two
extra points [0 : 0 : ±1 : 1]).
(3) For parabolic line, the corresponding operator moves all points on the dual line (con-
sider ∂y = Rx +By).
In the following proposition we summarize further properties of the above correspondence.
Proposition 3.2. Let ξ = aRx + bRy + cRz + fBx + gBy + hBz be an infinitesimal operator of
the Möbius group.
(1) The corresponding action of the one-parameter group is not loxodromic and therefore
Möbius equivalent (i.e., conjugated) either to rotation, or dilatation, or translation of the
plane (x̄, ȳ) if and only if
af + bg + ch = 0. (3.1)
(2) This action is Möbius equivalent to rotation if and only if (3.1) and a2+b2+c2 > f2+g2+h2
are true, the line with Plücker coordinates [a : b : c : f : g : h] being the set of polar points
of the circular orbits.
6 S.I. Agafonov
(3) This action is Möbius equivalent to dilatation if and only if (3.1) and a2 + b2 + c2 <
f2 + g2 + h2 are true, the line with Plücker coordinates [a : b : c : f : g : h] being the set of
polar points of the orbits.
(4) The action is Möbius equivalent to translation if and only if (3.1) and
a2 + b2 + c2 = f2 + g2 + h2 (3.2)
are true, the line with Plücker coordinates [a : b : c : f : g : h] being the set of polar points
of the orbits.
Proof. The type of subgroup transformations is determined by the eigenvalues of the matrix
representation for the generating operator ξ. Consider the characteristic polynomial for ξ
Ψ(λ) = λ2 +
1
4
(
a2 + b2 + c2 − f2 − g2 − h2
)
+
i
2
(af + bg + ch)
and the matrices for non-loxodromic Möbius representatives Rz, Bz and ∂y = Rx +By. ■
4 Polar curve splits into 3 non-coplanar lines
Recall that limit circles of a hyperbolic pencil correspond to intersection points of the hyperbolic
line with the Darboux quadric under the stereographic projection. Vertexes of an elliptic pencil
are two points common to all the circles of the pencil, the vertexes correspond to the intersection
of the Darboux quadric with the line, dual to the polar elliptic line. Considering a parabolic
pencil as a limit case of elliptic, one calls the point corresponding to the point of tangency of
the Darboux quadric with the parabolic line also the vertex. The vertex of a parabolic pencil is
the common point for all circles of this parabolic pencil.
The reader may visualize the pencil fixed by a line L thinking of its circles as cut on the
Darboux quadric by the pencil of planes containing the dual line L∗.
Let us list hexagonal 3-webs with non-planar polar lines. There are 9 types of such webs up
to Möbius transformation.
Figure 1. Hexagonal circular 3-webs. 3 hyperbolic pencils (left), 1 hyperbolic and 2 elliptic pencils
(center), 1 elliptic and 2 hyperbolic pencils (right).
Take 3 polar lines intersecting inside the Darboux quadric so that each of the three lines
contains the point dual to the plane of the other two polar lines, i.e., each pencil has a circle or-
thogonal to all the circles of the other two pencils. A representative of this web orbit, having one
limit circle at infinity, is shown in Figure 1 on the left. This type was described by Lazareva [8].
Hexagonal Circular 3-Webs with Reducible Polar Curves of Degree Three 7
Replacing two pencils by their orthogonal, we get the web in the center of Figure 1. In the
projective model, we replace two polar lines by their dual ones. This web was also described by
Lazareva [8].
Figure 2. Hexagonal circular 3-webs. 1 hyperbolic and 2 elliptic pencils (left), 1 parabolic, 1 elliptic,
and 1 hyperbolic pencil (center), 1 hyperbolic and 2 parabolic pencils (right).
Wunderlich [15] mentioned the following construction, used later also by Balabanova and
Erdoǧan, to produce hexagonal 3-webs: take two dual polar lines and supplement it by a third in-
tersecting that dual pair. There are four webs in the list, obtained in this way (see also [1]): with
two hyperbolic and one elliptic pencils on the right of Figure 1; with one hyperbolic and two ellip-
tic pencils on the left of Figure 2; with one hyperbolic, one elliptic, and one parabolic pencils in
the center of Figure 2; and with one hyperbolic and two parabolic pencils on the right of Figure 2.
Another web with two elliptic and one hyperbolic pencils is depicted on the left of Figure 3.
Its projective model has two elliptic polar lines, lying in a plane tangent to the Darboux quadric
at a point, and a hyperbolic line passing through the points (different from the above tangency
point), where duals to elliptic lines intersect the Darboux quadric. Its elliptic pencils share one
common vertex at infinity, the other two vertices are also the limit circles of the hyperbolic
pencil. This web was described by Erdoǧan [5].
Figure 3. Hexagonal circular 3-webs. 1 hyperbolic and 2 elliptic pencils (left), 3 elliptic pencil (center),
1 hyperbolic, 1 elliptic and 1 parabolic pencil (right).
In the center of Figure 3, there is a web with three elliptic pencils, the vertices of the pencils
are two of three fixed points. It is historically the first hexagonal circular 3-web described in
the literature [4]. The chosen representative has a vertex at infinity.
8 S.I. Agafonov
Erdoǧan [5] found a web with one hyperbolic, one elliptic and one parabolic pencil, arranged
so that the vertex P of the parabolic pencil coincides with one vertex of the elliptic pencil, while
the other vertex E of the elliptic pencil coincides with one of the limiting circles of the hyperbolic,
the common circle of the elliptic and the parabolic pencils being orthogonal to the circle passing
through the second limiting circle of the hyperbolic pencil and the points P , E. On the right of
Figure 3 is a Möbius representative of this type web with P at infinity. On the projective model,
we have 3 pairwise distinct points on the Darboux quadric: E, P and H. The hyperbolic polar
line Lh spears the quadric at E and H, the parabolic polar Lp touches the quadric at P and
intersects Lh, and the elliptic polar Le is dual to the line through P and E (and intersects Lp).
Finally, we present a family of hexagonal 3-webs formed by 3 pencils, whose Möbius orbits
are parameterized by one parameter. Two pencils are parabolic with distinct vertexes, and
the third is elliptic, whose dual to the polar line meets the Darboux quadric at these vertexes.
A representative of the family is shown in Figure 4, the vertexes being the origin and the infinite
point. In this normalization one can fix the direction of one parabolic line, the direction of the
other is arbitrary. Any web in this normalization is symmetric by dilatation x∂x + y∂y.
Figure 4. Hexagonal circular 3-web. 1 elliptic and 2 parabolic pencils.
Surprising is not only the fact that the “largest” family was explicitly described only in 1977
by Lazareva [8], even more amazing is that the family falls within the general construction (see
Introduction), which seems to have appeared first in the paper of Wunderlich [15] in 1938!
In what follows we refer to the above described webs as Blaschke–Wunderlich–Balabanova–
Erdoǧan–Lazareva list. To prove that there are no other classes, we will strongly use Lemma 2.2.
The singularity of the described type occurs when two of the three lines, tangent to Darboux
quadric at a point and meeting each its own polar line, coincide and the point is not a vertex
or limit circle of any pencil.
Proposition 4.1. Consider the family of lines such that
(1) they meet two fixed lines L1 and L2, and
(2) they are tangent to the Darboux quadric.
If the tangent points are on a circle, then either L1 intersects L2, or both L1 and L2 are tangent
to the Darboux quadric, or one meets the dual of the other at some point on the Darboux quadric.
Moreover, in the case of skew L1, L2 tangent to the Darboux quadric at two points p1, p2, the
curve of touching points splits into 2 circles, their planes containing the line p1p2 and bissecting
the angles between two planes P1, P2, where Pi is the plane through Li and p1p2.
Proof. Let [a : b : c : f : g : h] be Plücker coordinates of a line L touching the Darboux quadric.
Then, by Proposition 3.2, they satisfy equations (3.1) and (3.2). Therefore, a2+ b2+ c2 ̸= 0 and
we can normalize these coordinates to a2 + b2 + c2 = f2 + g2 + h2 = 1. One easily calculates
Hexagonal Circular 3-Webs with Reducible Polar Curves of Degree Three 9
the point p = (x, y, z) = (cg − bh, ah − cf, bf − ag) where the line L touches the quadric. Due
to normalization, one can rewrite this as
a = hy − gz, b = fz − hx, c = gx− fy. (4.1)
To simplify calculations, we can bring the Plücker coordinates of L1 to simple form by Möbius
transformation. We suppose that L1 and L2 are skew since the case of L1, L2 intersecting is
obvious.
If L1 is hyperbolic, we can choose a representative as L1 = [0 : 0 : 1 : 0 : 0 : 0]. Then one has
m ̸= 0 since the lines L1, L2 = [u : v : w : k : l : m] do not intersect. (We used the fact that the
Plücker coordinates of intersecting lines are orthogonal with respect to the bilinear symmetric
form defining the Plücker quadric.) Let us set m = 1. Moreover, the coordinates of L2 satisfy
the Plücker equation ku+ lv+w = 0 thus giving w. Since L intersect L1 and L2, we have h = 0
and ka+ lb+ c+ uf + gv+wh = 0, respectively. The above two equations cut a curve from the
three-dimensional variety of lines touching the quadric.
The touching points p trace a curve on the Darboux quadric. This curve can be computed as
follows. Equations (3.1) and (4.1) imply fx+gy+hz = 0, with h = 0 we have g = −fx/y. Now
normalization f2 + g2 + h2 = 1 gives
(
x2 + y2
)
f2 = y2, which means f is not identically zero.
Therefore, intersection condition ka + lb + c + uf + gv + wh = 0 is equivalent to kxz + lyz +
uy − vx− x2 − y2 = 0. This equation cuts the curve of tangent points on the Darboux quadric
(i.e., unit sphere centered at the origin). If this curve is in a plane z = Ax + By + C, then by
direct calculation one gets C2 = 1. One can choose C = 1 and then, by further calculations, we
get A = −k, B = −l, l = u, k = −v. Thus L2 = [u : v : 0 : −v : u : 1] lies in the plane tangent to
the Darboux quadric at (0, 0,−1), which is the point where the dual to L2 spears the Darboux
quadric. Note that the plane equation is z = vx− uy + 1.
If L1 is hyperbolic, we choose L1 = [0 : 0 : 0 : 0 : 0 : 1]. Then w ̸= 0 and we can
normalize L2 = [u : v : 1 : k : l : m] and the rest of the reasoning goes in a similar way.
If L1 is parabolic, we choose L1 = [0 : −1 : 0 : 1 : 0 : 0]. For skew L1, L2, we have −l+u ̸= 0.
Normalizing l − u = 1 gives l = u + 1. Further we rewrite (4.1) as f = bz − cy, g = cx − az,
h = ay− bx and, proceeding as before, obtain by calculation that L2 satisfy (3.1), which means
it is tangent to the Darboux quadric.
