parabolic pencils of circles
 | this activity is a page of geogebra-book elliptic functions & bicircular quartics & . . .(28.04.2023) |
this activity is also a page of geogebra-book geometry of some complex functions october 2021
A parabolic pencil of circles consists of all circles that touch a given circle at a given point
- the base point of the pencil.
The parallels to the -axis are such a parabolic "pencil of circles" in terms of möbius geometry:
The "circles" here are straight lines, which pass through the point and touch there.
This is best recognised with the help of the stereographic projection.
Each pencil of parallels is a parabolic pencil of circles with as its base point .
A Möbius transformation, which maps the points to three different points ,
transforms the pencil of straight lines parallel to the -axis into a parabolic pencil of circles, which
maps the parallels onto circles, which touch the circle through in .
Conversely, every parabolic pencil of circles can be transformed into a pencil of parallel lines
by a Möbius transformation.
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
- .