Apollonian circles

Some Apollonian circles. Every blue circle intersects every red circle at a right angle. Every red circle passes through the two points Template:Mvar, and every blue circle separates the two points.

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

Apollonian circle, the angle bisectors in X yield $\frac{|C{T}_i|}{|D{T}_i|}=\frac{|C{T}_o|}{|D{T}_o|}=r$, due $\mathrm{\angle }{T}_{i}X{T}_o=\frac{180^{\circ }}{2}=90^{\circ }$ and Thales's theorem X is located on a half circle with diameter $CD$

The Apollonian circles are defined in two different ways by a line segment denoted Template:Mvar.

Each circle in the first family (the blue circles in the figure) is associated with a positive real number Template:Mvar, and is defined as the locus of points Template:Mvar such that the ratio of distances from Template:Mvar to Template:Mvar and to Template:Mvar equals Template:Mvar, $$\left\{X\right|\frac{d(X,C)}{d(X,D)}=r\}.$$ For values of Template:Mvar close to zero, the corresponding circle is close to Template:Mvar, while for values of Template:Mvar close to Template:Math, the corresponding circle is close to Template:Mvar; for the intermediate value Template:Math, the circle degenerates to a line, the perpendicular bisector of Template:Mvar. 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 Template:Math, and is defined as the locus of points Template:Mvar such that the inscribed angle Template:Math equals Template:Mvar, $$\left\{X\right|C\hat{X}D=\theta \}.$$

Scanning Template:Mvar from 0 to π generates the set of all circles passing through the two points Template:Mvar and Template:Mvar.

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

Bipolar coordinates

Template:Main 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 Template:Mvar that sees points Template:Mvar under a given oriented angle of vectors i.e. $$\mathrm{isopt}(\theta )=\left\{X\right|\measuredangle (\overrightarrow{XC},\overrightarrow{XD})=\theta +2k\pi \}.$$ Such an arc is contained into a red circle and is bounded by points Template:Mvar. The remaining part of the corresponding red circle is Template:Math. When we really want the whole red circle, a description using oriented angles of straight lines has to be used: $$\text{full red circle}=\left\{X\right|\measuredangle (\overrightarrow{XC},\overrightarrow{XD})=\theta +k\pi \}$$

Pencils of circles

Both of the families of Apollonian circles are pencils of circles. Each is determined by any two of its members, called generators of the pencil. Specifically, one is an elliptic pencil (red family of circles in the figure) that is defined by two generators that pass through each other in exactly two points (Template:Mvar). The other is a hyperbolic pencil (blue family of circles in the figure) that is defined by two generators that do not intersect each other at any point.

Radical axis and central line

Any two of these circles within a pencil have the same radical axis, and all circles in the pencil have collinear centers. Any three or more circles from the same family are called coaxial circles or coaxal circles.

The elliptic pencil of circles passing through the two points Template:Mvar (the set of red circles, in the figure) has the line Template:Mvar as its radical axis. The centers of the circles in this pencil lie on the perpendicular bisector of Template:Mvar. The hyperbolic pencil defined by points Template:Mvar (the blue circles) has its radical axis on the perpendicular bisector of line Template:Mvar, and all its circle centers on line Template:Mvar.

Inversive geometry, orthogonal intersection, and coordinate systems

Circle inversion transforms the plane in a way that maps circles into circles, and pencils of circles into pencils of circles. The type of the pencil is preserved: the inversion of an elliptic pencil is another elliptic pencil, the inversion of a hyperbolic pencil is another hyperbolic pencil, and the inversion of a parabolic pencil is another parabolic pencil.

It is relatively easy to show using inversion that, in the Apollonian circles, every blue circle intersects every red circle orthogonally, i.e., at a right angle. Inversion of the blue Apollonian circles with respect to a circle centered on point Template:Mvar results in a pencil of concentric circles centered at the image of point Template:Mvar. The same inversion transforms the red circles into a set of straight lines that all contain the image of Template:Mvar. Thus, this inversion transforms the bipolar coordinate system defined by the Apollonian circles into a polar coordinate system. Obviously, the transformed pencils meet at right angles. Since inversion is a conformal transformation, it preserves the angles between the curves it transforms, so the original Apollonian circles also meet at right angles.

Alternatively, the orthogonal property of the two pencils follows from the defining property of the radical axis, that from any point Template:Mvar on the radical axis of a pencil Template:Mvar the lengths of the tangents from Template:Mvar to each circle in Template:Mvar are all equal. It follows from this that the circle centered at Template:Mvar with length equal to these tangents crosses all circles of Template:Mvar perpendicularly. The same construction can be applied for each Template:Mvar on the radical axis of Template:Mvar, forming another pencil of circles perpendicular to Template:Mvar.

More generally, for every pencil of circles there exists a unique pencil consisting of the circles that are perpendicular to the first pencil. If one pencil is elliptic, its perpendicular pencil is hyperbolic, and vice versa; in this case the two pencils form a set of Apollonian circles. The pencil of circles perpendicular to a parabolic pencil is also parabolic; it consists of the circles that have the same common tangent point but with a perpendicular tangent line at that point.

Physics

Apollonian trajectories have been shown to be followed in their motion by vortex cores or other defined pseudospin states in some physical systems involving interferential or coupled fields, such photonic or coupled polariton waves. The trajectories arise from the Rabi rotation of the Bloch sphere and its stereographic projection on the real space where the observation is made.

See also

• Apollonius quadrilateral

References

• Template:Citation.
• Template:Citation. Dover reprint, 1979, Template:Isbn.

Further reading

• Template:Citation.
• Template:Citation.
• Template:Citation. Dover reprint, 1990, Template:Isbn.

External links

• Template:Mathworld
• David B. Surowski: Advanced High-School Mathematics. p. 31