To check the last claim, we normalize the configuration so that L1 and L2 are tangent to the
Darboux quadric at points (0, 0,±1), look for a plane containing a circle of tangent points in
the form y = Ax, find two solutions for A, and verify the geometry by calculation. ■
Definition 4.2. Motivated by Lemma 2.2, we will refer to the circle of tangent points described
in Proposition 4.1 whose polar point is not an intersection of L1, L2 as singular circle.
Remark 4.3. It is easy to see that the curve of touching points is of degree four. In the
hypothesis of the proposition for the case of skew L1, L2 not tangent to the Darboux quadric,
this curve also splits. One component is a circle and the other has just one real point, where
dual to the elliptic line meets the hyperbolic.
Corollary 4.4. For hexagonal circular 3-webs formed by three pencils with non-planar polar
lines, any two L1, L2 of polar lines are either dual or obey geometrical restriction described by
Proposition 4.1. Moreover, the polar point of a singular circle, defined by two polar lines, belongs
to the third one.
Proof. Let L1, L2 be skew but not dual and such that the curve of points, where lines Lmeeting
both L1, L2 touch the Darboux quadric, is not planar. Then by Lemma 2.2 any such line meets
also the third polar line L3. Since L1, L2 are skew, L3 cannot intersect neither of L1, L2.
Thus L1, L2, L3 belong to one ruling of some quadric Q and the touching lines L form the other
10 S.I. Agafonov
ruling. But then L1, L2, L3 are also tangent to the Darboux quadric. This contradicts our
initial assumption. ■
Let (p, q), (r, s) ∈ R2 be vertices of an elliptic pencil, then the pencil circles form the family
I(x, y) :=
[(p− x)(r − x) + (q − y)(s− y)]2[
(p− x)2 + (q − y)2
][
(r − x)2 + (s− y)2
] = const.
The circles are the integral curves of the ODE
ωe := d(I) = f(x, y)dx+ g(x, y)dy = 0.
The circles of the hyperbolic pencil with limit circles at (p, q), (r, s) are orthogonal to the circles
of the above elliptic one. They are the integral curves of the ODE
ωh := g(x, y)dx− f(x, y)dy = 0.
Now we work out the cases with parabolic pencils. Let (p, q) ∈ R2 be the vertex of a parabolic
pencil and [r : 1 − r] ∈ P a direction orthogonal to the line tangent to all circles of the pencil.
Then the pencil circles form the family
Ĩ(x, y) :=
(x− p)2 + (y − q)2
r(p− x) + (r − 1)(y − q)
= const.
The circles are the integral curves of the ODE ωp := d
(
Ĩ
)
= 0. For the exceptional direc-
tion (1,−1), the pencil circles family is
Ī(x, y) :=
(x− p)2 + (y − q)2
(x− p)− (y − q)
= const
and the corresponding EDO ωp̄ := d
(
Ī
)
= 0.
Observe that differential forms σi, describing a 3-web of circles formed by 3 pencils, are
algebraic. Thus Lemma 2.2 remains valid also over complex numbers in passing from RP3
to CP3. By circles here we understand sections of the complex Darboux quadric by complex
planes. The complexification simplifies the proof of the following theorem.
Theorem 4.5 ([12]). Any hexagonal circular 3-webs formed by three pencils with non-coplanar
polar lines is Möbius equivalent to one from the Blaschke–Wunderlich–Balabanova–Erdoǧan–
Lazareva list.
Proof. We consider all types of non-coplanar polar line triples L1, L2, L3.
• Three hyperbolic pencils. By Corollary 4.4, all lines intersect at one point p. This point
cannot be outside the Darboux quadric. In fact, applying a suitable Möbius transformation,
we send this point to an infinite one, say px = [1 : 0 : 0 : 0] and the plane of L1, L2 to the
plane Z = 0. Then, by Corollary 4.4, the polar line L3 joins px and pz = [0 : 0 : 1 : 0], the polar
point of the plane Z = 0. Thus L3 cannot be hyperbolic as supposed. The point p cannot be
on the Darboux quadric either: we can send it to p = (1, 0, 0) and the plane of L1, L2 to the
plane Z = 0. Now Corollary 4.4 implies that L3 joins p and pz. Thus L3 is parabolic and not
hyperbolic.
Therefore, p is inside the Darboux quadric and one can send it to the origin (0, 0, 0). Corol-
lary 4.4 implies that any of the lines L1, L2, L3 contains the point dual to the plane of the other
two lines. Thus L1, L2, L3 are orthogonal and we have the web shown on the left of Figure 1.
Note that there is only one such web up to Möbius transformation.
Hexagonal Circular 3-Webs with Reducible Polar Curves of Degree Three 11
• One elliptic and two hyperbolic pencils. We can suppose that L3 is an elliptic polar line
and that L3 is the infinite line in the plane Z = 0. Due to Corollary 4.4, the polar lines L1, L2,
being hyperbolic, must intersect. Then the plane of L1, L2 is dual to some point on L3 and
therefore contains the line X = Y = 0 dual to L3 and can be assumed to be the plane Y = 0.
Suppose that none of L1, L2 is dual to L3. If both L1, L2 intersect L3, then the intersection
point is px = [1 : 0 : 0 : 0]. Applying Corollary 4.4 to the pair L1, L3, we conclude that L2,
joining px and the polar point of the plane of L1, L3 does not meet the Darboux quadric and
is not hyperbolic as supposed. Thus we can suppose that L1, L3 are skew and that L1, by
Corollary 4.4, contains (0, 0,−1). Since L1 is not dual to L3, we infer by Corollary 4.4 that L2
contains the dual point of the singular circle for L1, L3. This point lies in the tangent plane
to the Darboux quadric at (0, 0, 1), therefore L2, being non-parabolic, cannot meet L3 and
cannot also contain (0, 0, 1). Then it passes through (0, 0,−1), which contradicts the geometry
restriction imposed by Corollary 4.4.
So we can assume that L2 is dual to L3, i.e., it is the lineX = Y = 0. Corollary 4.4 prevents L1
to be skew with L3. Thus it meets L3 and therefore intersect L2 in a point inside the Darboux
quadric. We obtain the hexagonal web equivalent to the one on the right of Figure 1.
• One hyperbolic and two elliptic pencils. By Corollary 4.4, elliptic lines, say L2, L3, intersect.
We can assume that the hyperbolic line L1 is the coordinate axis X = Y = 0.
First consider the case when two lines, L1, L2, are dual. Then L2 is the infinite line in the
plane Z = 0 and we can assume the intersection line of L2 and L3 being the point py = [0 : 1 :
0 : 0]. Then L3 cannot be skew with L1 due to Corollary 4.4: the polar of the singular circle
determined by L1, L3 is finite and cannot lie on the infinite line L2. Thus L3, being elliptic,
intersect L1 outside the Darboux quadric and we get the web equivalent to the one shown on
the left of Figure 2.
Now suppose that no pair L1, Li is skew. Then L2 and L3 intersect L1. Since the triple L1,
L2, L3 is not coplanar, all three lines intersect at one point outside the Darboux quadric, which
can be taken as pz = [0 : 0 : 1 : 0]. By Corollary 4.4, the line L3 contains the dual point of
the singular circle determined by L1, L2 and the line L2 contains the dual point of the singular
circle determined by L1, L3. This fixes L1, L2, L3 up to rotation around z-axis and gives the
web shown in the center of Figure 1.
Finally, consider the case with skew but not dual L1, L2. Applying Möbius transformation,
we send the intersection point of L2 and L3 to px = [1 : 0 : 0 : 0] (preserving the position of L1).
Then L3 joins px and the polar point of the singular circle determined by L1, L2. We obtain
the web, equivalent to one on the left of Figure 3.
• Three elliptic pencils. We treat this case using the complex version of Corollary 4.4. First,
we conclude that, being non-coplanar, all three polar lines intersect at one point, which we can
send to py = [0 : 1 : 0 : 0]. None of 3 real planes, containing a pair of polar lines, can miss
completely the real Darboux quadric. In fact, if the real plane of L1, L2 do not intersect the
real Darboux quadric then the polar point of the complex singular circle of L1, L2 is inside the
real Darboux quadric and L3, being elliptic, cannot contain this point. None of 3 real planes,
containing a pair of polar lines, can intersect the real Darboux quadric. If the real plane of L1, L2
cuts the real Darboux quadric, the polar point of the real singular circle of L1, L2 is outside the
real Darboux quadric and the real plane of L1, L3 do not meet the real Darboux quadric. Thus
all 3 planes are tangent to the Darboux quadric. A representative of such web is shown in the
center of Figure 3.
• One parabolic and two hyperbolic pencils. By Corollary 4.4, all 3 polar lines must intersect
in one point. The intersection point cannot be inside the Darboux quadric since one line is
parabolic. It also cannot be outside: we can send it to px = [1 : 0 : 0 : 0] and the parabolic line,
joining px and the point dual to the plane of hyperbolic lines, will miss the Darboux quadric.
Therefore, this point is on the Darboux quadric. One can move the plane of hyperbolic lines
12 S.I. Agafonov
to Z = 0. Then the parabolic line contains pz = [0 : 0 : 1 : 0] and none of the hyperbolic lines
can pass through the point dual the singular circle of the other two lines. Thus there is no
hexagonal web with non-coplanar parabolic and two hyperbolic polar lines.
• One parabolic, one hyperbolic and one elliptic pencil. By Corollary 4.4, the parabolic line
meets the other two. If the hyperbolic line intersects the elliptic at some point p outside the
Darboux quadric, we can move the hyperbolic line to X = Z = 0 and p to py = [0 : 1 : 0 : 0].
Then the point ps, dual for the singular circle of hyperbolic and elliptic lines, is infinite and the
third polar line, joining py and ps cannot be parabolic.
Therefore, hyperbolic and elliptic lines are skew and Corollary 4.4 fixes the configuration up
to Möbius transformation: if these lines are dual we get the type shown in the center of Figure 2,
otherwise the type on the right of Figure 3.
• Two parabolic and one hyperbolic pencils. By Corollary 4.4, the hyperbolic line meets both
parabolic lines.
If the parabolic lines are skew, then the corresponding two singular circles have their polar
points on the line dual to the one joining the points of tangency of parabolic lines with the
Darboux quadric. This dual line is elliptic, therefore this configuration is not possible.
If the parabolic lines intersect outside the Darboux quadric, then we can bring the plane of
their intersection to Z = 0. Now the hyperbolic line contains pz = [0 : 0 : 1 : 0] by Corollary 4.4
and, intersecting the both elliptic lines, must meet them at their common point. Then it misses
the Darboux quadric and is not hyperbolic.
