3-web-of-circles: summary
In 2013 F. NILOV published 5 new examples of 3-webs-of-circles in an article. (Lit. note see below)
All 5 examples are related to conics.
In this book chapter we have generalized the examples to bicircular quartics; in Moebius geometry,
conics are special cases of this class of curves.
This page is intended to give an overview of the possible cases of 3-webs-of-circles ;
maybe it will open up a glimpse of the possible onessolutions for this probably
still unsolved problem by W. BLASCHKE (1938) W. Blaschke, G. Bol, 1938. Geometrie der Gewebe. Springer.
The methods for constructing the 3-webs-of-circles are based on a simple and well-known fact
derived from the conic sections:
Conics and bicircular quartics are enveloped by various sets of doubly tangent circles.
This includes tangents to conics.
Bicircular quartics like conics are characterized by focal points.
(*) If one of these focal points is reflected on the double-touching circles of a sheet, the mirror images lie
either on a circle - the directional circle - or on a line - the directional line.
Conversely, for every point on a directrix there is a unique double-touching circle with property (*)
Unfortunately, we have not been able to provide a general, geometrically plausible proof of this very useful fact.
Even the CAS module from geogebra is of no use in the search for an understandable reason. Known and new 3-webs-of-circles
In 1938 W. BLASCHKE posed the question of all hexagonal-webs consisting of circles.
W. Blaschke, G. Bol, 1938. Geometrie der Gewebe. Springer
The question of all hexagonal webs of straight lines was solved with the theorem of GRAF & SAUER:
Straight-line hexagonal webs always consist of the tangents of a plane curve of 3rd class.
Sechs-Eck-Gewebe aus Geraden
| What is a
hexagonal web ? |  | Diffeomorphism
- - - > - - ->
Here the
diffeomorphism
is the complex tan-function. |  |
BLASCHKE's problem for circles, however, seems to be unsolved until now and is evaluated as not simple.
In 1938, WALTER WUNDERLICH reported "Über ein besonderes Dreiecksnetz aus Kreisen", an article which is
still the central reference for this problem.
Surprisingly, the problem is solved in space: spatial 3-webs-of-circles always lie on DARBOUX cyclides.
These surfaces are the spatial moebius geometric counterparts of bicircular quartics.
Sechs-Eck-Gewebe 3D
The possible webs are all recorded. H. Pottmann, L. Shi, M. Skopenkov, Darboux cyclides and webs from circles
The only exception: the 3-webs-of-circles on spheres or planes! We listed 3-webs-of-circles for three pencils of circles in 1982 Sechs-Eck-Gewebe aus Kreisbüscheln.
In the meantime, further works have been published on this subject
A.M. Shelekhov, Classification of regular three-webs formed by pencils of circles, J. Math. Sciences 143:6 (2007) 3607-3629..
The essential idea for the characterisation of these webs of pencils of circles is the insight that the place in which
the circles from the different pencils touch each other must decay into circles (point-circles included).
The point of contact of two pencils of circles is always a special case of the bicircular quartics:
either moebius-geometrically the transform of a CASSINI-quartic,
or a quartic decaying into the product of two circles. Berührorte oder CASSINI-Kurven
For the general 3-webs-of-circles in BLASCHKE's problem, the question of the place of contact of circles
seems to play an essential role. The place of contact limits the open area,
in which at most the hexagonal-web-condition can be fulfilled!
The special hexagonal web of circles of W. WUNDERLICH consists of double-touching circles of a
2-part bicircular quartic.
Such a quartic has 4 different conzyclic focal points, 4 pairwise orthogonal symmetry circles and for each symmetry
there exists a family of double-touching circles with this symmetry.
One of these families is located in the "interior" of the quartic - by this we mean the quartic side which contains
the focal points. The associated circle of symmetry is the principal axis: this is the circle, on which the 4 focal points lie.
The 3 other families of circles lie on the outside; with the circles of these 3 families the hexagonal webs can be formed.
Exactly 2 circles of each of these 3 families go through each point in the exterior.
Therefore, different hexagonal webs can be created.
The 3 families have to be connected to the 3 different symmetries; if 2 pairs of the circles of the same symmetry
are formed on the outside, no hexagonal web is possible with these.
The construction of the hexagonal web from the 3 families of circles uses the 3 directional circles,
which are aligned to a given focal point with respect to the respective symmetry.
Midpoint conic sections also have families of double-touching circles on the outside. The tangent-lines belong to them.
The statement about the different symmetries cannot be applied to the tangents. Nevertheless
one can use the 2 tangents through any point in the exterior for hexagonal webs:
the applet below shows that in the limiting case - 2 focal points coincide and the 2-part quartic becomes a
midpoint conic section - from two different directional circles to different symmetries
the double-counting directional line for the tangents is created.
The coinciding focal point is thereby chosen as . limit: 2-part quartic ---> conic section
In 2013, FEDOR NILOV published 5 new examples of hexagonal webs of circles.
NEW EXAMPLES OF HEXAGONAL WEBS OF CIRCLES
All 5 examples are based on conic sections.
The following overview shall show that the examples can be extended to general bicircular quartics.
Possibly, a new foundation for the solution of BLASCHKE's problem results from this.
