Method of Images
Apollonian Circles¶
- https://en.wikipedia.org/wiki/Apollonian_circles
If you look at the blue circles, you would probably find that It's pretty similar to the figures on the handout.
Let's take a look on the definition of the first family of Apollonian circles
It's not difficult to find it is actually the same.
It is the most important thing in Apollonian circles that it have a principle
Every circles in the first family (blue circles) can be written as the linear combination of two different circles in the first family.
This Lemma is not hard to proof. By the definition above we have
for every circles in Apollonian circles family, we have the form of above formula. Namely, linear combination of two source point (in Electromagnetics usually line charges or point charges) can represent all of the circles.
note that these source points can also be view as circles with
Line Charge Problem¶
Type 1¶
Given
we can write the circle equation of the potential
Type 2¶
Given
simply assume two circles formula as
Lemma of Apollonian circles tell as that every circle in the first family can be written as the linear combination of these two circle.
and for the
It's easy to solve
Sphere Problem¶
for sphere problem, we commonly can get the relationship
In this type of problem, usually given
as the Lemma mentioned above we know that there exists a bisection(中垂線) of
The bisection could be written as the linear combination of two circle equations, surface of sphere and the point
guess
now we can fine
and for the value of