Therefore, the parabolic lines are tangent to the Darboux quadric at the same point. Since the
polar lines are not coplanar the hyperbolic line contains this point. We can bring the hyperbolic
line to X = Y = 0. Then Corollary 4.4 implies that the parabolic lines are dual and we obtain
the web type shown on the right of Figure 2.
• Two parabolic and one elliptic pencils. By Corollary 4.4, the elliptic line meets both
parabolic lines.
If the parabolic lines are skew, then the corresponding two singular circles have their dual
points on the line dual to the one joining the points of tangency of parabolic lines with the
Darboux quadric. The third polar line must contain these point, it is elliptic and we get the
web type presented in Figure 4.
If the parabolic lines intersect outside the Darboux quadric, then we can bring the plane of
their intersection to Z = 0. By Corollary 4.4, the elliptic line contains pz = [0 : 0 : 1 : 0] and,
intersecting the both elliptic lines, must meet them at their common point. Thus we get the
type shown in Figure 4.
If the parabolic lines are tangent to the Darboux quadric at the same point, then the third
line, being elliptic, cannot pass through this point. Therefore, it is coplanar with the parabolic
lines.
• Three parabolic pencils. Suppose that two polar lines L1 and L2 intersect. If the intersection
point is outside the Darboux quadric, then we bring the plane of L1, L2 to Z = 0. Since the
third polar line L3 does not lie in this plane it contains pz = [0 : 0 : 1 : 0] by Corollary 4.4.
Therefore, it is skew with at least one of L1, L2. Let it be L1. Then by Corollary 4.4, the line
L2 contains both dual point of two singular circles of L1, L3 which is obviously not possible.
If L1, L2 touch the Darboux quadric at the same point, then the line L3 is skew with at least
one of L1, L2. Again, this is precluded by Corollary 4.4.
Therefore, L1, L2, L3 are pairwise skew. Consider two singular circles C1, C2 of L1, L2,
their polar points p1, p2 and the family of lines L touching the Darboux quadric and meeting
both L1, L2. The third line L3, being parabolic, cannot contain both points p1, p2. If p1 /∈ L3,
then by Lemma 2.2 the family of lines L touching the Darboux quadric at points of C1 must meet
also L3. Therefore, the lines L constitute one ruling of a quadric touching the Darboux quadric
along C1. Therefore, L1, L2, L3 belong to the second ruling. If p2 ∈ L3, then, sending the
Hexagonal Circular 3-Webs with Reducible Polar Curves of Degree Three 13
plane of C1 to Y = 0 and the points, where L1, L2 touch the Darboux quadric, to (0, 0,±1), we
see that p2 = [1 : 0 : 0 : 0] since C2 is orthogonal to C1 and passes through (0, 0,±1). Then L3
must touch the Darboux quadric also at one of the points (0, 0,±1), which contradicts the
initial assumption that all 3 lines are skew. Thus p2 /∈ L3. Then the family of lines L touching
the Darboux quadric at points of C2 must meet also L3. This is not possible as the lines L,
constituting a ruling of a quadric that touch the Darboux quadric along C1 cannot meet the
orthogonal circle C2. ■
5 Polar curve splits into conic and straight line
First we describe the types then we prove that the list is complete. Some types are one-
parametric families and we denote the parameter value by c. The polar conic will be given
either by an explicit parametrization of the circle equations and we reserve u for the parameter,
or by indicating the conic equations. The former representation gives also a parametrization of
the polar conic: to a circle ϵ
(
x2 + y2
)
+ αx+ βy + γ = 0 (where ϵ = 0 or ϵ = 1, the case ϵ = 0
giving a line) corresponds the polar point with the tetracyclic coordinates [α : β : γ− ϵ : −γ− ϵ].
The parameter for circles in the pencil will be denoted by v.
To check hexagonality, one computes the Blaschke curvature using the formula in Appendix A
as follows. The polar conic gives a one-parameter family of circles on the unit sphere. The
stereographic projection from the “south” pole
X =
2x
1 + x2 + y2
, Y =
2y
1 + x2 + y2
, Z =
1− x2 − y2
1 + x2 + y2
,
transforms this family into a family of circles in the plane parametrized by the points of the
conic. The ODE for circles, defined by the conic,
P 2 +A(x, y)P +B(x, y) = 0, P =
dy
dx
,
is obtained by differentiating and excluding the coordinates of the conic points. The slope R
comes from the pencil of circles.
For some types, we add an additional geometric detail in the “title” to separate the types.
1. Polar conic plane does not cut Darboux quadric, hyperbolic pencil. The pencil
with limit circles at the origin and in the infinite point gives circles x2 + y2 = v, the polar conic
is the circle X2
0 + Y 2
0 + 4cX0Z0 = 0, U0 = 0, c > 0, defining the family
x2 + y2 − 4c
c2u2 + 1
x+
4c2u
c2u2 + 1
y = 1,
the circles of the family enveloping the cyclic
(
x2 + y2
)2 −
(
x2 + y2
)
(4cx+ 2)− 4c2y2 + 4cx+ 1 = 0
as shown on the left in Figure 5.
2. Polar conic plane does not cut Darboux quadric, hyperbolic pencil, webs
symmetric by rotations. The pencil with limit circles at the origin and at the infinite point
gives circles x2+y2 = v, the polar conic is the circle X2
0 + Y 2
0 =
4Z2
0
c2
, U0 = 0, defining the family
x2 + y2 +
2 cos(u)
c
x+
2 sin(u)
c
y = 1,
the circles of the family enveloping the cyclic
c2
(
x2 + y2
)2 −
(
2c2 + 4
)(
x2 + y2
)
+ c2 = 0,
which splits into two concentric circles as shown in the center of Figure 5.
14 S.I. Agafonov
Figure 5. Types 1, 2, 3.
Figure 6. Types 4, 5, 6.
3. Polar conic plane cuts Darboux quadric, hyperbolic pencil. The pencil with limit
circles at the origin and at the infinite point gives circles x2 + y2 = v, the polar conic is the
circle x20 + y20 = 4cx0, z0 = 0, c > 0, defining the family
x2 + y2 +
4c
c2u2 + 1
x− 4c2u
c2u2 + 1
y = −1,
the circles of the family enveloping the cyclic
(
x2 + y2
)2
+
(
x2 + y2
)
(4cx+ 2)− 4c2y2 + 4cx+ 1 = 0,
as shown on the right of Figure 5.
4. Polar conic plane cuts Darboux quadric, hyperbolic pencil, webs symmet-
ric by rotations. The pencil with limit circles at the origin and at the infinite point gives
circles x2 + y2 = v, the polar conic is the circle x20 + y20 = 4
c2
, z0 = 0, defining the family
x2 + y2 − 2 cos(u)
c
x− 2 sin(u)
c
y = −1,
the circles of the family enveloping the cyclic
c2
(
x2 + y2
)2
+
(
2c2 − 4
)(
x2 + y2
)
+ c2 = 0,
which splits into two concentric circles as shown on the left in Figure 6.
Hexagonal Circular 3-Webs with Reducible Polar Curves of Degree Three 15
Hexagonal circular 3-webs with reducible polar curves of degree three 17
Figure 7. Types 7, 8, 9.
8. Polar conic plane cuts Darboux quadric, parabolic pencil.
The pencil with the vertex at the infinite point gives lines
y = v,
the polar conic
x20 − z0 = 1, y0 = 0,
gives the family of circles (
x− 1
u
)2
+ y2 =
u2 − 1
u2
.
The circles envelopes the ellipse
x2 + 2y2 = 2
as shown in the center in Figure 7.
9. Polar conic plane cuts Darboux quadric, parabolic pencil, webs symmetric by
translations.
The pencil with the vertex at the infinite point gives lines
y = v,
the polar conic is
x20 + (1−
√
2)z20 − 2
√
2z0 = 1 +
√
2, y0 = 0,
defining the family of circles
(x+ u)2 + y2 = 1,
the circles touching the lines
y = ±1
as shown on the right in Figure 7.
10. Polar conic plane tangent to Darboux quadric, hyperbolic pencil.
The pencil with limit circles at (1, 0) and (−1, 0)) gives circles
x2 + y2 + 1 = vx,
Figure 7. Types 7, 8, 9.
5. Polar conic plane cuts Darboux quadric, elliptic pencil, webs symmetric by
homotheties. The pencil with vertexes at the origin and at the infinite point gives lines
y = vx, the polar conic is y20 + cz20 = c, x0 = 0, c /∈ [0, 1], defining the family of circles
x2 + (y + u)2 = (1 − 1/c)u2, the circles enveloping the lines x2 = (c − 1)y2, real for c > 1, as
shown in the center in Figure 6. The webs of the family are symmetric by homotheties with the
center in the origin.
6. Polar conic plane cuts Darboux quadric, elliptic pencil, polar conic and dual
to polar line intersect in 2 points on the Darboux quadric. The pencil with vertexes at
the origin and at the infinite point gives lines y = vx, the polar conic is 2cy0 = z20 − 1, x0 = 0,
c > 0, defining the family of circles
x2 + y2 +
(u− 1)
cu
y +
u− 1
u+ 1
= 0,
the circles enveloping the cyclic
c2
(
x2 + y2
)2
+
(
4cy − 2c2
)(
x2 + y2
)
+ (2y + c)2 = 0
as shown on the right in Figure 6.
7. Polar conic plane cuts Darboux quadric, elliptic pencil. The pencil with vertexes
at the origin and at the infinite point gives lines y = vx, the polar conic is 2cy0z0 + 1 = z20 ,
x0 = 0, c > 0, defining the family of circles
x2 + y2 − c+ u
cu
y +
c+ u
c− u
= 0,
as shown on the left in Figure 7.
8. Polar conic plane cuts Darboux quadric, parabolic pencil. The pencil with the
vertex at the infinite point gives lines y = v, the polar conic x20−z0 = 1, y0 = 0, gives the family
of circles
(
x− 1
u
)2
+ y2 = u2−1
u2 . The circles envelopes the ellipse x2 + 2y2 = 2 as shown in the
center in Figure 7.
9. Polar conic plane cuts Darboux quadric, parabolic pencil, webs symmetric
by translations. The pencil with the vertex at the infinite point gives lines y = v, the polar
conic is x20+
(
1−
√
2
)
z20−2
√
2z0 = 1+
√
2, y0 = 0, defining the family of circles (x+u)2+y2 = 1,
the circles touching the lines y = ±1 as shown on the right in Figure 7.
10. Polar conic plane tangent to Darboux quadric, hyperbolic pencil. The pencil
with limit circles at (1, 0) and (−1, 0) gives circles x2 + y2 + 1 = vx, the polar conic is x20 +
(1− c)y20 = c, z0 = −1, c > 0, c ̸= 1, defining the family of lines x0x+ y0y = 1, the lines of the
family enveloping the conic cx2 + c
c−1y
2 = 1 with foci (1, 0) and (−1, 0) as shown on the left in
Figure 8.
