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pencils of circles: differential equations

Author:Walter Füchte
 this activity is a page of geogebra-book elliptic functions & bicircular quartics & . . .(27.04.2023)
differential equation of a parabolic pencil of circles:
The complex vector indicates the direction in which the circles in f touch each other. solution:
  • , es ist .
differential equation of a elliptic pencil of circles:
solution:
  • maps the parallels to the -axis to the rays through f,
  • the parallels to the -axis on the concentric circles around . the moebiustransformation maps to and to . for this gives .
The geometric idea uses the inscribed angle theorem and the properties of the tangents in the intersections of an segment of a circle with the circle. And: The multiplication of two complex numbers and can be interpreted as a stretchrotation: .

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