contents of this geogebra-book
 | this activity is a page of geogebra-book elliptic functions & bicircular quartics & . . .(06.02.2023) |
Elliptic functions are meromorphic doubly-periodic complex functions,
which of a complex differential equation of the type
- with
elliptic functions and pencils of circles
The circles of an elliptical circle pencil are solution curves of the differential equation, ,
where the focal points are the base points of the circle pencil.
The differential equation describes a parabolic pencil of circles with as the point of contact.
Any elliptic differential equation of the above type can be written as the "product"
of two circle pencil differential equations, apprehend, even in different ways, depending on the position of the foci.
The solution curves of the elliptic differential equation are bisectors of the intersecting circles
from the 2 circle pencils of the product.
This is also the case when foci coincide.
The picture at the top left shows the elliptical directional field that results from the bisectors field
of the two circle pencils.
Hint: For 2 complex numbers is angle bisector!
focal points: normalform
elliptic functions and bicircular quartics
If the absolute invariant of the 4 focal points of an elliptic differential equation is real,
or focal points coincide, then for suitable confocal bicircular quartics are
solution curves of the differential equation.
If the 4 focal points are different, then
- for the focal points are concyclic, the quartics are 2-part;
- for 2 of the focal points pairs lie mirror-inverted on 2 orthogonal circles, the quartics are 1-part.
confocal bicircular quartics: 4 cases in normalform
If 2 of the focal points coincide into one, and one transforms this to ,
the result is a confocal midpoint cone section.
If 3 focal points coincide in one, one obtains confocal parabolas with this as a .
Not included above are 2 special cases:
- 4 different focal points with : the focal points are concyclic and have harmonic position, there are 2-part bicircular solution curves and, at an angle of 45° to them, 1-part bicircular solution curves. Square case with diagonals.
- Hexagonal case: On the Möbius sphere, the focal points can be arranged as the corners of a regular tetrahedron. Through each point (apart from the focal points) six 1-part bicircular quartics go through as solution curves; intersection angle: multiples of 60°.
hexagonal webs ???
bicircular quartics and hexagonal webs of circles
W. BLASCHKE's problem: (1938)
- Determine all hexagonal webs that can be formed from 3 families of circular arcs!
3-web of circles: well known - - - - - new and unknown
Left: Hommage á WALTER WUNDERLICH. In 1938 Walter Wunderlich examined 2-part bicircular quartics and
showed that these quartics have 3 families of double-touching circles,
from which a "besonderes Dreiecksnetz aus Kreisen" ("a special 3-web-of-circles") can be constructed.
To each of these 3 families of circles belongs a symmetry circle.
The 2-part bicircular quartics have 4 paired orthogonal symmetry circles.
The construction of these 3-web-of-circles uses the foci and the associated directic circles.
The constructions are successful in finding 3-web-of-circles also in the case of midpoint conic sections
and their double-touching circles, to which the tangents also belong: möbiusgeometrically is a
double-counting focal point and a curve point!
In 2013, FEDOR NILOV presented new 3-web-of-circles ("NEW EXAMPLES OF HEXAGONAl WEBS OF CIRCLES"):
For conic sections, these examples include not only the double-touching circles but also the pencils of circles
belonging to the focal points.
We will present a general overview of 3-web-of-circles in the last chapter of this geogebra-book.
Included are some circle webs that are probably unknown so far like the 3-web-of-circles shown above on the right.
The examples of FEDOR NILOV are included as special cases.