16 S.I. Agafonov
Figure 8. Type 10, 11, 12.
Figure 9. Type 13, 14, 15.
11. Polar conic plane tangent to Darboux quadric, hyperbolic pencil, polar line
meets polar conic. The pencil with limit circles at (1, 0) and (−1, 0) gives circles x2+y2+1 =
vx, the polar conic is y20 − cx0y0 + cy0 + x0 = 0, z0 = −1, c ≥ 0, defining the family of
lines x0x+ y0y = 1, the lines of the family enveloping the parabola
(cx− y)2 − (2c2 + 4)x− 2cy + c2 = 0
with focus at (1, 0) as shown in the center in Figure 8.
12. Polar conic plane tangent to Darboux quadric, hyperbolic pencil, polar line
contains the point where polar conic plane touches the Darboux quadric. The pencil
with limit circles at the origin and infinity gives circles x2 + y2 = v, the polar conic is x20 + y20 −
2y0 = 0, z0 = −1 defining the family of lines x0x + y0y = 1, the lines of the family enveloping
the parabola x2 + 2y = 1 with focus at (0, 0) as shown on the right in Figure 8.
13. Polar conic plane tangent to Darboux quadric, hyperbolic pencil, web sym-
metric by rotations. The pencil with limit circles at the origin and infinity gives circles
x2 + y2 = v, the polar conic is x20 + y20 = 1, z0 = −1 defining the family of lines x0x+ y0y = 1,
the lines of the family enveloping the circle x2 + y2 = 1 as shown on the left in Figure 9.
14. Polar conic plane tangent to Darboux quadric, elliptic pencil. The pencil with
vertexes at (1, 0) and (−1, 0) gives circles
(
x2 + y2 − 1
)2
(
x2 + y2 − 2x+ 1
)(
x2 + y2 + 2x+ 1
) = v,
the polar conic is cx20 + (c + 1)y20 + 1 = 0, z0 = −1, defining the family of lines x0x + y0y = 1,
the lines of the family enveloping the conic x2
c + y2
c+1 = −1 with foci (1, 0) and (−1, 0) as shown
in the center in Figure 9.
Hexagonal Circular 3-Webs with Reducible Polar Curves of Degree Three 17
15. Polar conic plane tangent to Darboux quadric, parabolic pencil. The pencil
with vertex at the origin gives circles x2 + y2 = 2vy, the polar conic is x20 + y20 = 1, z0 = −1
defining the family of lines x0x+y0y = 1, the lines of the family enveloping the circle x2+y2 = 1
as shown on the right in Figure 9.
Theorem 5.1. The webs of different types in the above classification list are not Möbius equiv-
alent. The webs of the same type with different normal forms are not Möbius equivalent.
Proof. The webs from different types are not Möbius equivalent: discrete geometric invariants
indicated in the descriptions, such as 1) presence of infinitesimal symmetry and 2) the mutual
position of polar line, polar conic and Darboux quadric, effectively separate the types.
To see that the different normal forms within a family are not Möbius equivalent, one com-
putes the subgroup Gp of PSO(3, 1) respecting the positions of the polar conic plane and the
polar line in the chosen normalization and checks that the Gp-orbits of the canonical forms are
different.
For the first 4 types, Gp is generated by Rz and by reflections in the coordinate planes.
For the 5th, 6th, 7th type, Gp is generated by Bz and by reflections in the coordinate planes.
For the 10th, 11th and 14th types, Gp is discrete and generated by reflections in the planes
x = 0 and y = 0. ■
As in the case of 3 pencils, Lemma 2.2 effectively selects candidates among 3-webs that can
be hexagonal. The singularities of the type described by the Lemma arise when either 1) a line,
joining 2 different points on the polar conic, touches the Darboux quadric at a point pt, while
the polar conic plane is not tangent the Darboux quadric or 2) a line, joining a point on the
polar line with a point on the polar conic, touches the Darboux quadric at a point pt while the
tangent plane to the Darboux quadric at pt is not tangent to the polar conic.
In the former case, the points pt trace a circle, which is the intersection of the polar conic
plane with the Darboux quadric. Then, by Lemma 2.2, the polar of the polar conic plane lies
on the pencil polar line.
In the latter case, consider a point pr running over the polar line. For non-planar polar
set, pr meets the polar conic plane πc only at one point. Therefore, the one-parameter family
of cones tangent to the Darboux quadric and having their vertexes at pr cuts the plane πc in
a one-parameter family of conics cr. Each conic cr intersect the polar conic cp at 4 (possibly
complex or multiple) points. If the polar conic cp is not a member of the family {cr}, at least
one of these 4 intersection points is moving along cp as pr runs over the polar line. In fact, if
intersection points are stable then {cr} is a pencil of conics containing cp.
Choose one such moving intersection point pi. Lines lr tangent to cr at pi form a one-
parameter family, or congruence of lines. All the objects in this construction are considered as
complex but the polar conic and the polar line must have equations with real coefficients and
the real part of the polar conic cannot lie completely inside the Darboux quadric.
Proposition 5.2. If the polar curve of a hexagonal circular 3-web is non-planar and splits into
a line and a smooth conic, then for the web complexification hold true
(1) the polar of the polar conic plane lies on the polar line, if this plane is not tangent to the
Darboux quadric and
(2) the congruence of lines lr is a pencil with the vertex on the polar conic, if the polar conic
is not a member of the family {cr}.
Proof. The first claim follows directly from Lemma 2.2: the points, where bisecant lines touch
the Darboux quadric, trace a circle whose polar point must lie on the polar line. For conic planes
missing the Darboux quadric, the complex version works.
18 S.I. Agafonov
To derive the second claim, observe that the line prpi touches the Darboux quadric at a sin-
gular point treated by Lemma 2.2. Thus the tangency point must trace a circle on the Darboux
quadric and the polar point p0 of the circle must lie on the polar conic. Then the plane, tangent
to the Darboux quadric and passing through prpi, contains p0. This plane cuts the plane πc of
the polar conic along a line lr, passing through the pi and tangent to the corresponding conic cr.
Hence {lr} is the pencil with vertex at p0. ■
Theorem 5.3. If the polar curve of a hexagonal circular 3-web splits into a smooth conic and
a straight line, not lying in a plane of the conic, then the web is Möbius equivalent to one from
the above presented list of types 1–15.
Proof. The polar conic plane either completely misses the Darboux quadric, or cuts it in a circle,
or is tangent to it. The polar line is either hyperbolic, or elliptic, or parabolic. Thus we have 9
cases to consider, each case defining a set of webs (possibly empty).
• Polar conic plane misses the Darboux quadric. Applying a suitable Möbius transformation,
we can send the polar conic plane to infinity. Then by Proposition 5.2 the polar line contains
the origin of the affine chart and therefore meets the Darboux quadric at 2 points. Thus
the polar line is hyperbolic. Applying a rotation around the origin, we map these two points
to (0, 0,±1). In the affine coordinates x = X
Z , y = Y
Z on the polar conic plane U = 0, the
conics cr are x2 + y2 = r, where r is considered as a complex parameter. These conics are real
only for real non-negative r. If the polar conic coincides with one of the conics cr, then we get
Type 2, symmetric by Rz.
If the polar conic is not one of cr, then, by Proposition 5.2, the points, where lines of pencil
with the vertex at some point p0 = (x0, y0) ∈ C2 touch the circles cr, run over the polar conic.
The line of the pencil y = y0 + k(x − x0) corresponding to the parameter k ∈ C, is tangent to
the conic cr for
r =
x20k
2 − 2x0y0k + y20
k2 + 1
at the point
p(k) = (x(k), y(k)) =
(
k(x0k − y0)
k2 + 1
,
y0 − x0k
k2 + 1
)
.
The points p(k) run over the complex conic x2 + y2 = x0x + y0x. This conic is real if and
only if x0 and y0 are real. It is smooth if and only if p0 ̸= (0, 0). We got the first web of the
list, Type 1. This conic is the circle, passing through the origin O = (0, 0) and p0 and having
its center at the midpoint of the segment Op0. (We obtained a theorem of scholar geometry.)
Using Rz, we normalize p0 to y0 = 0, x0 > 0.
For infinite p0, different from the cyclic points, the line l(k) of the pencil y = ax+ k, a ∈ C
touches a unique conic of the family {cr} at the point p(k) =
(
− ak
a2+1
, k
a2+1
)
. The points p(k)
are collinear: x(k) + ay(k) = 0 and we cannot obtain a smooth polar conic in this way. Finally,
if p0 is cyclic, the lines from the pencil touch the conics cr at the very point p0 and we get no
conic at all.
• Polar conic plane cuts the Darboux quadric, hyperbolic pencil. We send the conic plane
to Z = 0. Now the polar line contains the infinite point [0 : 0 : 1 : 0] by Proposition 5.2. The
subgroup of the Möbius group, preserving the plane Z = 0, is generated by rotations around Z-
axis and the boosts Bx and By. Using this subgroup, we normalize the polar line to x = y = 0.
The conics cr are circles in the plane z = 0 with the center at the origin. Repeating the
arguments that we used above for conic planes missing the Darboux quadric, we get Types 3
and 4.
Hexagonal Circular 3-Webs with Reducible Polar Curves of Degree Three 19
• Polar conic plane cuts the Darboux quadric, elliptic pencil. We normalize the conic plane
to X = 0. Then by Proposition 5.2 the point [1 : 0 : 0 : 0] lies on the polar line. The
polar line, being elliptic, meets the plane X = 0 at a point p outside the unit circle. Möbius
transformations, preserving the plane X = 0, are generated by rotations around X-axis and the
boosts By and Bz. Using these transformations, we send the point p to [0 : 1 : 0 : 0 : 0]. Now the
polar line is U = Z = 0 and the conics cr have equations z2 + y2
r = 1 in the affine coordinates.
If the polar conic coincides with one of these conics, then r is real and we get Type 5.
Otherwise, by the second claim of Proposition 5.2, the lines lr meet at some point p0 ∈ cp.
If p0 = (0, y0, z0) is finite, a line of the pencil of lines z = z0 + k(y − y0) with vertex at p0 is
tangent to the conic cr with
r =
y20k
2 − 2y0z0k + z20 − 1
k2
.
Thus the points, where the lines lr touch cr, are parametrized by k via
y =
y20k
2 − 2y0z0k + z20 − 1
k(y0k − z0)
, z =
1
z0 − y0k
.
Excluding k, we get the conic y0z
2 − z0yz + y − y0 = 0. This conic is real only for real y0, z0
and is smooth if and only if y0
(
z20 − 1
)
̸= 0. If z20 < 1, we normalize to z0 = 0 applying Bz and
get Type 6. If z20 > 1, we send p0 to infinity applying Bz.