We cannot provide exact proofs for the existence of the new hexagonal circle-webs,
but there is some evidence that these generalisations are correct. Enumeration
I: Known reference hexagonal-webs of circles: Most of the known hexagonal-webs of circles refer to the article
"Uber ein besonderes Dreiecksnetz aus Kreisen" (1938) von Walter Wunderlich
a: 2-part bicircular quartic and 3 families of double-touching circles with 3 different symmetries
b: midpoint conic sections with one set of double-touching circles and 2 sets of tangents
| a:
Cartesian
Oval |  | a:
|  |
| b:
ellipse
|  | b:
hyperbola
|  |
II: a: new by F. NILOV (a) "The tangent lines to a circle counted twice and a parabolic pencil of circles with the vertex
at the center of the circle";
b: new: the tangents to a centre conic section counted twice and the elliptic pencil of circles through
the conic section focal points create a hexagonal web.
In the boundary transition the example (a) of NILOV is created.
| new
by
F. NILOV
(a)
|  | b:
new |  |
| new
by
F. NILOV
(b)
|  | b:
new
|  |
III: a: new: 2-part bicircular quartic with 2 families of double-touching circles
and an elliptical pencil of circles through a pair of focal points;
the 3 families must belong to 3 different symmetries!
b: new: the same with a hyperbolic pencil of circles around a focal pair; 3 different symmetries.
| a:
new: |  | b:
new: |  |
c: example new by F. NILOV (c) midpoint-cone intersection: a family of double-touching circles,
a family of tangents and a pencil of straight lines through a focal point:
| new:
by
F. NILOV
(c)
hyperbola
|  | new:
by
F. NILOV
(c)
ellipse
|  |
d: no hexagonal web of circles: If 2 families of the double-touching circles of a quartic or a conic section are replaced
by 2 pencils of circles through or around the focal points, no hexagonal web is created;
even if the 3 families of circles belong to different symmetries!
An instructive example for this:
A midpoint conic section with a familiy of tangents, the circles of the elliptic pencil of circles through
the two focal points and a pencil of straight lines through one of the focal points seem to be
create a hexagonal web of circles.
If one checks the hexagonal web condition by computation, it turns out that the hexagonal figure closes
only approximately, but not exactly.
Even with the eye, discrepancies can only be seen with high magnification and at the edges of the web.
Kein 6-Eck-Netz für Ellipsen
| none
hexagonal web:
d:
|  | none hexagonal web!
one replaces
the elliptic pencil
of circles
through the focal points
by the
hyperbolic pencil
of circles,
so there is even
n o hexagonal web !!!
|  |
Enumeration 2
IV: known:
This example involves 3 families of circles with a common circle of symmetry:
all circles are orthogonal to a fixed circle. (Which, by the way, can also be imaginary!).
If one adds to 2 pairs of double-touching circles with the same symmetry any
hyperbolic pencil of circles with the same symmetry, the result is a hexagonal web of circles.
| case IV: |  |
Reasoning:
If one projects the circles stereo-
graphically onto the Möbius sphere
and projecting them into the
symmetry plane, the
bicircular quartic becomes a conic section,
the double-touching circles
become tangents and the
pencil of circles becomes
a pencil of straight lines.
|
In all these cases, the hyperbolic pencil cannot simply be replaced by the orthogonal pencil.
in most cases no hexagonal web is created.
This is a counter-example for the web transformation problem formulated by F. NILOV:
it is not possible in every case to replace a pencil of circles involved in a hexagonal web of circles by the orthogonal pencil.
All the more remarkable are the following examples, which arise as a generalisation of F. NILOV's new web (e)
to bicircular quartics.
V: a: new: 2-part bicircular quartic for which a focal circle is a vertex circle:
2 families of double-touching circles inside the quartic and an elliptic pencil of circles through
the focal points lying on the same side.
b: new: 1-part quartic, 2 families of double-touching circles on one side and
the elliptic pencil of circle through the focal points on the same side.
precondition: focal circle = vertex circle. This circle is part of the points of contact!
| new
for 2-part
quartic |  | new
for 1-part
quartic
|  |
c: new by F. NILOV (e): the double-touching circles lying inside an ellipse and the elliptical pencil of circles
through the focal points, provided: the focal circle is a vertex circle is: eccentricity .
| new
by
F. NILOV (e)
|  | new:
this is also a
hexagonal web,
if the
elliptical pencil
of circles
through the
parallels to the main axis
is replaced.
|  |
This change is also applicable to the examples above: also the elliptical pencil of circles through the two
other focal points of the quartic lead to hexagonal webs!
| new: |  |
VI: new by F. NILOV (d): "The tangent lines to a parabola counted twice and a parabolic pencil of circles with limiting point
at the focus and an arbitrary point on the directrix"
This parabola example appears to be rather singular: all attempts to use the directrices for other
bicircular quartics to form hexagonal web of circles have failed.
| new:
by
F. NILOV
(d)
|  | A peculiarity of this example:
The tangents to the parabola from any
point on the directrix are orthogonal:
the directrix is also orthoptic curve
of the parabola!
The orthogonal tangents through
the point on the directrix
belong to the
points of contact of the web.
|
