elliptic/hyperbolic pencils of circles
An elliptical pencil of circles consists of all circles through 2 base points, which we also call focal points.
The orthogonal circles form the polar hyperbolic pencil of circles.
In the applet above, the pencil of rays through w0 is an elliptical pencil of circles, the 2nd focal point is .
The concentric circles around w0 are the polar hyperbolic pencil.
The circles arise from the axis-parallel straight lines as images under the complex function
In the applet below, w0 and w are the focal points.
The circles are images of the axis-parallel straight lines under the complex function
In general, pencils of circles and their loxodromes
- i.e. the curves, which intersect the circles of the pencil at a constant angle -
are characterised by a differential equation and thus by a vector field of the type
Here the complex solution function is analytical, or meromorphic.
The zeros , which we call focal points, can coincide ( - then there is a parabolic circle pencil - ).
One can interpret the circles of a hyperbolic pencil of circles dynamically as circular waves, which propagate
from a source in the direction of the circles of the orthogonal elliptical pencil to an sink.
The source and the sink are the focal points of the wave motion.
We call these vector fields linear.
For explanation we refer to the representation of the Möbius group by the complex special orthogonal group SO(3,)
and its LIE algebra . geogebra-book Möbiusebene, special the chap. Kreisbüschel und lineare Vektorfelder
If 2 such vector fields are superimposed, "quadratic vector fields" are created whose solution curves
can be confocal conic sections or confocal bicircular quartics.
Focal points are the zeros of the linear vector fields.
The solution curves in these cases are angle bisectors of the intersecting circles from the two pencils of circles.
links:
geogebra-book möbiusebene
geogebra-book Leitlinien und Brennpunkte