For infinite point p0 = [0 : Y0 : Z0 : 0], we conclude that Y0 ̸= 0 and Z0 ̸= 0, otherwise the
tangency points pi do not trace a conic. Thus we can set p0 = [0 : y0 : 1 : 0], where y0 ̸= 0. In the
affine coordinates u = U
Z , y = Y
Z in the plane X = 0, the conic cr is 1 +
y2
r = u2. A line from the
pencil y = ku + y0 is tangent to the conic cr if and only if r = k2 − y20. The points, where the
lines lr touch cr, are parameterized by k via y =
y20−k2
y0
, u = − k
y0
. This is a parametrization of
the polar conic of Type 7 y0u
2+ y = y0, which becomes y0U
2− y0Z
2+Y Z = 0 in homogeneous
coordinates.
• Polar conic plane cuts the Darboux quadric, parabolic pencil. We normalize the conic plane
to Y = 0. Then the point [0 : 1 : 0 : 0] is on the polar line. The plane Y = 0 is stable under
rotations Ry along the y-axis and under boosts Bx and Bz. Rotating, if necessary, around the
y-axis, we bring the polar line to z = −1, x = 0. In the affine coordinates on the plane Y = 0,
the conics cr have equations
x2 +
r − 1
r
z2 − 2
r
z − r + 1
r
= 0.
If the polar conic coincides with one of cr, then r is real. The chosen position of polar conic
plane and polar line is stable by action of the group generated by Bz. Applying it, one can
normalize to r = 1√
2
and we get Type 9.
Otherwise, consider first the pencil of lines z = z0 + k(x − x0) with finite vertex at p0 =
(x0, 0, z0). A line from the pencil is tangent to the conic cr with
r =
x20k
2 − 2x0(z0 + 1)k + (z0 + 1)2(
x20 − 1
)
k2 − 2x0z0k + z20 − 1
.
Thus the points, where the lines lr touch cr, are parametrized by k via
x =
k(x0k − z0 − 1)
k2 − x0k + z0 + 1
, z =
(
x20 − 1
)
k2 − x0(2z0 + 1)k + z0(z0 + 1)
k2 − x0k + z0 + 1
.
Excluding k, we get the conic
(z0 + 1)x2 − x0xz + z2 − x0x+ (1− z0)z − z0 = 0. (5.1)
20 S.I. Agafonov
This conic is real only for real x0, z0 and is smooth if and only if z0 ̸= −1. The chosen position
of polar conic plane and polar line is stable by action of the group generated by Bz and Bx−Ry.
Consider the orbits of points in the plane Y = 0. The orbit dimension is two for points outside
the union of the line U +Z = 0 and the circle X2+Z2 = U2, and is one on this union except for
their common point. Thus the finite representatives of the orbits are [0 : 0 : 0 : 1], [0 : 0 : 1 : 1]
and [0 : 0 : −1 : 1]. For the first two points, the conics (5.1) lie inside the Darboux quadric and
there is no real circles. For the point [0 : 0 : −1 : 1], the conic (5.1) is not smooth.
For infinite vertexes p0, the pencil z = k gives a non-smooth conic. Thus the pencil can
be chosen as x = az + k. A line from the pencil is tangent to the conic cr with r = (k−a)2
k2−a2−1
.
The points, where the lines lr touch cr, are x = k−a
a2−ak+1
, z = k2−ak−1
a2−ak+1
. Excluding k, we get the
conic x2 − axz − ax− z − 1 = 0, which is real only for real a. Taking into account the action of
the group generated by Bz and Bx −Ry, we set a = 0 and get Type 8.
• Polar conic plane tangent to Darboux quadric, hyperbolic pencil. If the hyperbolic line
does not contain the point where the polar conic plane touches the Darboux quadric, then we
sent this point to (0, 0,−1) and the polar line to Y = Z = 0. In the affine coordinates on the
plane z = −1, the conics cr have equations (x− r)2 +
(
1− r2
)
y2 +1− r2 = 0. If the polar conic
coincides with one of cr, then the web curvature
KB =
4
(
r2 − 1
)2(
x2 − 1
)(
rx2 + ry2 +
(
r2 − 3
)
x+ r
)(
x2 + y2 − 2rx+ 1
)4
x4y3
(
x2 + 1− 2rx
)6
vanishes only for r = ±1, the conic cr being non-smooth for these values.
A line from the pencil y = y0 + k(x− x0) with finite vertex at p0 = (x0, y0,−1) is tangent to
the conic cr with
r =
(
x20 + 1
)
k2 − 2x0y0k + y20 + 1
2k(x0k − y0)
.
Thus the points, where the lines lr touch cr, are parametrized by k via
x =
x0
(
x20 − 1
)
k3 − y0
(
3x20 − 1
)
k2 + x0
(
3y20 + 1
)
k − y0
(
y20 + 1
)
k
((
x20 − 1
)
k2 − 2x0y0k +
(
y20 − 1
)) ,
y =
2(x0k − y0)(
x20 − 1
)
k2 − 2x0y0k +
(
y20 − 1
) .
Excluding k, we get the cubic
(
x20 − 1
)
y3 +
(
y20 − 1
)
x2y − 2x0y0xy
2 + 2y0x
2 + 2y0y
2 − 2x0y0x+
(
x20 − y20
)
y = 0. (5.2)
This cubic splits into a smooth real conic and a line in 3 cases:
(1) for y0 = 0, (5.2) factors as y
(
x2 +
(
1− x20
)
y2 − x20
)
= 0 and we get Type 10,
(2) for x0 = 1, y0 ̸= 0, (5.2) factors as (x− 1)
((
y20 − 1
)
xy − 2y0y
2 + 2y0x+
(
y20 − 1
)
y
)
= 0,
(3) for x0 = −1, y0 ̸= 0, (5.2) factors as (x+ 1)
((
y20 − 1
)
xy + 2y0y
2 + 2y0x−
(
y20 − 1
)
y
)
= 0.
The cases 2) and 3) give Type 11, the substitution x→ −x reducing one to the other.
For infinite vertexes p0, the pencil x = k gives a non-smooth conic. Thus the pencil can be
chosen as y = ax + k. A line from the pencil is tangent to the conic cr with r = k2+a2+1
2ak . The
points, where the lines lr touch cr, are
x =
k
(
k2 − a2 + 1
)
a
(
k2 − a2 − 1
) , y =
2k
a2 − k2 + 1
.
Hexagonal Circular 3-Webs with Reducible Polar Curves of Degree Three 21
Excluding k, we get the cubic y3 + a2x2y + 2axy2 + 2ax+
(
1− a2
)
y = 0. The cubic splits into
a line and a conic only for a = 0 or for a2 ± 2ia − 1 = 0. The former case gives non-real and
non-smooth conic y2 + 1 = 0, the latter – the non-real conic xy ± i
(
y2 + 2
)
= 0.
If the hyperbolic line contains the point where the polar conic plane touches the Darboux
quadric, then we sent this point to (0, 0,−1) and the polar line to X = Y = 0. In the affine
coordinates on the plane z = −1, the conics cr are concentric circles. The case of concentric
circles was considered above. We get Types 12 and 13.
• Polar conic plane tangent to Darboux quadric, elliptic pencil. If one vertex of the elliptic
pencil coincides with the point where the polar conic plane touches the Darboux quadric, then
the polar curve is planar.
Thus we can sent the tangent point to (0, 0,−1) and the vertexes to (±1, 0, 0). The conics cr
have equations
(
r2 + 1
)
x2 + (y + r)2 = r2 + 1. If the polar conic coincides with one of cr, then
the web curvature
KB = −64
(
r2 + 1
)2
x
(
y2 + 1
)(
rx2 + ry2 +
(
3 + r2
)
y − r
)(
2ry − x2 − y2 + 1
)4
(
x2 − y2 − 1
)4(
r2 − x2 + 1
)6
vanishes only for r = ±i, the conic cr being non-smooth for these values.
A line from the pencil y = y0 + k(x− x0) is tangent to the conic cr with
r =
(
x20 − 1
)
k2 − 2x0y0k +
(
y20 − 1
)
2(x0k − y0)
.
The points, where the lines lr touch cr, are
x =
2k(x0k − y0)(
x20 + 1
)
k2 − 2x0y0k + y20 + 1
,
y = −x0
(
x20 − 1
)
k3 − y0
(
3x20 − 1
)
k2 + x0
(
3y20 + 1
)
k − y0
(
y20 + 1
)
(
x20 + 1
)
k2 − 2x0y0k + y20 + 1
.
Excluding k, we get the cubic
(
y20 + 1
)
x3 − 2x0y0x
2y +
(
x20 + 1
)
xy2 − 2x0x
2 − 2x0y
2 +
(
x20 − y20
)
x+ 2x0y0y = 0.
This cubic splits into a conic and a line in 4 cases:
(1) for y0 = ±i, the cubic equation factors as
(±i− y)
(
±2ix0x
2 −
(
1 + x20
)
xy ∓ i
(
1 + x20
)
x+ 2x0y
)
= 0,
(2) for y0 = ±ix0, the cubic equation factors as
(x± iy)
((
1− x20
)
x2 ∓ i
(
x20 + 1
)
xy − 2x0x± 2ix0y + 2x20
)
= 0,
(3) for x0 = 0, the cubic equation factors as x
((
y20 + 1
)
x2 + y2 − y20
)
= 0,
(4) for x0 = ±1, the cubic equation factors as (x∓ 1)
((
y20 +1
)
x2 +2y2 ∓ 2y0xy±
(
y20 − 1
)
x−
2y0y
)
= 0.
In the cases 1) and 2) the conic is not real, the case 3) gives Type 14, for the case 4) the web
curvature does not vanish.
22 S.I. Agafonov
For infinite vertexes p0, the pencil x = k gives a non-smooth conic. Thus the pencil can be
chosen as y = ax + k. A line from the pencil is tangent to the conic cr with r = a2−k2+1
2k . The
points, where the lines lr touch cr, are
x = − 2ak
k2 + a2 + 1
, y =
k
(
k2 − a2 + 1
)
k2 + a2 + 1
.
Excluding k, we get the cubic
a2x3 − 2ax2y + xy2 +
(
1− a2
)
x+ 2ay = 0.
The cubic splits into a line and a conic in 2 cases:
(1) for a = 0 the cubic equation factors as x
(
y2 + 1
)
= 0 and the conic is non-smooth
(2) for a = ±i the cubic equation factors as (x ± iy)
(
x2 ± ixy − 2
)
= 0, and the conic is not
real.
• Polar conic plane tangent to Darboux quadric, parabolic pencil. For non-planar polar curves,
the points, where the polar line and polar conic plane touch the Darboux quadric, are different.
