Английская Википедия:Apollonian circles
Шаблон:Short description Шаблон:About
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 Greek geometer.
Definition
The Apollonian circles are defined in two different ways by a line segment denoted Шаблон:Mvar.
Each circle in the first family (the blue circles in the figure) is associated with a positive real number Шаблон:Mvar, and is defined as the locus of points Шаблон:Mvar such that the ratio of distances from Шаблон:Mvar to Шаблон:Mvar and to Шаблон:Mvar equals Шаблон:Mvar, <math display=block>\left\{X \ \Biggl| \ \frac{d(X,C)}{d(X,D)} = r\right\}.</math> For values of Шаблон:Mvar close to zero, the corresponding circle is close to Шаблон:Mvar, while for values of Шаблон:Mvar close to Шаблон:Math, the corresponding circle is close to Шаблон:Mvar; for the intermediate value Шаблон:Math, the circle degenerates to a line, the perpendicular bisector of Шаблон: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 Шаблон:Math, and is defined as the locus of points Шаблон:Mvar such that the inscribed angle Шаблон:Math equals Шаблон:Mvar, <math display=block>\left\{X \ \Bigl| \ C\hat{X}D = \theta\right\}.</math>
Scanning Шаблон:Mvar from 0 to π generates the set of all circles passing through the two points Шаблон:Mvar and Шаблон:Mvar.
The two points where all the red circles cross are the limiting points of pairs of circles in the blue family.
Bipolar coordinates
Шаблон: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 Шаблон:Mvar that sees points Шаблон:Mvar under a given oriented angle of vectors i.e. <math display=block>\operatorname{isopt}(\theta) = \left\{X \ \Biggl| \ \measuredangle \biggl( \overrightarrow{XC}, \overrightarrow{XD} \biggr) = \theta +2k\pi \right\}.</math> Such an arc is contained into a red circle and is bounded by points Шаблон:Mvar. The remaining part of the corresponding red circle is Шаблон:Math. When we really want the whole red circle, a description using oriented angles of straight lines has to be used: <math display=block>\text{full red circle} = \left\{X \ \Biggl| \ \measuredangle \biggl( \overrightarrow{XC}, \overrightarrow{XD} \biggr) = \theta + k\pi \right\}</math>
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 (Шаблон: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.[1]
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.[2]
The elliptic pencil of circles passing through the two points Шаблон:Mvar (the set of red circles, in the figure) has the line Шаблон:Mvar as its radical axis. The centers of the circles in this pencil lie on the perpendicular bisector of Шаблон:Mvar. The hyperbolic pencil defined by points Шаблон:Mvar (the blue circles) has its radical axis on the perpendicular bisector of line Шаблон:Mvar, and all its circle centers on line Шаблон: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 Шаблон:Mvar results in a pencil of concentric circles centered at the image of point Шаблон:Mvar. The same inversion transforms the red circles into a set of straight lines that all contain the image of Шаблон: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,[3] the orthogonal property of the two pencils follows from the defining property of the radical axis, that from any point Шаблон:Mvar on the radical axis of a pencil Шаблон:Mvar the lengths of the tangents from Шаблон:Mvar to each circle in Шаблон:Mvar are all equal. It follows from this that the circle centered at Шаблон:Mvar with length equal to these tangents crosses all circles of Шаблон:Mvar perpendicularly. The same construction can be applied for each Шаблон:Mvar on the radical axis of Шаблон:Mvar, forming another pencil of circles perpendicular to Шаблон: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.[4]
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.[5] 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
Notes
References
External links
- Шаблон:Mathworld
- David B. Surowski: Advanced High-School Mathematics. p. 31
Шаблон:Ancient Greek mathematics
- ↑ Шаблон:Harvtxt.
- ↑ MathWorld uses “coaxal,” while Шаблон:Harvtxt prefer “coaxial.”
- ↑ Шаблон:Harvtxt, p. 59.
- ↑ Шаблон:Harvtxt.
- ↑ Шаблон:Cite journal
