construction of directric circles
 | this activity is a page of geogebra-book elliptic functions & bicircular quartics & . . .(05.08.2023) |
translation is in progress
An attempt to justify the directric-circle construction for bicircular quartics
Bicircular quartics are simple to characterize in terms of möbius geometry:
These quartics have 4 foci, coincident ones included.
We try to carry out and justify the directric-circle construction for the cases (1) - (3).
At several places we have regretted that for these basic constructions we have not found computational
justifications for these basic constructions.
We lack the appropriate comprehensible calculus for these circular geometric statements.
Attempts to get light into the dark with geogebra-CAS failed - mostly because of the unclear formulas.
We provide geometric reasons here.
| (1) | 4 conzyklic , pairwise different foci. The quartic has 4 pairwise orthogonal symmetry circles, one of them imaginary. By a Möbius transformation one reaches that the axes and the unit circle are symmetry circles, and the foci lie on the -axis: with and (as a rule) |
| (2) | 4 different foci, which are mirror images of 2 pairs on 2 orthogonal circles. By a Möbius transformation one receives the axes as symmetry circles and one can receive with and for the focal points. |
| (3) | 2 single and one double focal points. The axes can be chosen as symmetry axes, the foci on the -axis: and 0 as double focal point, with ; in principle one can be chosen . Mirrored at the unit circle one receives a center cone section. |
| (4) | A single and a triple focal point: if the latter is chosen as , the quartic is a parabola. |
| (5) | Two double foci or one quadruple focal point: the quartic is the product of two non-touching or two touching circles. |
Given in the applet are a focal point f, a vertex s and and for the cases (1) - (3).
The implicit equation of the bicircular quartic is then
- and
- on the -axis: , on the -axis (both calculated in real terms) and on the unit circle: (both calculated in real).
Die Leit-Kreis-Konstruktion
Falls alle Brennpunkte auf einem gemeinsamen Kreis liegen (Fälle (1), (3) und (4)), so bezeichnen wir
diesen Kreis als "Hauptachse".
Grundeigenschaft der Leitkreise:
Man zeichne einen der 4 Brennpunkte (im Folgenden mit f bezeichnet) und eine der Symmetrien einer
bizirkularen Quartik aus. (Die Hauptachsensymmetrie ausgenommen, Wellen zeigt eine Konstruktionsmöglichkeit für diesen Fall )
Zu der Symmetrie gehört eine Schar von die Quartik doppelt-berührenden Kreisen.
- Spiegelt man den Brennpunkt f an den Kreisen der Schar, so liegen die Spiegelpunkte auf einem Kreis, dem Leitkreis bezüglich f.
- Zu jedem Punkt q auf diesem Leitkreis gehört genau ein doppelt-berührenden Kreis der Schar mit der genannten Eigenschaft.