Thus we can normalize the polar conic plane to z = −1 and the polar line to x = 0, z = 1. This
configuration is preserved by Bz. The conics cr have equations r
4x
2 + y = 1
r . If the polar conic
coincides with one of cr, then the web curvature
KB = −4096r5xy2
(
ry + 2x2 + 2y2
)(
ry − x2 − y2
)4
(
x2 − y2
)4(
r2 − 4x2
)6
vanishes only for r = 0, the conic cr being non-smooth for this value.
A line from the pencil y = y0 + k(x − x0) is tangent to the conic cr with r = k2+1
y0−x0k
. The
points, where the lines lr touch cr, are
x =
2k(x0k − y0)
k2 + 1
, y =
x0k
3 − y0k
2 − x0k + y0
k2 + 1
.
Excluding k, we get the cubic
x3 + xy2 − 2x0x
2 − 2x0y
2 +
(
x20 − y20
)
x+ 2x0y0y = 0.
This cubic splits into a conic and a line in 2 cases:
(1) for y0 = αx0, where α
2 ± 2iα− 1 = 0, the cubic equation factors as
(x± iy)
(
±ixy − x2 + 2x0x∓ 2ix0y − 2x20
)
= 0,
(2) for x0 = 0, the cubic equation factors as x
(
x2 + y2 − y20
)
= 0.
In the case 1), the conic is not real, the case 2) gives Type 15 after rescaling by Bz.
For infinite vertexes p0, the pencil y = k gives a non-smooth conic. Thus the pencil can be
chosen as x = ay + k. A line from the pencil is tangent to the conic cr with r = −a2+1
ak . The
points, where the lines lr touch cr, are
x =
2k
a2 + 1
, y =
k
(
1− a2
)
a
(
1 + a2
) .
Excluding k, we get the line
(
a2 − 1
)
x+ 2y = 0.
Thus we have considered all the cases with non-planar polar curve, the theorem is proved. ■
Corollary 5.4. Suppose that the polar curve of a hexagonal circular 3-web is reducible algebraic
of degree three. Then the polar curve is either planar or the web is Möbius equivalent to one
described by Theorems 4.5 and 5.3.
Hexagonal Circular 3-Webs with Reducible Polar Curves of Degree Three 23
6 Hexagonal circular 3-webs with Möbius symmetries
A Möbius transformation is a symmetry of a circular 3-web if it maps any web circle to a web
circle. The Möbius group in RP3 can be realized as PGL2(C), or equivalently, as the group
of fractional-linear transformations of z = x + iy, where (x, y) are cartesian coordinates in the
plane. A generator of any 1-dimensional subalgebra can be brought to the Jordan normal form
by adjoint action. The generator can be chosen either as
(
0 1
0 0
)
, or
(
λ 0
0 −λ
)
, where λ = α+ iβ is
some complex number with Re(λ) = α, Im(λ) = β.
One way to obtain symmetric hexagonal 3-webs is provided by the Wunderlich construction
(see [15] and Introduction).
Another easy way to produce hexagonal circular 3-webs is to choose two orbits of polar points
such that one is a conic and the other is a coplanar straight line, these two orbits forming a polar
curve. This construction may degenerate if there are 3 orbits which are coplanar straight lines.
For translations (x, y) 7→ (x, y + u), the orbit of a polar point for a circle (x− a)2 + y2 = r,
parametrized by u as follows
[
−2a : −2u : a2 + u2 − r − 1 : −a2 − u2 + r − 1
]
, is a conic in the
plane −X+a(Z+U) = 0. For a nonvertical line y = ax, the orbit is a line Z+U = X+aY = 0,
parametrized by u via [a : −1 : −u : u]. Thus we immediately get the following hexagonal
3-webs symmetric by translations.
T1. 3 families of parallel lines. Möbius orbits of such webs form a two-parametric family.
The polar curve splits into 3 coplanar lines. Observe that any web of this family has a 3-
dimensional symmetry group.
T2. Polar curve splits into conic and coplanar line. There is only one Möbius class of
such webs, any representative is formed by horizontal lines y = const and by the orbit of a circle
which can be chosen as the unitary one centered at the origin.
T3. Wunderlich’s type. There are several types, depending on the position of generating
curves.
(1) Nondegenerate type. Webs are formed by vertical lines and by two different orbits of
circles. The family of orbits is two-parametric: by translation and rescaling we can fix one
orbit.
(2) Coinciding circle orbits. There is only one Möbius class of this type. It was already
obtained as Type 9 in the classification of Theorem 5.3.
(3) Generated by a circle and a line. The family of orbits is one-parametric: by translation
and rescaling we can fix the generating circle.
For dilatations (x, y) 7→ (ux, uy), the orbit of a polar point for a circle (x−a)2+(y− b)2 = r,
parametrized by u via
[
−2au : −2bu : a2+b2−r−u2 : −
(
a2+b2−r
)
−u2
]
, is a conic in the plane
bX − aY = 0 if (a, b) ̸= (0, 0) and a2 + b2 ̸= r, the hyperbolic line X = Y = 0 if (a, b) = (0, 0),
and the parabolic line bX−aY = U −Z = 0 if a2+ b2 = r. For a line ax+ by+ c = 0 with c ̸= 0,
the orbit is the parabolic line bX − aU = Z + U = 0, parametrized by u via [au : bu : c : −c].
The lines with c = 0 are invariant. Invoking the classification of the webs with 3 pencils, we list
the following hexagonal 3-webs symmetric by dilatations.
D1=T1. 3 families of parallel lines.
D2. 2 coplanar parabolic lines and hyperbolic line intersecting them. Family of
parallel lines, the parabolic pencil with circles tangent to a line of the family through the origin,
and the family of concentric circles with the center at the origin. There is only one Möbius class
of such webs.
D3. 2 dual parabolic lines and hyperbolic line through their common point. 2 or-
thogonal families of parallel lines and the family of concentric circles with the center at the
origin. There is only one Möbius class of such webs, we have already presented it on the right
of Figure 2.
24 S.I. Agafonov
D4. Polar curve splits into conic and coplanar hyperbolic line. Family of circles
obtained by the dilatations from one not passing through the origin and not having its center
at the origin and the family of concentric circles with the center at the origin. Möbius orbits of
such webs form a one-parameter family.
D5. Polar curve splits into conic and tangent parabolic line. Family of circles
obtained by dilatations from one not passing through the origin and not having its center at the
origin and the family of parallel lines orthogonal to the orbit of centers of the circles. Möbius
orbits of such webs form a one-parameter family.
D6. Wunderlich’s type.
(1) Nondegenerate type, polar curve with 2 conics and elliptic line. Webs are formed by pencil
of lines centered at the origin and by two different orbits of circles at general position. The
family of Möbius orbits is 3-parametric: one can choose the centers of circles on a fixed
circle centered at the origin and normalize by rotations.
(2) Polar curve with conic and elliptic line. Webs are formed by pencil of lines centered at the
origin and orbit of a circle. The family of Möbius orbits is 1-parametric, it is Type 5 in
the classification of Theorem 5.3.
(3) Polar curve with conic, hyperbolic and elliptic line. Webs are formed by pencil of lines
centered at the origin, orbit of a circle, and the family of concentric circles with the center
at the origin. The family of Möbius orbits is 1-parametric.
(4) Polar curve with conic, parabolic and elliptic line. Webs are formed by pencil of lines
centered at the origin, orbit of a circle, and a family of parallel lines. The family of
Möbius orbits is 2-parametric.
(5) Polar curve with hyperbolic line, elliptic line dual to hyperbolic, and parabolic line inter-
secting them. Webs are formed by the pencil of lines centered at the origin, the family of
concentric circles with the center at the origin, family of parallel lines. There is only one
Möbius type. This web is shown in the center of Figure 2.
(6) Polar curve with 2 parabolic lines touching Darboux quadric at the same point and copla-
nar elliptic line. Webs are formed by the pencil of lines centered at the origin and 2 families
of parallel lines. The family of Möbius orbits is 1-parametric.
(7) Polar curve with 2 parabolic lines and elliptic line whose dual joins the touching points of
parabolic lines with Darboux quadric. Webs are formed by the pencil of lines centered at
the origin, a family of parallel lines and a parabolic pencil with the vertex at the origin.
The family of Möbius orbits is 1-parametric. A representative of this web is shown in
Figure 4.
For rotations (x, y) 7→ (x cos(t) + y sin(t),−x sin(t) + y cos(t)), the orbit of a polar point for
a circle (x−a)2+y2 = r, parameterized by t via
[
−2a cos(t) : −2a sin(t) : a2−r−1 : −a2+r−1
]
,
is a circle in the plane
(
a2 − r + 1
)
Z +
(
a2 − r − 1
)
U = 0 if a ̸= 0. For a line ax + by + c = 0
with c ̸= 0, the orbit of its polar is also a circle [a cos(t)− b sin(t) : a sin(t) + b cos(t) : c : −c] in
the plane Z + U = 0. The circles with a = 0 are invariant, the orbit of a polar point of a line
with c = 0 is an elliptic line. Thus one can easily list the following hexagonal 3-webs symmetric
by rotations.
R1. Polar curve is a circle symmetric by rotations around z-axis and coplanar
elliptic line. The family of Möbius orbits is 1-parametric.
R2. Wunderlich’s type. One foliation is formed by orbits of points by rotations and the
other two are images of two circles by rotation, these circles can coincide or “degenerate” into
straight lines. Examples of such webs with coinciding generators are shown in Figure 5 (center),
Figure 6 (left), and Figure 9 (left). The reader easily describes different types and computes
corresponding Möbius orbit dimension of the webs.
Hexagonal Circular 3-Webs with Reducible Polar Curves of Degree Three 25
Theorem 6.1. Hexagonal circular 3-web with 1-dimensional Möbius symmetry is Möbius equiv-
alent to one of the above described T-, D-, or R-types.
Proof. A circular 3-web, symmetric by a given 1-parametric subgroup of the Möbius group and
not obtained by the Wunderlich construction, is fixed by a choice of three curves, each being
either a circle or a straight line, one from each foliation. Two circles can coincide: an orbit of
circle still gives a (singular) 2-web. Moreover, one can move around these curves by the stabilizer
of the infinitesimal generator of the subgroup.
• Webs symmetric by translations ∂y. Consider a point such that none of the leaf tangents
is parallel to the field ∂y. If a generating curve C1 of some foliation is a circle, then there are
2 circles from the orbit of C1 passing through this point. Thus there are two locally defined
direction fields ∂x + P±∂y tangent to these 2 circles. Globally they are not separable: one
direction swaps for the other upon running along C1.
