Circles of Apollonius

The circles of Apollonius are any of several sets of circles associated with Apollonius of Perga, a renowned Greek geometer.

Most of these circles are found in planar Euclidean geometry, but analogs have been defined on other surfaces; for example, counterparts on the surface of a sphere can be defined through stereographic projection.

The main uses of this term are fivefold: A circle is usually defined as the set of points P at a given distance r (the circle's radius) from a given point (the circle's center).

Apollonius discovered that a circle could be defined as the set of points P that have a given ratio of distances k = ⁠d1/d2⁠ to two given points (labeled A and B in Figure 1).

Let d1, d2 be non-equal positive real numbers.

Let C be the internal division point of AB in the ratio d1 : d2 and D the external division point of AB in the same ratio, d1 : d2.

Then, Therefore, the point P is on the circle which has the diameter CD.

Again by the converse of the angle bisector theorem, the line

The Apollonius pursuit problem is one of finding whether a ship leaving from one point A at speed vA will intercept another ship leaving a different point B at speed vB.

The minimum time in interception of the two ships is calculated by means of straight-line paths.

If the ships' speeds are held constant, their speed ratio is defined by μ.

If both ships collide or meet at a future point, I, then the distances of each are related by the equation:[1] Squaring both sides, we obtain: Expanding: Further expansion: Bringing to the left-hand side: Factoring: Dividing by

: Completing the square: Bring non-squared terms to the right-hand side: Then: Therefore, the point must lie on a circle as defined by Apollonius, with their starting points as the foci.

The circles defined by the Apollonian pursuit problem for the same two points A and B, but with varying ratios of the two speeds, are disjoint from each other and form a continuous family that cover the entire plane; this family of circles is known as a hyperbolic pencil.

Another family of circles, the circles that pass through both A and B, are also called a pencil, or more specifically an elliptic pencil.

These two pencils of Apollonian circles intersect each other at right angles and form the basis of the bipolar coordinate system.

Within each pencil, any two circles have the same radical axis; the two radical axes of the two pencils are perpendicular, and the centers of the circles from one pencil lie on the radical axis of the other pencil.

In Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane.

By solving Apollonius' problem repeatedly to find the inscribed circle, the interstices between mutually tangential circles can be filled arbitrarily finely, forming an Apollonian gasket, also known as a Leibniz packing or an Apollonian packing.

[2] This gasket is a fractal, being self-similar and having a dimension d that is not known exactly but is roughly 1.3,[3] which is higher than that of a regular (or rectifiable) curve (d = 1) but less than that of a plane (d = 2).

The Apollonian gasket was first described by Gottfried Leibniz in the 17th century, and is a curved precursor of the 20th-century Sierpiński triangle.

[4] The Apollonian gasket also has deep connections to other fields of mathematics; for example, it is the limit set of Kleinian groups;[5] see also Circle packing theorem.

is defined as the unique circle passing through the triangle vertex

that maintains a constant ratio of distances to the other two vertices

is defined as the unique circle passing through the triangle vertex

that maintains a constant ratio of distances to the other two vertices

All three circles intersect the circumcircle of the triangle orthogonally.

The line connecting these common intersection points is the radical axis for all three circles.

The two isodynamic points are inverses of each other relative to the circumcircle of the triangle.

This line is perpendicular to the radical axis, which is the line determined by the isodynamic points.

Figure 1. Apollonius' definition of a circle.
Proof of Apollonius' definition of a circle
Figure 2. A set of Apollonian circles. Every blue circle intersects every red circle at a right angle, and vice versa. Every red circle passes through the two foci, which correspond to points A and B in Figure 1.
Apollonius' problem may have up to eight solutions. The three given circles are shown in black, whereas the solution circles are colored.
Figure 4. A symmetrical Apollonian gasket, also called the Leibniz packing, after its inventor Gottfried Leibniz