Apollonian circles

In geometry, Apollonian circles are two families (pencils) of circles such that every circle in the first family intersects every circle in the second family orthogonally, and vice versa. These circles form the basis for bipolar coordinates. They were discovered by Apollonius of Perga, a renowned ancient Greek geometer.

Definition

The Apollonian circles are defined in two different ways by a line segment denoted CD.

Each circle in the first family (the blue circles in the figure) is associated with a positive real number r, and is defined as the locus of points X such that the ratio of distances from X to C and to D equals r,

{

X

|

d

(

X

,

C

)

d

(

X

,

D

)

=

r

}

.

{\displaystyle \left\{X\ {\Biggl |}\ {\frac {d(X,C)}{d(X,D)}}=r\right\}.}

For values of r close to zero, the corresponding circle is close to C, while for values of r close to ∞, the corresponding circle is close to D; for the intermediate value r = 1, the circle degenerates to a line, the perpendicular bisector of CD. The equation defining these circles as a locus can be generalized to define the Fermat–Apollonius circles of larger sets of weighted points.

Each circle in the second family (the red circles in the figure) is associated with an angle θ, and is defined as the locus of points X such that the inscribed angle ∠CXD equals θ,

{

X

|

C

X

^

D

=

θ

}

.

{\displaystyle \left\{X\ {\Bigl |}\ C{\hat {X}}D=\theta \right\}.}

Scanning θ from 0 to π generates the set of all circles passing through the two points C and D.

The two points where all the red circles cross are the limiting points of pairs of circles in the blue family.

Bipolar coordinates

A given blue circle and a given red circle intersect in two points. In order to obtain bipolar coordinates, a method is required to specify which point is the right one. An isoptic arc is the locus of points X that sees points C, D under a given oriented angle of vectors i.e.