Consider one of the foliations and the corresponding direction field ∂x + P∂y. Since the
foliation is symmetric, the slope P does not depend on y. Since all the leaves are circles of
the same curvature, the foliation has a first integral (P ′)2
(P 2+1)3
, where P ′ = dP
dx . The differential
equation (P ′)2
(P 2+1)3
= A2 = const has general solution
P (x) =
A(x− x0)√
1−A2(x− x0)2
,
the zero value of A corresponding to the foliation by straight parallel lines. Let the slopes of
the other two web foliations be Q and R, and the connection form be γ = α(x)dx + β(x)dy.
Hexagonality condition dγ = 0 implies β(x) = const, which amounts to
k =
P ′
(P −Q)(P −R)
+
Q′
(Q− P )(Q−R)
+
R′
(R− P )(R−Q)
= const. (6.1)
If all foliations are formed by circles and not by straight lines, then the above identity is not
possible. In fact, each of the slopes P , Q, R, considered as function of complex x, has two
ramification points, for example, P has singularities at x± = x0 ± 1/A. If at least one of the
6 ramification points is not coinciding with one of the others, then, supposing it be x+ of P and
expanding the expression (6.1) for x = x+ + t2 by t at t = 0, one sees that it has a simple pole
and therefore cannot be constant. Therefore, each ramification point of any slope coincides with
a ramification point of another slope.
The ramification points correspond to the group orbits (lines x = const) tangent to the
foliation circles. If only two circles are tangent along some orbit x = const, then, applying
Lemma 2.2, we conclude that either this orbit belongs to the third foliation and the web is of
Wunderlich’s type or the two generating circles coincide. In the latter case, the ramification
points of the third foliation again must coincide with the common ramification points of the
first two. This means that all orbits of generating circles coincide and we do not have a 3-web.
Therefore, either all leaves of all foliations are straight lines or one of the foliation, say the
one corresponding to R, is formed by straight lines with slope R = const and the other two are
formed by orbits of the same circle and Q = −P . Then it is immediate that k = 0 and R = 0.
We get the type T2. The webs of Wunderlich’s type, which are hexagonal, are excluded from
consideration by the coordinate choice.
• Webs symmetric by dilatations x∂x + y∂y. Stabilizer of the 1-dimensional algebra spanned
by x∂x+y∂y is generated by rotation y∂x−x∂y and dilatation x∂x+y∂y. The orbit of the polar
of the generating curve is either a conic, or a hyperbolic line, or a parabolic line. Parabolic lines
touch the Darboux quadric at one of the stationary point of the dilatation.
We use modified polar coordinates rectifying the symmetry
u = arctan
(y
x
)
, v =
1
2
ln
(
x2 + y2
)
. (6.2)
26 S.I. Agafonov
A curve x 7→ (x, y(x)) is a circle if and only if y′′′ = 3y′(y′′)2
1+(y′)2 , therefore integral curves of a sym-
metric vector field ∂u + P (u)∂v are circles if and only if
P ′′ =
3P
P 2 + 1
(P ′)2 − P
(
P 2 + 1
)
. (6.3)
The integral curves are straight lines if and only if y′′ = 0, which is equivalent to P ′ = P 2+1.
Hence
(
apply z 7→ 1
z
)
the integral curves are circles of a parabolic pencil with the vertex at the
origin if and only if P ′ = −
(
P 2 + 1
)
thus giving P (u) = tan(u− u0) and P (u) = − tan(u− u0)
respectively. The second-order equation (6.3) has a first integral
(P ′)2
(
P 2 + 1
)3 − 1
P 2 + 1
= A = const,
allowing also to integrate it
P (u) =
√
A+ 1 tan(u− u0)√
1−A tan2(u− u0)
.
To the hyperbolic pencil with the circles centered at the origin corresponds the solution P ≡ 0.
For hexagonal webs, the connection form is γ = α(u)du + β(u)dv. Hexagonality condi-
tion dγ = 0 implies β(u) = const, giving again (6.1) (where the slopes of the other foliations
are Q and R). Observe that the choice of local coordinates u, v excludes webs of Wunderlich’s
type and we have to show that hexagonal are only the types D1, D2, D3, D4, D5. This can
be done as follows. The types D1, D2, D3 are webs with three pencils, the case being settled
earlier. Therefore, we study the webs whose polar curve includes at least one conic.
The approach used for translation works also in this case if we pass to complex webs: the
singular points of solutions P , Q, R become complex if the corresponding generating circle on the
Darboux quadric separates the stable points (0, 0,±1) of the dilatation. Considering behavior
of the expression k at singular points of P , Q, R, we see that necessarily a singular point of
slope for one foliation must coincide with a singular point for another. Singular points are the
points (possibly complex) where the group orbit touches the generating circle, or lies either on
a line of one foliation or on the common tangent to circles of parabolic pencil with the vertex
at (0, 0).
Applying Lemma 2.2 as in the case of translation, we infer that, for webs of non-Wunderlich’s
type, having two foliations with conic polar curves and coinciding singular points, these polar
curves must coincide and the third polar must be a line.
Consider the points where the circles from the common orbit are tangent. Lemma 2.2 implies
that for non-Wunderlich’s type the leaves of the third foliations must be also tangent to the
circles and we get either type D4 or D5.
Finally, if two components of the polar curve are lines and the third is a conic then the lines
must be parabolic for non-Wunderlich’s type. In fact, considering the points where circles of
the hyperbolic pencil are tangent to circles corresponding to conic, we conclude by Lemma 2.2
that the third foliation lines are also tangent to the circles at these points but then the singular
points of the “conic” solution to (6.3) can not be compensated.
Thus the polar lines are parabolic. The expression k cannot be constant if the poles of Q, R,
corresponding to these lines, do not coincide with singular points of P , which represent a conic
in the polar curve. Therefore, the generating circle of this “conic” solution does not separate
the stationary point of the dilatation and the singular points are real. Then by Lemma 2.2 the
web cannot be hexagonal.
• Webs symmetric by rotations y∂x−x∂y. Stabilizer of the 1-dimensional subalgebra spanned
by y∂x − x∂y is generated by rotations y∂x − x∂y and dilatations x∂x + y∂y. The polar orbit for
a generating curve is either a circle or an elliptic line.
Hexagonal Circular 3-Webs with Reducible Polar Curves of Degree Three 27
We again use polar coordinates (6.2) to describe symmetric vector fields ∂v + P (v)∂u. The
integral curves of such vector field are circles (in coordinates x, y, of course) if and only if
P ′′ =
3P
P 2 + 1
(P ′)2 + P
(
P 2 + 1
)
. (6.4)
The integral curves are straight lines if and only if P ′ = −P
(
P 2 + 1
)
. The integral curves are
circles passing through the origin if and only if P ′ = P
(
P 2+1
)
. Solutions of these two equations
are P (v) = 1√
e2(v−v0)−1
and P (v) = 1√
e−2(v−v0)−1
, respectively. The second-order equation (6.4)
has a first integral
(P ′)2
(
P 2 + 1
)3 +
1
P 2 + 1
= A2 = const,
allowing also to integrate it
P (v) =
√
A2 − 1 tanh(v − v0)√
1−A2 tanh2(v − v0)
, A2 ̸= 1.
The value A2 = 1 gives the special solutions with generating circles passing through the sta-
tionary points of the symmetry, i.e., elliptic pencil with the lines passing through the origin
with P ≡ 0. Analysis of the behavior of k at singular points and use of Lemma 2.2, similar to
the ones performed above, show that only the type R1 is hexagonal, the Wunderlich types being
excluded by the choice of variables. (In fact, multiplying by i the independent variable of the
differential equation (6.3) reduces it to (6.4).)
• Loxodromic symmetry. Finally, we show that there is no hexagonal 3-webs symmetric
by loxodromic vector field y∂x − x∂y + κ(x∂x + y∂y) for any κ ̸= 0. Note that the Wunderlich
construction does not give circular webs as the symmetry orbits are spirals. We use the following
coordinates:
s =
κ
2
ln
(
x2 + y2
)
− arctan
(y
x
)
, t = κ arctan
(y
x
)
+
1
2
ln
(
x2 + y2
)
,
the variable t being invariant by the symmetry. Integral curves of a symmetric vector field
∂t + P (t)∂s are circles (in coordinates x, y) if and only if
P ′′ =
3P
P 2 + 1
(P ′)2 +
(
P 2 + 1
)
(κP + 1)(P − κ)
(
κ2 + 1
)2 . (6.5)
This equation has a first integral
(P ′)2
(
P 2 + 1
)3 +
2κP − κ2 + 1(
κ2 + 1
)(
P 2 + 1
) = A = const.
This integral does not allow to integrate (6.5) in elementary functions but allows to study the
behavior of solutions at singular points. A singular point emerges when a symmetry orbit is
tangent to a circle (or a line) of corresponding foliation.
Consider possible singularity types. If we exclude from consideration the stationary points
of symmetry, then the symmetry vector field touches a generic circle at two points and the
tangency is simple. The corresponding solution P has two singularities if the tangency points
belong to different orbits and only one if the points lie on the same orbit. There are circles, for
which two tangency points merge to give only one singularity. If the generating curve is a line
or a circle through the origin, then the solution has only one singularity. Finally, there are two
28 S.I. Agafonov
constant solutions, namely P = −1/κ corresponding to the invariant hyperbolic line X = Y = 0
and P = k corresponding to the invariant elliptic line Z = U = 0.
If a solution P has two singularities at t1, t2, then A ̸= 0 and the singularity is of the same
type as for the non-loxodromic cases
P (t) =
1
4
√
4A
√
t− ti
+
{
analytic function of
√
t− ti
}
.
If a solution P is generated by a circle tangent to the symmetry trajectory t = t0 and the
tangency is of second order, then there is only one singularity of the following type at t = t0
P (t) = c(t− t0)
− 2
3 + · · · , where c ̸= 0 and the omitted terms are not essential for our analysis.
The condition of double tangency is equivalent to A = 0. The corresponding generating circle
(x − a)2 + y2 = r2 verifies the relation r2 = (k2+1)a2
k2
. For an orbit t = t0, there is at most one
such circle.
Let P , Q, R be solutions giving a hexagonal 3-web. These solutions, as well as the coordi-
nates s, t, are defined only locally but we can prolong them along a symmetry orbit, along a leaf
of some of the 3 foliations, or along any curve, as long as we do not meet a singular point of one
of P , Q, R. The condition of hexagonality (6.1) remains satisfied along any such prolongations.
Therefore, we cannot meet a singularity of only one of P , Q, R, they emerge necessarily at least
in pairs.
Suppose there is a symmetric hexagonal 3-web. Consider a non-singular point. There are 3
leaves passing through it. Each leaf can be considered as the generating curve of the respective
foliation. At least one of the corresponding solutions P , Q, R has a singular point. Let us run
along the respective leaf until we meet a singularity t1 of P , Q or R. Then at least two of P , Q, R
are singular at t1 and the orbit t = t1 is tangent to at least two web leaves at a singular point ps.
