. . . and ellipse & hyperbola
 | this activity is a page of geogebra-book elliptic functions & bicircular quartics & . . .(12.02.2023) |
Midpoint conic sections have 3 families of double-touching circles:
one of these families is in the interior, which is the conic section side containing the foci.
The two pairs on the outside are the -axisymmetrical double-touching circles
and the tangents. We count the tangents among the double-touching circles, since they
are möbius-geometrically circles through - and can be interpreted as both a curve point and a focal point.
We count the set of tangents twice: for a 2-part bicircular quartic let 2 of the foci coindent.
If this limit point is chosen as , then 2 families of double-touching circles merge into the group of tangents,
the corresponding directices circles merge into the directrix-circle of the tangents.
This clarifies a little the fact that the tangents (double-counted) and the double-touching circles on the outside
can create a 3-web-of-circles.
2 double-touching circles pass through each point on the outside of a midpoint conic section.
However, no 3-web-of-circles can be constructed from 2 of these circles and one tangent per point.
In general, the following applies to 2-part quartics: the 3 double-touching circles per point can only be
be extended to a 3-web-of-circles, if they belong to different symmetries!