Since the symmetry orbit is not a circle, Lemma 2.2 implies that either exactly two leaves at ps
are tangent and therefore coincide or all three leaves are tangent at ps.
In the former case, suppose that the coinciding leaves are CQ and CR. Then the leaf CP
at ps is different from CQ = CR at ps. Let us go along CP keeping track of Q, R until one of
the two leaves corresponding to Q and R touches CP at some p̄s. Such point exists until all the
foliations are formed by straight lines. Then by Lemma 2.2 this leaf coincide with CP at p̄s, thus
all three generating curves of the web coincide and there is no 3-web. If all 3 generating curves
are straight lines, then CP is necessary the line through the origin and P = κ. Applying the
map z 7→ 1/z, we transform the lines CQ and CR into circles and the above argument applies.
If all three leaves are tangent at ps, then at least one leaf, say CP , is different from any of
the other two. Let us go again along CP keeping track of Q, R until one of the two leaves
corresponding to Q and R touches CP . Let it be CQ. Repeating the above used argument we
conclude that such point p̄s exists and CP coincides at p̄s with CQ. Then CP coincides with CQ
also at ps. Now either all 3 generating curves CP , CQ, CR coincide at ps and we do not have
3-web, or CR is different from CP = CQ at ps. Now we repeat the trick with prolongation,
this time along CR, and conclude that CP = CQ = CR at ps. Thus all three generating curves
coincide and we can get at most 2-web. ■
7 Concluding remarks
7.1 Circular hexagonal 3-webs on surfaces
Pottmann, Shi, and Skopenkov [11] classified circular hexagonal 3-webs on nontrivial Darboux
cyclides: such surfaces carry up to 6 one-parameter families of circles, 3 families can be picked
up in 5 different ways to form a hexagonal web. In fact, as was proven by Lubbes [9], if through
a general point of a surface in R3 pass at least 3 circles then the surface is either a plane, or
a sphere, or a Darboux cyclid.
Hexagonal Circular 3-Webs with Reducible Polar Curves of Degree Three 29
7.2 Erdoǧan’s approach to Theorem 4.5
The first attempt to prove Theorem 4.5 appeared in [6]. The idea was to choose a Möbius
normalization sending one of the vertexes to infinity, to set y = 0, and to obtain “sufficient
number of equations” to fix the pencil configurations. The author claimed that the curvature
equation for y = 0 is a polynomial one of degree 6 in x, though the calculation itself was not
present. Nowadays, armed with a powerful computer (32GB of RAM is enough) and a symbolic
computation system like Maple, one can perform this computations and check that the degree
may be much higher (in fact, up to 18) for some choices of polar line types.
Anyway, brute computer force does work: with the above mentioned equipment the author
of this paper managed to derive the classification results. The treating has the following steps:
(1) choosing an initial Möbius normalization,
(2) computing the curvature,
(3) isolating and factoring the highest homogeneous part of the curvature equation,
(4) Möbius renormalization adjusted to the geometric information obtained in the previous
step and repeating from the step 2 until one makes the curvature vanish.
7.3 Boundaries of regular domain for hexagonal 3-webs
To avoid heavy computation of the curvature in proving Theorem 4.5, the author of [12] suggested
to use the structure of web singular set (see Lemma 2.2). The presented proof was not correct.
The author argued that the web equation u3 = F (u1, u2), relating first integrals ui of the web
foliations Fi, may be rewritten as u3 = f(α(u1) + β(u2)) and used this form on the curve
of singular points Γ1. This argument is definitely wrong as the functions α, β typically have
singularities on Γ1: a simple counterexample is the web equation u3 = u1u2.
7.4 Hexagonal 3-subwebs
Consider the autodual tetrahedron with vertexes at [1 : 0 : 0 : 0], [0 : 1 : 0 : 0], [0 : 0 : 1 : 0]
and [0 : 0 : 0 : 1]. Lines joining the vertexes of this tetrahedron give a 6-web A6 with 6 pencils
of circles. Any 3-subweb of this 6-web is hexagonal: any 3 lines are either coplanar or give
a polar curve of a hexagonal 3-web. By direct computation (better use computer!) one checks
the following claim.
Proposition 7.1. The rank of the 6-web A6 is maximal, i.e., is equal to 10.
Another remarkable feature of this autodual 6-web is that the infinitesimal operators, corre-
sponding to the pencils in the sense of Proposition 3.2, form the basis Rx, Ry, Rz, Bx, By, Bz
of the Lie algebra of the Möbius group so that the commutator of any two of them is either zero
or an operator of the basis.
There is also autodual 4-web A4 whose polar curve is the union of 4 pencils corresponding
to Rz, Bz, Bx − Ry, Rx + By. This web has similar properties: any its 3-subweb is hexagonal,
the commutator of any two of the four operators is either zero or an operator of the set, its
rank is maximal. One finds more examples with hexagonal subwebs among symmetric webs of
Wunderlich’s type (see Section 6).
7.5 Conjecture
The polar curve of a hexagonal circular 3-web is an algebraic curve such that each its irreducible
component is either a twisted cubic or a planar curve of degree at most 3. The author thanks
an anonymous referee for providing an example of hexagonal circular 3-web with polar twisted
cubic.
30 S.I. Agafonov
A Appendix: Curvature equation
(
A2 − 4B
)(
R2 +AR+B
)[
(AR+ 2B)Axx +A
(
R2 −B
)
Axy −BR(A+ 2R)Ayy
− (A+ 2R)Bxx +
(
A2 − 2R2 − 2B
)
Bxy +R
(
A2 − 2B +AR
)
Byy
+
(
4B −A2
)
Rxx +A
(
A2 − 4B
)
Rxy +B
(
4B −A2
)
Ryy
]
+
(
A2 − 4B
)2
(A+ 2R)R2
x + (A+ 2R)
(
A2 − 4AR− 4R2 − 8B
)
B2
x+
−
(
2A3R2 +A2R3 + 7A2BR+ 4ABR2 + 4BR3 + 4AB2 − 4B2R
)
A2
x
+
(
B −R2
)(
2A3R+A2R2 +A2B + 4BR2 + 4B2
)
AxAy
−A
(
A2 − 4B
)2
(A+ 2R)RxRy
+
(
2A3R−A4 + 4A2R2 − 8AR3 − 8R4 + 8A2B − 16BR2 − 8B2
)
BxBy
+BR
(
2A3R+ 7A2R2 + 4AR3 +A2B + 4ABR− 4BR2 + 4B2
)
A2
y+
−R
(
A4 −A3R− 6A2R2 − 4AR3 − 8A2B − 8ABR+ 8B2
)
B2
y
+B
(
A2 − 4B
)2
(A+ 2R)R2
y +
(
A2 − 4B
)(
A2R−AR2 −AB − 8BR
)
AxRx
+
(
3A3R+ 13A2R2 + 8AR3 +A2B + 12ABR− 4BR2 + 12B2
)
AxBx
+ 2
(
A2 − 4B
)(
A2 +AR+R2 − 3B
)
BxRx +
(
5A2R3 −A4R−A3R2 + 4AR4
+A2BR+ 4ABR2 − 4BR3 − 4AB2 − 4B2R
)
[AxBy +AyBx]
+
(
A2 − 4B
)(
A2R2 +A2B + 2ABR− 2BR2 − 2B2
)
[AxRy +AyRx]+
−A
(
A2 − 4B
)(
A2 +AR+R2 − 3B
)
[BxRy +ByRx]
+B
(
A2 − 4B
)(
A2R−AR2 −AB − 8BR
)
AyRy
+
(
4B −A2
)(
A3R+A2R2 −A2B − 6ABR− 6BR2 + 2B2
)
ByRy+
−R
(
2A4R+ 5A3R2 + 3A2R3 −A2BR+ 4ABR2 + 4BR3 + 8AB2 + 20B2R
)
AyBy
= 0.
Acknowledgements
This research was supported by FAPESP grant # 2022/12813-5. The author thanks the anony-
mous referees for valuable suggestions.
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[3] Blaschke W., Einführung in die geometrie der Waben, Birkhäuser, Basel, 1955.
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Hexagonal Circular 3-Webs with Reducible Polar Curves of Degree Three 31
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1 Introduction
2 Hexagonal 3-webs, Blaschke curvature, singularities
3 Projective model of Möbius geometry
4 Polar curve splits into 3 non-coplanar lines
5 Polar curve splits into conic and straight line
6 Hexagonal circular 3-webs with Möbius symmetries
7 Concluding remarks
7.1 Circular hexagonal 3-webs on surfaces
7.2 Erdogan's approach to Theorem 4.5
7.3 Boundaries of regular domain for hexagonal 3-webs
7.4 Hexagonal 3-subwebs
7.5 Conjecture
A Appendix: Curvature equation
References
|
| id | nasplib_isofts_kiev_ua-123456789-213533 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T17:53:29Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Agafonov, Sergey I. 2026-02-18T11:27:06Z 2025 Hexagonal Circular 3-Webs with Reducible Polar Curves of Degree Three. Sergey I. Agafonov. SIGMA 21 (2025), 043, 31 pages 1815-0659 2020 Mathematics Subject Classification: 53A60 arXiv:2306.11707 https://nasplib.isofts.kiev.ua/handle/123456789/213533 https://doi.org/10.3842/SIGMA.2025.043 The paper reports the progress with the classical problem, posed by Blaschke and Bol in 1938. We present new examples and new classifications of natural classes of hexagonal circular 3-webs. The main results are the classification of hexagonal circular 3-webs with reducible polar curves of degree 3 and the description of hexagonal circular 3-webs admitting a one-parameter Möbius symmetry. This research was supported by FAPESP grant # 2022/12813-5. The author thanks the anonymous referees for their valuable suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Hexagonal Circular 3-Webs with Reducible Polar Curves of Degree Three Article published earlier |
| spellingShingle | Hexagonal Circular 3-Webs with Reducible Polar Curves of Degree Three Agafonov, Sergey I. |
| title | Hexagonal Circular 3-Webs with Reducible Polar Curves of Degree Three |
| title_full | Hexagonal Circular 3-Webs with Reducible Polar Curves of Degree Three |
| title_fullStr | Hexagonal Circular 3-Webs with Reducible Polar Curves of Degree Three |
| title_full_unstemmed | Hexagonal Circular 3-Webs with Reducible Polar Curves of Degree Three |
| title_short | Hexagonal Circular 3-Webs with Reducible Polar Curves of Degree Three |
| title_sort | hexagonal circular 3-webs with reducible polar curves of degree three |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/213533 |
| work_keys_str_mv | AT agafonovsergeyi hexagonalcircular3webswithreduciblepolarcurvesofdegreethree |